Drag Reduction by Gas Layers and Streamlined Air Cavities Attached to Free

Home , Leidenfrost effect

Drag reduction by gas layers and streamlined air cavities attached to free-

falling spheres

In Fulfillment of the Requirements

For the Degree of

Doctor of Philosophy

King Abdullah University of Science and Technology

Thuwal, Kingdom of Saudi Arabia

November 2019 EXAMINATION COMMITTEE PAGE

The dissertation of Aditya Jetly is approved by the examination committee.

Committee Chairperson: Prof William Roberts

Committee Members: Prof Sigurdur Thoroddsen, Prof Noreddine Ghaffour, Prof Glen McHale, Dr. Ivan Uriev Vakarelski

ABSTRACT

The general objective of this thesis is to conduct experiments on sphere free-falling in liquid that advance our understanding of the drag reduction on solids moving in liquid by means of lubricating gas layers and attached streamlined air cavities.

Part I of the thesis investigates the effect of thin air layers (~ 1 – 2 μm), naturally sustained on superhydrophobic surfaces, on the terminal velocity and drag force of metallic spheres free- falling in water. By comparing the free-fall of unmodified spheres and superhydrophobic spheres, it is seen that the thin air layers reduced the drag force on the spheres by up to 80 %, at Reynolds numbers 105 to 3×105, owing to an early drag crisis transition.

Part II of the thesis investigates the drag reduction by means of the dynamic

Leidenfrost vapor-layer sustained on the surface of heated metallic spheres free-falling in a fluorocarbon liquid, FC-72 (perfluorohexane). Analysis of the extended free-fall trajectories and acceleration, based on the sphere dynamic equation of motion, enables the accurate evaluation of the vapor-layer-induced drag reduction, without the need for extrapolation. The drag on the Leidenfrost sphere in FC-72, can be as low as CD =

0.04±0.01, which was an order of magnitude lower than the values for the no-vapor-layer spheres in the subcritical Reynolds number range.

Part III of the thesis examines a recently demonstrated phenomenon of the formation of stable-streamlined gas cavity following the impact of a non-wetting sphere on water. The sphere encapsulated in a teardrop-shaped gas cavity was found to have near- zero hydrodynamic drag due to the self-adjusting streamlined shape and the free-slip

4 boundary condition on the cavity interface. In this case the streamlined cavity is attached just above the sphere’s equator, instead of entirely wrapping the sphere. Nevertheless, this sphere with attached cavity has near-zero-drag and predetermined free-fall velocity in compliance with the Bernoulli law of potential flow. Last, the effect of surfactant addition to the water solution is investigated. The shape and fall velocity of the cavities was unaffected by the addition of low-surface-modulus synthetic surfactants, but was destabilised when a solution containing high-surface-modulus surfactants were used.

ACKNOWLEDGMENTS

There are many people without whom I would not have been able to accomplish this work and I am extremely grateful for their support and encouragement.

I would first like to express my deepest appreciation towards our Senior Research

Scientist Dr. Ivan Vakarelski who has been more than a supervisor supporting me throughout my time at KAUST. Thanks for having the faith in me and inspiring me to accomplish and achieve all the goals. I could not have envisioned having a better mentor during my PhD study. I would also be grateful for my thesis advisor Prof. Sigurdur

Thoroddsen for his guidance, ideas and continuous support throughout the time at KAUST.

A special thanks to various people who have helped me during my time at KAUST.

I would like to thank Dr. Erqiang Li, Mujataba Mansoor, Nadia Kouryetam, Vivek, Tariq

Alghamdi, Ziqiang Yang, Kenneth Langley, Andres, Fede & Arief and the entire High

Speeds Fluids Imaging Lab for all the help during the PhD.

I would also like to thank my parents and my brother for all the belief that they instilled in me during the PhD and helping me reach the goals set. The last 5 years I met a number of great people in KAUST, some of whom graduated and some of whom are still with me. Bayan Wasfi, Vasudeo Babar, Afaque Momin, Abdulrahman Alzarie, Adnan

Khan, Mohsin Mukkaddam, Sarah, Enjey, Abdulaziz Khan, Lina Alghamdi, Teyyuba ,

Yeshi Di, Somak, Alivia, Biswarup, Devendra, Graduate affairs and the KAUST

Community.

Finally, I would like to express my gratitude to the almighty

TABLE OF CONTENTS

Examination Committee Page ...... 2 Copyright Page ...... 3 Abstract ...... 4 Acknowledgments ...... 6 List of abbreviations ...... 9 List of Figures ...... 10 List of Tables ...... 15

Chapter 1: General Introduction ...... 16 1.1 Drag on a Bluff Body Moving in Liquid...... 16 1.2 Drag Reduction Methods ...... 19 1.3 Drag Reduction by Gas Layers and Air Cavities ...... 21

Part I Drag crisis moderation by thin air layers sustained on superhydrophobic spheres falling in water ...... 23

Chapter 2: Introduction to Part I ...... 24 2.1 Surface Wettability ...... 24 2.2 Prior Studies on Plastron Drag Reduction...... 25 Chapter 3: Experimental Setup and Methods for Part I ...... 30 3.1 Experimental Setup ...... 30 3.2 Sphere Preparation ...... 32 3.3 Plastron Characterization ...... 34 3.4 High-Speed Imaging ...... 35 3.5 Determination of the Drag Coefficient...... 37 Chapter 4: Experimental Results for Part I ...... 39 4.1 Velocity vs Time Dependences ...... 39 4.2 Drag Coefficient vs Reynolds Number Dependence ...... 48 Chapter 5: Conclusions for Part I ...... 51

Part II Giant Drag Reduction on Leidenfrost Spheres Evaluated from Extended Free- fall Trajectories ...... 52

Chapter 6: Introduction to Part II ...... 53 6.1 The Leidenfrost Effect ...... 53 6.2 Prior Works on Leidenfrost Drag Reduction ...... 56 Chapter 7: Experimental Setup and Methods for Part II ...... 58 7.1 Experimental Setup ...... 58 7.2 Experimental Protocol ...... 59 Chapter 8: Experimental Results for Part II ...... 62 8.1 Velocity vs Time Dependences ...... 62 8.2 Falling Sphere Equation of Motion ...... 72 8.3 Drag Coefficients vs Reynold Number ...... 77 Chapter 9: Conclusions for Part II...... 86

Part III Stable-streamlined cavities following the impact of non-superhydrophobic spheres on water ...... 87

Chapter 10: Introduction to Part III ...... 88 10.1 Sphere impact on liquid...... 88 10.2 Prior works on stable cavities...... 91 Chapter 11: Experimental Setup and Methods for Part III ...... 92 11.1 Experimental Setup ...... 92 11.2 Sphere Preparation ...... 93 11.3 Surfactant Solutions Preparation ...... 95 Chapter 12: Experimental Results for Part III ...... 97 12.1. Formation of the cavity ...... 97 12.2 Cavity Shape, Fall Velocity & Drag ...... 105 12.3 Effect of Surfactant ...... 116 Chapter 13: Conclusions for Part III ...... 122

Chapter 14: General Conclusions...... 124 Bibliography ...... 127

LIST OF ABBREVIATIONS

Ds -Diameter of the sphere (mm)

3 ρs -Density of the sphere (kg/m )

ST -Stainless Steel

TC -Tungsten Carbide

ZO -Zirconium Oxide

CMC -Critical Micelle Concentration

CD -Drag Coefficient

Re -Reynolds Number

R - Radius of the Sphere (mm)

Lp -Length of Projectile (mm)

Rp -Radius of Projectile (mm)

Es -Surface Modulus (mN/m)

FC72 -Perfluorohexane

LIST OF FIGURES

Figure 1 - Drag Force Curve for a solid sphere. The points 4 - 5 depict the Drag Crisis (©NASA) ...... 18 Figure 2- (A) Schematic of the experimental setup showing the liquid tank with sphere release magnetic holder mechanism on top. High-speed camera is in the front of the tank with halogen lamps providing the back-light illumination. (B) Photograph of actual laboratory experimental set-up with the 2.5 tall tank. The high-speed camera is seen in the background...... 31 Figure 3 – (a) Photograph of a 50 mm superhydrophobic steel sphere shortly after the sphere is immersed in water. The silver shiny appearance of the sphere is a signature of the thin air layer sustained on the superhydrophobic surface. (b) The same sphere after four hours in still water. The loss of the shiny appearance is due to the plastron gradual dissolution in water. (Jetly et al. 2018 [77]) ...... 33 Figure 4- An AFM 3D image of the Glaco coated surface topology taken over 60×60 μm area. In this case four cycles of Glaco coating were repeated. (b) AFM force curve vs piezo displacement, ΔX taken above the plastron covered substrate. The force curve branches correspond to: (A to C) piezo extension approaching the surface toward the cantilever (red line): (A) free cantilever approaching the surface; (B) cantilever jump-in toward the surface upon the tip contact with the plastron; (C) cantilever in hard-wall contact with the surface; (D to F) piezo retraction pulling the tip away from the surface (blue line): (D) cantilever tip in hard-wall contact with the substrate; (E) the cantilever detaches from the surface; (F) free cantilever moving away from the surface. The plastron layer thickness, δ is estimated as the distance between the jump-in position and the zero force hard-wall contact position. Notice that parts of the force curve is out of the force detection range of – 3 mN to + 3 mN...... 36 Figure 5 - Comparison of spheres fall velocity progression of the sphere velocity with time during the fall in the 2.5 meter water tank for superhydrophobic (red squares) or hydrophilic (blue circles) steel sphere with diameters: (a) 20 mm (b) 25 mm (c) 40 mm (d)

45 mm (e) 50 mm and (f) 60 mm. The solid line for each data set is formal fit using the exponential form...... 42 Figure 6 - Comparison of spheres fall velocity progression of the sphere velocity with time during the fall in the 2.5 meter water tank for superhydrophobic (red squares) or hydrophilic (blue circles) tungsten carbide sphere with diameters: (g) 20 mm (h) 25 mm (i) 30 mm (j) 35 mm (k) 40 mm (l) 45 mm and (m) 50 mm tungsten carbide sphere. The solid line for each data set is formal fit using the exponential form...... 46 Figure 7- Dependence of the drag coefficient on the Reynolds number for steel and tungsten carbide spheres falling in water. Open blue triangles are for hydrophilic steel spheres (20, 25, 30, 40, 45, 50 and 60 mm). Open blue squares for hydrophilic tungsten carbide spheres (20, 25, 30, 35, 40, 45 and 50 mm). Solid red triangles and solid red squares are for the respective superhydrophobic steel and tungsten carbide spheres freshly immersed in water and covered by a plastron. Both sets of results are calculated using eqn 3.4 with the sphere velocity and acceleration taken at the end of the 2.5 m water tank fall. Literature values for smooth solid steel spheres falling in an “infinite” tank measured by White [14] are shown as crosses ...... 50 Figure 8 – (a) Leidenfrost effect example of a drop levitating on a hot plate. (b) Inverted Leidenfrost effect wherein the solid body is heated at high temperatures thereby being enclosed within a vapor layer or gas layer which helps to reduce drag ...... 54 Figure 9 - High-speed camera snapshots of a 20 mm steel sphere held stationary in FC -

72: (a) when the sphere temperature, TS is above the Leidenfrost temperature TL ~ 110 ⁰C and (b) shortly after passing the Leidenfrost temperature with explosive boiling. (c, d) High-speed camera snapshot silhouette images of a 20 mm steel sphere falling in FC-72:

(c) for sphere temperature, TS ~ 250 ⁰C above the Leidenfrost temperature, and (d) at temperature slightly below the Leidenfrost temperature Vakarelski et al. (2011) [81] .... 55 Figure 10-Photograph comparing the sizes of the experimental tanks. On left is the 3 meter tall and 14 cm in diameter cylindrical 50 liters tank and on right is the 2 meter tall, 20 × 20 cm, square cross section 80 liters tank with heating device used in the present study. The 2 meter tall and 8 cm in diameter cylindrical 10 liters tank used in the prior study is also included for comparison (middle). The high-speed camera used to monitor the sphere free- fall in the tank can be seen in the background...... 61

Figure 11 - Comparison of spheres fall velocity progression with time, during the fall of identical steel spheres in FC-72, of room temperature, TS = 21 ⁰C (blue circles) or under

Leidenfrost regime, TS = 250 ⁰C (red squares) for diameters: (a) 2R = 10 mm (b) 13 mm (c) 15 mm (d) 20 mm (e) 25 mm (f) 30 mm (g)40 mm (h) 45 mm (i) 50 mm and (j) 60 mm ...... 67 Figure 12 - Comparison of spheres fall velocity progression with time during the fall of identical tungsten carbide spheres in FC-72, for sphere of room temperature, TS = 21 ⁰C

(blue circles) or Leidenfrost spheres, TS = 250 ⁰C (red squares): for diameters (k) 2R = 10 mm (l) 15 mm (m) 20 mm (n) 25 mm (o) 40 mm and (p) 45 mm ...... 70 Figure 13- Comparison of the experimental sphere free-fall velocity progression with time with the Eq. (8.3) analytical solution of the BBO equation assuming a constant drag coefficient fall. (a) Fall of 2R = 25 mm steel sphere (squares) and 2R = 25 mm tungsten carbide (triangles) in FC-72 measured in the 3 meters tall, 14 cm in diameter tank. (b) Fall of the same spheres in water measured in a 2.5 meter tall, 40 × 40 cm cross-section tank

(experimental data from Jetly et al. 2018 [77]). The red lines are Eq. (8.3) fits, where CD and τ were determined from the experimental value of the terminal velocity, UT using Eq. (8.4) and Eq. (8.5) respectively...... 76

Figure 14 - Procedure to calculate CD. Sphere depth coordinate (a), the corresponding fall velocity (b) and drag coefficient (c) vs. time during the fall of a 2R = 25 mm steel Leidenfrost sphere (red squares) and 2R = 25 mm tungsten carbide Leidenfrost sphere (blue triangles) in FC-72 measured in the 3 meters tall tank. (a) The spheres depth coordinate, x(t) is measured from the high-speed videos using a 25 ms sampling rate. (b) The sphere velocity is calculated as moving median time differential of the coordinate over a time interval, Δt = 125 ms, using data smoothing function, e. g. U (t) = (x(t + 0.5Δt) - x(t - 0.5Δt))/ Δt. (c) The drag on the sphere calculated using Eq.(8.10), where the sphere acceleration, dU/dt was calculated as moving median of the velocity time differential, over Δt = 125 ms...... 80 Figure 15- Drag coefficient vs Reynold number for selected steel (ST, 2R = 15, 25, 40 mm) Leidenfrost spheres and tungsten carbide (TC, 2R = 10, 15, 25 mm) Leidenfrost spheres. Each data set is evaluated from the sphere fall trajectory data, as done in the Figure 14, and then plotted vs. the sphere instantaneous Reynolds number...... 81

CD values. Each data point is calculated using Eq. (8.10) with the sphere velocity and acceleration taken close to the end of the 3 m or 2 m tank fall. Open diamonds (blue) are for the no-vapor steel spheres, open triangles (blue) for the no-vapor tungsten carbide sphere. Solid diamonds (red) are for Leidenfrost steel spheres and solid triangles (red) for Leidenfrost tungsten carbide spheres. Open squares (blue) are for the no-vapor-spheres,

TS = 55 °C and solid squares (green) are data for Leidenfrost steel sphere, TS = 160 °C falling in FC-72 heated to 55 °C. Literature values for no vapor steel spheres falling in an “infinite” water tank measured by White [14] are shown as crosses...... 85 Figure 17 - Snapshots of the various stages in the formation of a stable cavity from the time of impact onto a liquid surface. The number below the frame indicates the depth of the sphere below the surface of water...... 101 Figure 18- Snapshots of the sphere attached-streamlined cavity for a 10 mm steel sphere for: (a) Plasma cleaned sphere with water contact angle,  < 30°, released from h0 = 4.2 meters above the water surface; (b) Ethanol cleaned sphere   60°, h0 = 3.2 m; (c)

Unmodified sphere   90°, h0 = 2.0 m; (d) Ultra Glaco coated sphere,   120°, h0 = 0.5 m ...... 102 Figure 19 - Examples of the depth trajectory vs time (a) and decent velocity vs time (b) for the case of a 10 mm unmodified steel sphere with attached cavity impacting from about 2 m height above the water surface (red squares and 10 mm unmodified tungsten carbide sphere impacting from about the same height of about 2 meter above the water surface (blue circles). Arrows mark the establishment of the steady streamlined cavity regime, with no more bubble shedding ...... 103 Figure 20 - High-speed camera snapshots of sphere-in-cavity formation for a 10 mm

Leidenfrost steel sphere, TS = 250 °C free-falling in PP1 (a) and for spheres with attached streamlined air-cavity formation free-falling in water using: (b) 20 mm zirconia sphere; (c) 15 mm steel sphere; (d) 10 mm tungsten carbide sphere. All spheres in (b-d) have water contact angle of about 90° and were released from about 2.0 m above the water surface. The cavity – sphere contact line is marked with white arrow in each image...... 104

Figure 21 - Snapshots comparing a 15 mm unmodified steel sphere with cavity formation free-falling in room temperature water (a) with 15 mm superhydrophobic steel sphere-in- cavity formation (b) free-falling in room temperature water. (c) Sphere in cavity formation for a 15 mm Leidenfrost steel sphere, TS = 400 °C in 95 °C water ...... 108 Figure 22 - Dependence of the ratio of the total sphere-with-cavity over the sphere volume,

VSC/VS on the sphere to liquid density ratio, ρS/ρ. Data shown are for 10 mm (green triangles), 15 mm (red squares) and 20 mm (blue circles) unmodified zirconia (ZO), steel (ST) or tungsten carbide (TC) spheres. The dotted line corresponds to neutral buoyancy of the sphere-with-cavity formation...... 109 Figure 23 - Cavity volume, as a function of (퐿 퐷2), cavity diameter D and length L, for the same sphere sizes as in Figure 22. The dotted line is best linear fit to the data, that gives 2 the relation VSC = 0.47LD ...... 110 Figure 24 - An example of a high-speed video-camera snapshot used to determine the volume of the sphere-with-cavity formation, VSC. b) Photograph of a 3D-printed solid projectile used to estimate the drag on a similar streamlined-shape solid body ...... 111 Figure 25- Variation of the sphere-in-cavity terminal velocity 푈 with sphere diameter 퐷푆 and sphere to liquid density ratio, ρS/ρ -1. The same sphere symbols are used as in Fig. 22. The dotted line is a linear best fit to the data, giving a coefficient C = 4.7 in Eq. (12.3)...... 114

Figure 26 - Variation of the drag coefficient CD with the Reynolds number Re for sphere- with-cavity for unmodified zirconia spheres of diameter DS = 10, 15 and 20 mm, L/R ~4.0

(solid red diamonds), unmodified steel spheres of diameter DS = 10, 15 and 20 mm, L/R ~

4.7 (solid red triangles) and unmodified tungsten carbide spheres, DS = 10 and 15, L/R ~ 6.3 (solid red circles). Data for solid spheres without a cavity (empty green squares) are taken from Jetly et al.[73] Data for superhydrophobic steel sphere-in-cavity formation, DS = 15, 20 and 25 mm, L/R ~4.5 (empty red triangles) and solid streamlined projectiles of similar shape, DP = 25 mm, LP/RP = 4.5 (solid blue squares), are taken from Vakarelski et al. [135]...... 115 Figure 27 - Snapshots comparing 10 mm unmodified steel spheres with cavity formation free-falling in (A) pure water with different surfactants: (B) SDS solution; (C) Shampoo solution; (D) SLES + CAPB + MAc solution; (E) Soap solution ...... 121

LIST OF TABLES

Table 1- Physical parameters for zirconium oxide (ZO), steel (ST) or tungsten carbide (TC) spheres with attached cavity formation falling at constant velocity in room temperature water, TW = 21 °C. All data are collected using unmodified spheres of   90° which were released from about 2.0 meter height above the water level in the tank for a tank filled with pure water...... 94 Table 2 - Short names, composition, surface tension and surface-dilation modulus of the surfactant solutions used. The surface tensions were measure with a Kruss tensiometer.

Data for the surface dilation modulus ES are taken from Denkov et al. 2005 [136]...... 96

Chapter 1: General Introduction

1.1 Drag on a Bluff Body Moving in Liquid

The movement of a body through another medium makes it experience a certain resistance force, which tries to impede its motion, this resisting force is known as the drag force. The drag force and methods to reduce drag is important for a wide range of practical applications from sport cars and fighter jets, to building tall architectural designs, to naval vessels and effective operation of microfluidics.

The drag experienced by a body moving inside liquid has two major components: skin friction or viscous drag and form or pressure drag. The skin friction drag is due to friction of the fluid over the surface of the body and is dependent on the viscosity of the fluid. The skin friction drag is the leading component for the drag on bodies of large surface area and is more pronounced at lower fluid velocities. The form drag arises from difference of the pressure distribution over the body moving in the fluid. The form drag is the leading component for the drag on bluff bodies moving at higher velocities.

The drag on a body moving in a fluid is usually expressed as universal dependence of the drag coefficient, CD on the Reynolds number, Re. For the case of a sphere moving in fluid the drag coefficient and the Reynold number are defined as:

2 2 퐶퐷 = 2퐹퐷/(휋푅 휌푈 ), (1.1)

2휌푅푈 푅푒 = (1.2) µ

Here FD is the drag force on the sphere, U is the sphere velocity, ρ is the fluid density, R is the sphere radius and µ is the fluid shear viscosity

The general dependence of the drag force on a sphere is given by the classical drag force curve shown in Figure 1. For very low Reynolds numbers (Re < 0.5) the sphere is in the Stokes’ flow regime, where CD = 24/Re. For Reynolds number between 0.2 < Re <

1000 the Schiller-Naumann [1] empirical relation for CD is valid:

24 퐶 = ( ) × 1 + 0.15푅푒0.687 (1.3) 퐷 푅푒

In the subcritical range 5×103 < Re < 3×105, it is well known that the flow separation occurs at about the equator ~ 90º relative to the front of the spheres. In this regime the form drag is prominent and the contribution of the skin friction drag is less than

5% towards the total drag and the drag coefficient is approximately constant in the ranges

0.44-0.55. Once the Reynolds number is further increased Re≈ 2.5 ×105 the boundary layer shifts to becoming turbulent and the flow separation point shifts downstream ~120º from the front of the sphere. The momentum of the turbulent boundary layer is stronger than for a laminar boundary layer which accommodates the attachment to the surface of the object.

The resulting wake becomes much narrower which leads to the pressure being compensated in the forward portion of the sphere thereby reducing drag. In this regime the drag coefficient decreases dramatically to a range of CD ≈ 0.1–0.2. This phenomenon is called the drag crisis.

Figure 1 - Drag Force Curve for a solid sphere. The points 4 - 5 depict the Drag Crisis (©NASA)

. In regard to flow around a sphere the various regimes that existed were:

 At low Re~0; the flow around the sphere is axisymmetric, steady and fully

attached.

 Re =5; the flow separation occurs

 Re~ 210; the axisymmetry of the wake is broken

 Re~270; wake becomes unsteady and planar symmetric and the vortex shedding

begins to occur

 Re > 350; planar symmetry is broken and wake turns both unsteady and

asymmetric.

 103 ≤ Re ≤ 4×105; drag coefficient of a no slip sphere is independent of the Re.

 Above Re ~ 2.5×105; Sharp drop in drag coefficient is observed to reach ~ 0.1.

This phenomenon is termed as the drag crisis.

1.2 Drag Reduction Methods

Drag reduction on bluff bodies moving at high Reynolds numbers (104 – 106) can be achieved by provoking early drag crisis. Fundamentally, all drag reduction mechanisms of a sphere moving in a Newtonian fluid involve moving the point of flow separation to the rear portion of the sphere surface to generate a narrower wake. The critical Reynolds number can be reduced as well by active control of the flow over the sphere. Recently, it has been shown that drag reduction can arise from mass transfer resulting from the melting of an ice surface in contact with water [2].

Putting surface features on the sphere surface can induce turbulence transition in the boundary layer at a lower Reynolds number compared to smooth surfaces. As shown in

Figure 1 the drag crisis on roughened surface sphere is triggered at significantly lower

Reynold number. A well-known example is the use of dimples on golf balls [3] which are shown to generate discrete vortices energizing the boundary layer flow. The dimple depth was one of the most significant parameters which contributes to the drag reduction and this has been studied both experimentally and numerically [4]. Studies have shown that the use of dimples in turbulent flow can reduce the drag by around 20% in comparison to flat surfaces [5]. To maximize the drag reduction due to dimples it was concluded that the flow must be smooth with minimal mixing and no flow separation if possible. Hot wire techniques were used to observe the shift in the separation point by the use of dimples on the surfaces.

The use of microstructures or riblets to contribute to the drag reduction has been studied by various researchers in the past [6, 7]. The use of riblets can delay the transition from laminar (viscous sub-layer) to turbulent boundary layer on flat surfaces [8]. This passive method of drag reduction stabilizes the flow near the wall by inhibiting the spanwise movement of the longitudinal vortices during the drag producing sweep and burst events [9]. It was found that the drag can effectively be reduced by up to 10 %. The effectiveness of the drag reduction due to the use of riblets is highly dependent on the flow direction wherein if the flow is not parallel to the riblets or the microstructure the drag can tend to increase. Techniques such as Smoke-wire and VITA (Variable Intensity Time

Averaging) have been used to visualize the flow over the riblets. It was also seen that a turbulent boundary layer on a surface exhibiting longitudinal ribs could develop a lower shear stress than that on a smooth surface [10-12].

Other efficient approaches include the addition of flexible polymer in the liquid solution which are shown to reduce drag, both on free-falling spheres and in pipe-flow transport relevant to industrial scale applications. It was seen that even a small addition of

5 parts per million (ppm) of polyacrylamide polymer into turbulent pipe flows could lead to an increase in flow speed by around 80 % for given pressure drop [13-16].

1.3 Drag Reduction by Gas Layers and Air Cavities

An alternative innovative approach to reduce drag on solid moving in liquid is by the introduction of a lubricating gas layer at the solid-liquid interface. The major physical concept here was the exchange of the no-slip boundary condition on a solid-liquid interface with a free-slip or stress-free boundary condition at the gas-liquid interface. This can be effective in reducing both of the major drag components. The various methods by which the gas layers can be induced on the solid surfaces include: bubble injection, super cavitation, superhydrophobic surfaces and Leidenfrost layers.

Active methods to induce the gas layer on the solid include: gas or bubble injection along the solid-liquid interface. Supercavitation is associated with high velocity of projectiles moving in the liquid. Supercavitation has shown to greatly reduce the drag on projectile moving in water, however this phenomenon takes place only at very high speeds, limiting the applications to military torpedoes [17-18]. These supercavitating torpedoes

(VA-111 Shkval) can accelerate to speeds in excess of 200 knots. Supercavitation at high velocity of projectiles moving in liquid and some combination of air injection can result in assisted or partial cavitation [19-21].

Bubble injection on solid surfaces has been shown to be efficient in reducing skin drag with laboratory prototypes as well as for real ships. Although this technology is still under development, sea trials with the actual hulls of commercial ships indicates that the use of air lubrication at certain conditions can reduce fuel consumption by up to 12% [22].

Studies for bubble drag reduction (BDR) both numerically and experimentally have taken into account the bubble sizes, bubble breakups and coalescence to calculate the lift and drag forces to demonstrate the drag reduction [23]. Earlier studies demonstrated that skin

21 friction drag-reduction only persisted for a few meters using BDR for Reynolds number as high as 220 x106 [24-25]. Studies achieved frictional drag reduction of over 80% by injecting air on surfaces [25]. For substrates using water hydrolysis and pyrolysis the gas generation demonstrated moderate drag reduction [25-26].

A passive way to induce gas layer that did not require additional energy input is the use of water repellent superhydrophobic surfaces which can naturally sustain a thin air layer when immersed in water. This air layer is also referred to as a plastron [27-29]. For the last two decades numerous studies have demonstrated that such an air layer can reduce drag for flow in micro-fluidic channel and on superhydrophobic plates [30]. However results for drag reduction by superhydrophobic coatings of bluff bodies moving in liquid such as falling spheres or prototype ship hulls, have been far fewer and the effects shown limited to marginal drag reduction [27-30]. This is explained in detail in Part I of the present thesis.

Recently the use of the Leidenfrost effect to sustain a continuous vapor layer on solid surfaces, heated to a temperature that exceeds the boiling point of the surrounding liquid, has proven to be an efficient method to study the effect of gas layers on the drag of solid objects. The drag reduction due to the use of Leidenfrost layers is explained in detail in Part II of this thesis.

Part I

Drag crisis moderation by thin air layers sustained on superhydrophobic spheres falling in water

Published in the SoftMatter (Jetly et al. 2018, see reference [77])

Chapter 2: Introduction to Part I

An innovative approach to reduce drag on bluff bodies is by the use of superhydrophobic surfaces. These surfaces retain thin layers of vapor on their surface which helps to reduce the drag force. The following chapter delves into drag reduction capabilities of the superhydrophobic coatings and surfaces.

2.1 Surface Wettability

The contact angle is the angle that a liquid droplet makes between itself and a surface. The maximum angle that a drop can make before it tends to roll off from the surface is termed as the advancing contact angle. In contrast the minimum angle that the drop would make is its receding angle. The difference between the advancing and receding contact angle is the contact angle hysteresis. The contact angle hysteresis can also be defined as the contact interface which can take a range of values without moving. The

Young’s equation demonstrates that there exists only one stable contact angle, but in real surface there exists wide range of stable angles. In this study the contact angles considered while defining surfaces based on the wettability was the static contact angle [161].

The contact angle depends on various factors such as the surface energy, surface roughness, surface preparation method and surface cleanliness [31-36]. On the other hand the contact angle hysteresis is the difference between the advancing and receding contact angle. Surfaces can be mainly classified as hydrophilic and hydrophobic. The difference between hydrophilic and hydrophobic surface lies in the contact angle that the water droplet makes with the surface. In the case of hydrophilic surfaces the contact angles are less than

90˚. On the other hand, hydrophobic surfaces are those surfaces where the contact angle ranges between 90˚ to 120˚.

Surfaces which possess micro-or-nano roughness in addition to being hydrophobic thereby making a water contact angle of over 150˚ and exhibit a low hysteresis between the advancing and receding contact angles are termed as superhydrophobic. These surfaces are typically water repelling in nature, like the Lotus leaf. On these surfaces the droplet of water sits on the micro-or-nano roughness and not on the surface itself. The use of superhydrophobic surfaces with entrained gas limits the amount of skin friction experienced by the surface thereby inducing drag reduction capabilities.

The superhydrophobic surfaces can exist over one of the two extremes: The Cassie-

Baxter state or the Wenzel State. The Cassie-Baxter describes the state wherein the roughnesses present on the surface does not penetrate into the fluid in which the surface is submerged within. Thus, in this state the drop sits on the roughness. In contrast Wenzel state showcases that if a surface was rough and the liquid tended to follow the contours on the surface then the roughness effect had to emphasize the intrinsic wetting tendency of either the roll-up of the liquid or the film formation [162]. The Wenzel state is not superhydrophobic and the contact angle is lower than for the former state [37].

2.2 Prior Studies on Plastron Drag Reduction

A passive way to induce a gas layer that does not require additional energy input is by the use of water repellent superhydrophobic coatings which can naturally sustain a thin air layer when immersed in water. This air layer is also referred to as a plastron. The term

“plastron” was first introduced into the literature in relation to superhydrophobic surfaces by Shirtcliffe et al., showcased how the plastron could fix the water-vapor interface and provide for an incompressible oxygen-carbon dioxide exchange surface [157]. In regard to drag reduction probably the first applied used of plastron was by McHale et al., wherein they found drag to be reduced by 5 to 15 % in the intermediate Reynolds number ranges

[29] and Shirtcliffe et al., wherein the superhydrophobic nanoribbons with hydrophobic coatings surface finished allowed for greater flows at lower pressure differences [158]. It was also seen that the topographic features on the surface should be large enough to provide an optimum thick plastron and should be dispersed in a manner that the circulation within the plastron is not suppressed [45].

The physical background of the gas layer drag-reduction is through the exchange of a no-slip, or stick boundary condition at the solid-liquid interface with a partial-slip boundary condition at the gas-liquid interface. This directly affects the fluid flow around the solid body and the related hydrodynamic drag. The air gas layer sustained on the superhydrophobic solid surface can thus affect both major components of the hydrodynamic drag, i.e. The skin friction and the form or pressure drag. It should be noted that the majority of prior work on superhydrophobic surface drag-reduction has been done for the flow in channels or for flow over flat surfaces in which case the skin friction drag is the leading component. There have been far fewer studies on drag reduction on bluff bodies moving in liquid, for which case, for practical conditions the form drag can have significant or the leading contribution.

A good review on earlier works on slip and related drag reduction on superhydrophobic surface was given by Rothstein in 2010 [27] including both theory

26 considerations and major experimental works. As mentioned above these were mostly works on flow in channels and flow over flat surfaces. Some early examples are the works of Ou et al. 2004 [38, 39] for laminar flow in superhydrophobic microchannel and of Choi et al. 2006 [40, 41] for flow in microscale fluidics. For turbulent flow Watanabe et al. 1999

[42] for flow in a 16 mm pipe diameter with a super-water-repellent surface, Gogte et al.

2005 [43] measured drag on an unstructured superhydrophobically coated hydrofoil in water tunnel and Daniello et al. 2009 [44] investigated flow in a rectangular cell with smooth and superhydrophobic PDMS walls. In all the cases the drag reduction was rather limited.

Most closely related with the present study is the work of McHale et al. 2009 [28,

29] on terminal velocity and drag reduction on superhydrophobic spheres falling in water.

In this study the free-fall of acrylic spheres with various superhydrophobic surface coatings, for the Reynolds number range of Re = 104 – 3 × 104 was investigated. A 5 to 15

% drag reduction was found, but only for thicker plastron layers (50 to 250 μm) produced on spheres coated with coarse grains of hydrophobic sands. No drag reduction was measured when the superhydrophobic coating was directly deposited on the smooth sphere surface to sustain a thin plastron layer as the one investigated in this study. In following works the drag reduction effects of the plastron on the sphere free-falling in a liquid was investigated using analytical and theoretical models [37, 45] as well as numerical simulations [46]. More recently McHale group used both meshes and conformable superhydrophobic PDMS membranes with the membranes being reported as providing the best drag reduction result [47] and in a pipe flow [48].

In the last two decades research on superhydrophobic coating drag-reduction effect has intensified, driven by the quest for energy saving and greener future [50-74]. The wetting characteristics of a superhydrophobic surfaces is known to be governed by the geometrical structure of the surface and the chemical composition [63]. In work by Coulson et al. an oleo-phobic PTFE plate was prepared by depositing polymer layers which had low-surface-energy onto the micro-roughened PTFE substrates. This deposition led to an increase in the contact angle for the decane onto the modified substrate [64]. It was seen that the water contact angle could reach a high value of 160° to 175° if they had microstructures present on the surface[66].

Some related studies on bluff body’s drag reduction are the works of Dong et al.

2013 [30] testing the effect of superhydrophobic coating on macroscopic model ship and

Zang et al. 2015 [75] doing a similar investigation with a superhydrophobic submarine model. In both cases notable drag reduction was observed, although the models were guided inside a test channel on a wire, which complicated the interpretation.

The purpose of the present work, compared to prior art on drag reduction of superhydrophobic spheres falling in water, is first to investigate the effect of a relevantly thin air layer (~ 1 - 2 μm) that is easy to produce using commercially available superhydrophobic coatings. Second we use a range of larger and heavier spheres to achieve

Reynold numbers, Re = 3 × 104 – 3 × 105, closer to the drag crisis transition where the drag on the sphere is expected to be most sensitive to the surface mobility condition. In the ideal case scenario it can be assumed that the surface tension influence outweighs the viscosity effect and so there is no deformation in the air-water interface. The thickness of the superhydrophobic layer is due to the structure of the surface supporting the air layer. The

28 micro structure roughnesses or features depress the air velocities with the plastron in addition to the viscous drag that arises from the flow over the micro structures. It is important to note that the interface rigidification by the surfactants is negligible in the inertia dominated regime. The thick plastron is expected to modify the separation points

[45].

Chapter 3: Experimental Setup and Methods for Part I

3.1 Experimental Setup

The schematic of the experimental setup used in the proposed research is given in

Figure 2A. The actual setup is also depicted in Figure 2B. The tank was manufactured at

ChangTong Science and Technology (TianJin) Co. LTD, China and was customized to be

2.5 m deep and having a cross-section of 40 × 40 cm with reinforced boundaries to prevent any leakage or breakage. A ladder was placed on the side to gain access to the top of the tank and drop the spheres. Back panel of the tank was covered with light diffusing sheets to ensure uniform diffusion of the light from the Halogen bulbs (500W) which were placed at the back of the tank.

A basket was placed at the bottom of the tank to ensure easy retrieval of the spheres at the end of the experiments. For the smaller sphere sizes, electromagnetic setup was used to hold the sphere above the surface of the liquid. Once the electromagnet was switched off the sphere fell straight in the center of the tank. The use of electromagnet ensured that no spin was induced on the sphere during its release. A furnace was used to heat the sphere during coating to ensure the coating was robust and stable as explained in Section 3.2.

Figure 2- (A) Schematic of the experimental setup showing the liquid tank with sphere release magnetic holder mechanism on top. High-speed camera is in the front of the tank with halogen lamps providing the back-light illumination. (B) Photograph of actual laboratory experimental setup with the 2.5 tall tank. The high-speed camera is seen in the background.

3.2 Sphere Preparation

3 The metal spheres were polished steel spheres (ρs = 7.8 g/cm ) of sphere size 2R =

3 20 to 60 mm, or polished tungsten carbide spheres (ρs = 14.8 g/cm ) of sphere size 2R =

20 to 50 mm. After washing with ethanol and water, the unmodified sphere surfaces became hydrophilic with a water contact angle <90°.

The surfaces of the metal spheres were made superhydrophobic by the application of a commercial glass coating: Glaco Mirror Coat Zero (Soft 99 Ltd, Japan), to which we will refer in this thesis as Glaco. Glaco is an alcohol based dispersion of silica nanoparticles

(∼40 nm) coated with a silane hydrophobic agent. Details on the sphere coating process and surface characterization can be found in our prior work [76].

In brief, the spheres are washed with Glaco and dried at a temperature of about 160

°C to help consolidate the coating. The coating process is repeated 2 to 4 times, resulting in excellent water repellency of the sphere surfaces with a water droplet contact angle of more than 160°. Fig. 3a is a picture of a superhydrophobic steel sphere shortly after immersion in water. The silver mirror shiny appearance of the sphere was due to reflection from the thin air layer sustained on the superhydrophobic sphere surface. As shown in Fig.

3b taken about four hours after the sphere was immersed in the water, the air layer dissolution with time results in the loss of the shiny appearance.

Figure 3 – (a) Photograph of a 50 mm superhydrophobic steel sphere shortly after the sphere is immersed in water. The silver shiny appearance of the sphere is a signature of the thin air layer sustained on the superhydrophobic surface. (b) The same sphere after four hours in still water. The loss of the shiny appearance is due to the plastron gradual dissolution in water. (Jetly et al. 2018 [77])

3.3 Plastron Characterization

The figure 4 depicts the topographic image of the Glaco coating following four deposition cycles taken using an atomic force microscopy (AFM, Asylum Research MFP-

3D). The typical valley to peak distances is less than 1 μm and we can assume that the effective thickness of the air plastron on our surfaces is of the same range of ~ 1 μm. In prior studies [29, 78] the characteristic roughness of the surface was a good upper limit estimate for the effective thickness of the plastron layer.

The thickness of the plastron was further evaluated by AFM force measurement performed over a flat superhydrophobic surface sample immersed in water. The cantilever used in the measurements (MikroMasch, CSC36/Cr-Au) had a nominal spring constant of

2 N/m and a gold coated tip of about 15 μm of height. Fig. 4 depicts a typical force vs. piezo displacement curve showcasing the interaction of the cantilever tip with the substrate during the AFM force measurement approach (red line) and retraction (blue line) cycle.

The scanning began with the tip being placed several microns above the plastron meniscus

(branch A in Fig. 4). The moment the tip contacted the air-water meniscus (B), the strong capillary force pulled on to the tip and this pull resulted in a jump-in which contacted the hard wall with the surface (C). The measurement began with the tip being kept in hard wall contact with the substrate (D) using the capillary adhesive force until it snapped back (E) from the surface.

The thickness of the plastron layer in our case was determined using the estimation of the distance between the tip, jump in position and the neutral force hard-wall contact position. The hard-wall contact position was depicted by the dashed line with the double

34 arrow in the figure 4 wherein δP ~ 1 μm. The placement of the tips over different samples provided us with an approximation of the plastron thickness which ranged from 0.3 μm to

2.0 μm. Most cases the thickness of the plastron layer was observed to be approximated at

1 μm. In comparison to the previous work [29] where the plastron layer thickness was 50 to 250 μm thick, our plastron layers were much thinner (~1 μm) and attributed to having higher drag reduction.

3.4 High-Speed Imaging

The high speed camera used to record the free-falling spheres in the experiments mentioned in this thesis was the Photron SA-5. The Photron SA-5 camera has a CMOS sensor with a pixel size of 20µm.The maximum filming rate for the camera is 1 million frames per second [fps] at a smaller resolution of (64 × 16) pixels. The camera is capable of recording 7500 fps at the full resolution of (1024 × 1024). In these experiments the frame rate was set between 1000-2000 fps with a resolution of 512×1024 and the shutter speed between 1/2000 - 1/25000 depending on the experiments to be performed.

The Photron SA-5 camera comes with its own software package which helps in assisting in image and video processing. The software Photron FASTCAM viewer was used to record, process, trim and playback all the videos and images taken. The recording mode was set to end trigger to obtain the last seconds of the video recorded in loops. The progress of the sphere fall velocity during the free-fall in the tank was evaluated by image processing the high-speed videos usually by using the Photron camera software (Photron

FASTCAM Viewer, PFV Ver.3670).

Figure 4- An AFM 3D image of the Glaco coated surface topology taken over 60×60 μm area. In this case four cycles of Glaco coating were repeated. (b) AFM force curve vs piezo displacement, ΔX taken above the plastron covered substrate. The force curve branches correspond to: (A to C) piezo extension approaching the surface toward the cantilever (red line): (A) free cantilever approaching the surface; (B) cantilever jump-in toward the surface upon the tip contact with the plastron; (C) cantilever in hard-wall contact with the surface; (D to F) piezo retraction pulling the tip away from the surface (blue line): (D) cantilever tip in hard-wall contact with the substrate; (E) the cantilever detaches from the surface; (F) free cantilever moving away from the surface. The plastron layer thickness, δ is estimated as the distance between the jump-in position and the zero force hard-wall contact position. Notice that parts of the force curve is out of the force detection range of – 3 mN to + 3 mN.

3.5 Determination of the Drag Coefficient

The motion of sphere during the free-fall in a still Newtonian fluid is depicted by the classical Basset-Boussinesq-Oseen (BBO) equation, which for the high velocity range considered here can be written in the form: [16, 79-80]

2 푡 ⁄ 푑푈 퐶퐷 휋퐷 2 푑푈 3 2 푑푈 푑휏 푚푆 = (푚푆 − 푚푓)𝑔 − 휌 ( ) 푈 − 푘푚푓 − 퐷 √휋휌휇 ∫ 푑휏 (3.1) 푑푡 2 4 푑푡 2 0 √푡 − 휏

Where, g is the acceleration due to gravity, ms is the sphere mass, mf the mass of the fluid of the same volume as the sphere, k is the added mass coefficient (for a sphere k

~ 0.5). The left-hand side of the equation is the sphere inertia force. The right-hand side first, second, third and fourth terms are the submerged weight, drag, added mass and

Basset history force respectively.

If the drag coefficient dependence on the velocity is known for the entire range of the sphere fall the BBO equation can be solved numerically to predict the sphere velocity evolution with time. Respectively, if the velocity dependence on time is known, BBO equation can be used to estimate the instantaneous drag of the sphere falling with acceleration. For higher velocity range considered in our experiments the Basset history force has been shown to be insignificant (less than 1%) and can be omitted [16]. In such case from eqn 3.1 the drag coefficient can be approximated as:

4퐷 푑푈 퐶 = ((휌 − 휌)𝑔 − (휌 + 0.5휌) ) (3.2) 퐷 3휌푈2 푆 푆 푑푡

37 where, U and dU/dt are the experimentally determined velocity and sphere acceleration.

For a sphere falling at terminal velocity, UT this equation converges to the expression for steady free-fall velocity:

4퐷 퐶퐷 = 2 (휌푆 − 휌)𝑔 (3.3) 3휌푈푇

The equation (3.3) can be derived from the buoyancy, gravity and drag forces balance. For all of the spheres studied the experimental velocity vs. time dependence can be fitted well by the exponential form

−푡⁄휏 푈 = 푈푇(1 − 푒 ) (3.4)

This exponential form has been used before to predict the terminal velocity of sphere for the case where, the sphere is still accelerating at the bottom of the tank and has not reached the terminal velocity, UT [15]. From the best fit of this equation with the experimental data one can obtain the extrapolated value for the terminal velocity UT, and the characteristic time for reaching the approaching of the terminal velocity, . In Chapter

8 of this theses we derive alternative analytical expressions for the velocity vs time dependence (see equations 8.3 -8.5)

However, here we limit the use of eqn 3.4 only as a smoothing function for the experimentally observed U vs. t dependencies, during the sphere fall in the 2.5 meter tank.

Respectively, all the drag coefficients data point present later in results were calculated using eqn 3.2 by taking the experimental values of U = UMax measured close to the bottom of the 2.5 m tank and the corresponding dUMax/dt values evaluated from the experimental

U vs. t dependences (Section 4.1)

Chapter 4: Experimental Results for Part I

4.1 Velocity vs Time Dependences

For each of the spheres (ST or TC) the entire free-fall was recorded and analyzed using the setup previously mentioned in the Chapter 3 section 3.1 and following the protocol in section 3.2. Examples for the progression of the sphere velocity with time during the spheres fall in the 2.5 m water tank are depicted below. The Red data points for all the below velocity vs time dependence curves will be for the superhydrophobic spheres

(Vapor Layers) and the blue data points for the hydrophilic spheres (no-vapor-layers) free- falling in water. The solid lines represents the best fit curves for these points using equations 3.4.

a 3 2R = 20 mm 3 ρs = 7.8 g/cm Hydrophilic 2

Superhydrophobic

Velocity [m/s] Velocity 1

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 Time [s]

b 3 2R = 25 mm ρ = 7.8 g/cm3 s Hydrophilic

2 Superhydrophobic

Velocity [m/s] Velocity 1

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 Time [s]

c 4 2R = 40 mm 3 ρs = 7.8 g/cm Hydrophilic

Superhydrophobic

2 Velocity [m/s] Velocity

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Time [s]

d 6 2R = 45 mm 3 5 ρs = 7.8 g/cm Superhydrophobic 4

Hydrophilic

Velocity [m/s] Velocity 2

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Time [s]

e 6 2R = 50 mm Superhydrophobic 3 (Plastron) ρs = 7.8 g/cm

4 Hydrophilic (No gas layer)

Velocity [m/s] Velocity 2

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Time [s]

6 f 2R = 60 mm 3 Superhydrophobic ρs = 7.8 g/cm

4 Hydrophilic

Velocity [m/s] Velocity 2

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Time [s]

Figure 5 - Comparison of spheres fall velocity progression of the sphere velocity with time during the fall in the 2.5 meter water tank for superhydrophobic (red squares) or hydrophilic (blue circles) steel sphere with diameters: (a) 20 mm (b) 25 mm (c) 40 mm (d) 45 mm (e) 50 mm and (f) 60 mm. The solid line for each data set is formal fit using the exponential form.

g 4 2R = 20 mm 3 ρs = 14.8 g/cm 3 Hydrophilic

Superhydrophobic

2 Velocity [m/s] Velocity 1

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 Time [s]

h 4 2R = 25 mm 3 ρs = 14.8 g/cm Superhydrophobic

Hydrophilic

2 Velocity [m/s] Velocity 1

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Time [s]

i 4 2R = 30 mm Superhydrophobic 3 ρs = 14.8 g/cm 3 Hydrophilic

2 Velocity [m/s] Velocity 1

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Time [s]

j 6 2R = 35 mm 3 5 ρs = 14.8 g/cm Superhydrophobic

3 Hydrophilic

Velocity [m/s] Velocity 2

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Time [s]

k 6 2R = 40 mm Superhydrophobic 3 ρs = 14.8 g/cm

Hydrophilic

Velocity [m/s] Velocity 2

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Time [s]

l 6 Superhydrophobic

2R = 45 mm 3 ρs = 14.8 g/cm 4

Hydrophilic

Velocity [m/s] Velocity 2

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Time [s]

m 6 2R = 50 mm Superhydrophobic 3 ρs = 14.8 g/cm

4 Hydrophilic

Velocity [m/s] Velocity 2

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Time [s]

Figure 6 - Comparison of spheres fall velocity progression of the sphere velocity with time during the fall in the 2.5 meter water tank for superhydrophobic (red squares) or hydrophilic (blue circles) tungsten carbide sphere with diameters: (g) 20 mm (h) 25 mm (i) 30 mm (j) 35 mm (k) 40 mm (l) 45 mm and (m) 50 mm tungsten carbide sphere. The solid line for each data set is formal fit using the exponential form.

As demonstrated in Fig. 5 and Fig 6 the smaller sizes sphere (20 mm to 40 mm steel; 20 mm to 30 mm tungsten carbide) we measured almost identical velocities for the unmodified and superhydrophobic spheres. However, for the larger sphere (45 mm to 60 mm steel; 35 mm to 50 mm tungsten carbide) the plastron covered superhydrophobic spheres were falling significantly faster approaching the bottom of the tank. In Fig. 5e for the case of 50 mm steel sphere, the velocity of the superhydrophobic sphere close to the bottom of the tank is more than 50 % higher that the velocity of the unmodified sphere.

To assert for the larger spheres if the drag reduction was due to the plastron layer, the larger spheres were ethanol washed and then plasma cleaned to eliminate the superhydrophobic coating and the plastron present on their surfaces. The spheres were again allowed to free-fall and the trajectory was recorded as in the previous cases. It was seen that the fall velocity was similar to the unmodified hydrophilic spheres, thereby asserting the drag was reduced due to the presence of the thin plastron layer and not the roughness inherited by the spheres.

The drag coefficients (CD) for the free-falling spheres, both the unmodified spheres

(no gas layer) and the superhydrophobic spheres (plastron) were plotted in terms of the

Reynolds numbers (Re).The points were plotted against the literature values [14].The data points were calculated by analyzing each free-fall thereby measuring the velocity of the sphere and the acceleration close to the bottom of the 2.5 m tank using the equations previously mentioned.

4.2 Drag Coefficient vs Reynolds Number Dependence

Fig. 7 presents the results of our experiments in terms of the drag coefficient, CD vs

Reynolds number, Re dependences for the unmodified (no gas layer) and superhydrophobic surface (plastron) spheres. Each data point is calculated from the experimentally measured sphere velocity and acceleration close to the bottom of the 2.5 m tank using eqn (3.4) for the drag coefficient

The data in Fig. 7 shows an early drag crisis transition for the plastron covered superhydrophobic spheres, starting from Re ≈ 1.2 × 105, which is significantly lower than the transition for the case of smooth hydrophilic spheres, occurring at Re ≈ 2.5 × 105. Data for both steel and tungsten carbide spheres seems to fall on the same master curve with drag reduction reaching about 80 % for the larges spheres (50 – 60 mm steel, 40 -60 mm tungsten carbide). This range of drag reduction is similar to the one achieved by traditional methods to evoke early drag crisis as surface roughening [3, 15], polymers addition [14-

16] or mass transfer [2].

Earlier experiments with sphere covered with Leidenfrost vapor layer falling in heated 95 ⁰C, have demonstrated an early drag crisis transition at Reynolds numbers as low as Re > 2×104 and similar range was observed for Leidenfrost spheres falling in the perfluorocarbon liquids, FC-72 or PP1 wherein viscosity is close to that of water [77-80].

The extended Re range of drag reduction for Leidenfrost spheres in comparison to the superhydrophobic spheres studied here is expected because of the substantial difference in the thickness of the gas layers: ~ 50-100 μm for the Leidenfrost vapor layer [81-84] vs. ~

1 μm for the present air layer sustained on the superhydrophobic surface. Furthermore, the

Leidenfrost gas layer fully separates the liquid from the sphere surface, whereas the air plastron is pinned to the asperities of the superhydrophobic surface texture.

Even though the effect of drag reduction due to thin air layers (plastron) on superhydrophobic surfaces has been anticipated in the past the actual extent for high

Reynolds number’s closer to the drag crisis has not been demonstrated. Numerical simulations modelling the result of the Leidenfrost vapor layer experiments have shown that a Navier slip length model can be successfully used to relate the gas layer drag reduction to an effective slip length [84-85].

The simulations predict that the drag reduction effect is a very sensitive function of the Reynolds number and the scaled slip length, which in turn depends on the gas layer thickness and the vapor to liquid viscosity ratio. The present data provide further opportunities to develop and test such models by covering a range of much thinner gas layer thicknesses that have not been studied before, and which are relevant for a wide range of practical applications.

0.6 No gas layer

D St TC C St (White) 0.4

0.2 Plastron Drag Coefficient, Coefficient, Drag St TC

0 3 ×104 4 6 8 105 2 4 6 8 106

Reynolds number, Re

Figure 7- Dependence of the drag coefficient on the Reynolds number for steel and tungsten carbide spheres falling in water. Open blue triangles are for hydrophilic steel spheres (20, 25, 30, 40, 45, 50 and 60 mm). Open blue squares for hydrophilic tungsten carbide spheres (20, 25, 30, 35, 40, 45 and 50 mm). Solid red triangles and solid red squares are for the respective superhydrophobic steel and tungsten carbide spheres freshly immersed in water and covered by a plastron. Both sets of results are calculated using eqn 3.4 with the sphere velocity and acceleration taken at the end of the 2.5 m water tank fall. Literature values for smooth solid steel spheres falling in an “infinite” tank measured by White [14] are shown as crosses

Chapter 5: Conclusions for Part I

In this chapter we discussed the effect of thin air layers which were naturally sustained to textured superhydrophobic surfaces. The fall velocity and the drag on metallic spheres both hydrophilic and superhydrophobic spheres was measured. The fall of these spheres in a 2.5 meter deep tank enabled the investigation to extend into higher Reynolds number ranges which were close to the drag crisis. It was also shown that the drag on larger superhydrophobic spheres reduced dramatically by up to 50-80%. This drag reduction was attributed to the early triggering of the drag crisis at Re ~1.2 × 105, which was substantially lower than the corresponding drag crisis occurring for the smooth solid spheres. The experiments demonstrated that even a very thin air layer (~ 1 - 2 μm) can significantly affect the flow patterns around bluff bodies at the drag crisis transition range. These results complement early results obtained using thicker dynamic Leidenfrost vapor layers and could have important implication in the development of gas-layer-lubrication based drag reduction strategies.

Part II

Giant drag reduction on Leidenfrost spheres evaluated from extended free-fall trajectories

Published in the Experimental Thermal and Fluid Science (Jetly et al. 2018, see reference [138])

Chapter 6: Introduction to Part II

6.1 The Leidenfrost Effect

Leidenfrost effect is a phenomenon wherein a drop of liquid when placed on a very hot surface whose temperature is significantly above the boiling point of the liquid sustains a gas layer between the drop of liquid and the hot surface. The everyday-life example of this effect is drop of water or oil on a hot pan wherein the drop levitates above the hot surface thereby becoming extremely mobile due to the presence of the lubricating effect of the vapor layer. The figure 8 showcases the Leidenfrost effect.

In the experiments performed within our lab we make use of the inverted

Leidenfrost effect (figure 8b) wherein the solid body is heated to temperatures higher than the boiling point of the liquid. This high temperature heating leads to formation of a gas/vapor layer around the solid body which encloses the body and further helps to reduce drag.

A sphere at Leidenfrost temperature when dipped into a liquid in our case FC-72 exhibits two stages as the temperature of the sphere falls below the Leidenfrost temperature. This first stage of the Leidenfrost condition is evident from the continuous thin vapor layer formed around the sphere (Figure 9 a, c). Next as the temperature of the sphere falls below the Leidenfrost vapor temperature the periodic release of bubbles occurs which is marked by the explosive destruction of the bubbles around the sphere (Figure 9 b, d). The dynamic vapor layer stable around a 20 mm diameter steel sphere held stationary in FC-72, when the sphere is heated well above the Leidenfrost temperature of about

110⁰C which is shown in figure 9.

a b

Figure 8 – (a) Leidenfrost effect example of a drop levitating on a hot plate. (b) Inverted Leidenfrost effect wherein the solid body is heated at high temperatures thereby being enclosed within a vapor layer or gas layer which helps to reduce drag

a b

c d

Figure 9 - High-speed camera snapshots of a 20 mm steel sphere held stationary in FC -72: (a) when the sphere temperature, TS is above the Leidenfrost temperature TL ~ 110 ⁰C and (b) shortly after passing the Leidenfrost temperature with explosive boiling. (c, d) High-speed camera snapshot silhouette images of a 20 mm steel sphere falling in FC-72: (c) for sphere temperature, TS ~ 250 ⁰C above the Leidenfrost temperature, and (d) at temperature slightly below the Leidenfrost temperature Vakarelski et al. (2011) [81]

6.2 Prior Works on Leidenfrost Drag Reduction

An ultimate way to investigate the effect of the gas layer on the drag experienced by a solid moving in liquid is by the means of the Leidenfrost effect [76, 83, and 86]. The approach involves the liquid sustaining a continuous and robust lubricating layer on the surface of fast moving solids in water. The two main contributors of hydrodynamic drag being skin friction & form drag. The skin friction which arises from the fluid velocity and pressure induced drag referred to form drag. In general the flow around a sphere has been extensively studied both experimentally [87-89] and numerically [90-94]. In respect to drag reduction on underwater moving bodies by gas layers the drag was found to be reduced by the use of gas generation method using electric field or heating [95], or by gas generation inclusion on a three phase interface [96]

Recently the use of the Leidenfrost effect [76, 83, 86] has been shown to be a very efficient method to study gas layer drag reduction by sustaining a continuous vapor layer on the surface of solids which are heated to temperatures well above the boiling temperature of the surrounding liquid [81, 82, 84, 97, 98]. Initial experiments with heated

Leidenfrost spheres falling in liquid were performed using a perfluorocarbon liquid, FC-

72 mostly composed of perfluorohexane (C6F14) [81]. The FC-72, was preferred to water because of its much lower vaporization heat capacity (about 30 times lower) and lower boiling point of about 56 ⁰C, which makes it significantly easier to sustain a continuous vapor layer on the surface of the free-falling hot spheres. The study concluded the drag reduction in three Reynolds number regimes as being negligible for the low Reynolds

4 number (Re up to ~ 2×10 ) , Continuous reduction with CD going from 0.45 to 0.1 for

56 intermediate Reynolds numbers and low limiting drag coefficients (CD~0.07) for high Re

≥ 1×105.

By comparing the free-fall velocities inside FC-72 of room temperature spheres with sphere heated above the Leidenfrost temperature it was demonstrated that the

Leidenfrost vapor layer can reduce the hydrodynamic drag on sphere by up to 85% depending on the spheres Reynolds number [81]. The drag reduction is manifested as an early drag crisis transition, indicated by the wake separation point movement to the rear of the Leidenfrost spheres. Figure 9c and Figure 9d are snapshot silhouette images of a 20 mm steel sphere falling in FC-72, when the sphere temperature is above (Figure 9c) or slightly below (Figure 9d) the Leidenfrost temperature, demonstrating the delayed wake separation for the vapor-layer-shielded sphere.

In a later study [82] the Leidenfrost vapor layer effect was as well replicated for spheres falling in water. In this case a combination of superhydrophobic surface spheres and high water temperature, TW ~ 85 ⁰C to 95 ⁰C was needed to sustain a stable vapor layer during the free-falling. Similar approach was used in a modified Taylor-Couette experiment, which demonstrated that apart from the pressure component of the drag on falling sphere (form drag), the vapor layer can as well dramatically reduce the skin-friction component of the drag in turbulent flow conditions [98].

Most recently by using a range of high viscosity perfluorocarbon liquids it was shown that the Leidenfrost vapor layers drag reduction on falling sphere can occur for a very wide range of Reynolds number from as low as 600 up to 105, far below the traditional drag crisis transition range [84]. Related numerical simulations using a Navier slip model with viscosity dependent slip length were shown to fit the experimental data [85].

Chapter 7: Experimental Setup and Methods for Part II

7.1 Experimental Setup

Experiments were conducted in two liquid tanks as shown in Figure 10. The first is a 3 m tall and 14 cm in diameter cylindrical tank made of clear acrylic (50 L tank). The second tank is 2 m tall, and has a square cross-section area of 20 × 20 cm (80 L tank), with front and back walls of double-glazed glass windows. An electric heater was installed at the bottom of this tank that allowed the FC-72 liquid to be heated to 55 °C close to the boiling point of about 57 °C.

The fluid used (FC-72, 3M™ Fluorinert™ Electronic Liquids) is a clear odorless, fully-fluorinated liquid, mostly composed of perfluorohexane (C6F14). It has a boiling point

3 of about 57 °C, heat of vaporization, HC = 88 kJ/kg, and liquid density ρ = 1680 kg/m .

Using an Ubbelohde type capillary viscometer (Fungilab) we measured the FC-72 dynamic viscosity at room temperature of about 21 °C to be µ = 0.74 ± 0.01 mPa s (slightly higher than the producer given value of 0.64 mPa s) and at 55 °C we measure µ = 0.39 ± 0.01 mPa s

7.2 Experimental Protocol

In respect to the spheres of larger diameters, it was challenging to produce a sphere fall in the 3 meter tall tank without the sphere hitting the wall of the tank before it reached at the bottom of the tank due to the inherit deviation from the straight fall at ranges of

Reynolds number in this study [82]. Alternatively, the larger size spheres, ST of 2R = 30 to 60 mm and TC of 2R = 30 to 45 mm, experiments were conducted using the wider, 2 meter tall 20 × 20 cm cross-section tank in which it was possible to produce a free-fall without the sphere hitting the tank walls.

In the first investigation of the Leidenfrost vapor layers drag reduction on sphere free-falling in FC-72, a 2 meter tall and 8 cm in diameters cylindrical liquid tank was used

[81]. As the practical limiting factor for the tanks size was the available amount of FC-72 fluid, we chose to use taller, but narrower tanks to track the fall of the smaller spheres (10 to 25 mm), and a shorter but wider tank to track the fall of the larger spheres (30 to 60 mm).

The free-fall of spheres in the tank was recorded with a high speed camera (Photron

SA-5) using typical filming rates of 2000 fps. The sphere location vs time was determined by image processing the videos using the camera software (Photron FASTCAM Viewer,

PFV Ver.3262). The usual coordinate sampling time interval was 20 ms. From these data we evaluated the time dependence of the sphere fall velocity and acceleration used to estimate the drag coefficient. The sphere velocity was calculated from coordinate data by using a moving median value over Δt = 100 ms time interval as data smoothing function.

The sphere acceleration was respectively calculated from the velocity data as moving median over the same time interval.

In the Leidenfrost spheres experiments, before release in the tank, we heat the sphere to TS = 250 ⁰C in a controlled temperature furnace for at least 30 minutes. Using forceps the heated sphere is moved to the tank and carefully released from just below the

FC-72 liquid surface. Using sphere temperature vs. cooling time data for perfluorohexane

[84] we estimated that in all cases the sphere temperature drops by less than 15 ⁰C during the fall in the tank, which guarantees that over the entire fall the sphere remains in the

Leidenfrost regime. The Leidenfrost state was as well confirmed by visual observation of the sphere cooling at the bottom of the tank fallowing the fall. To account for the tanks wall effects we used the velocity correction formula given by Newton (1678):

2 2 1/2 푈⁄푈∞ = [1 − (퐷⁄퐷푇) ][1 − 0.5 (퐷⁄퐷푇) ] (7.1)

where, U is the measured velocity and U∞ is the corrected velocity for infinite flow domain, D = 2R is the sphere diameter and DT is the diameter of a cylindrical tank. For the a  a square cross-section tank, the effective diameter DT in Eq. (7.1) is calculated by equating the areas of the circle and the square, i. e. 퐷푇 = (2⁄√휋)푎

L = 2 M

DT = 8 cm L = 3 M D = 14 cm T L = 2 M 20×20 cm

Figure 10-Photograph comparing the sizes of the experimental tanks. On left is the 3 meter tall and 14 cm in diameter cylindrical 50 liters tank and on right is the 2 meter tall, 20 × 20 cm, square cross section 80 liters tank with heating device used in the present study. The 2 meter tall and 8 cm in diameter cylindrical 10 liters tank used in the prior study is also included for comparison (middle). The high-speed camera used to monitor the sphere free-fall in the tank can be seen in the background.

Chapter 8: Experimental Results for Part II

8.1 Velocity vs Time Dependences

The major set of experiments were conducted to compare the free-fall of room temperature spheres in FC-72, TS = 21 °C to the fall of identical spheres which were heated before release to the Leidenfrost sphere temperature, TS = 250 °C. In our prior FC-72

Leidenfrost drag reduction study [81] we established that the fall velocity of the sphere sharply increases once the sphere temperature is above about 140 °C, which secures a stable vapor layer on the falling sphere, and stays almost constant for TS between 140 °C and

280 °C. Respectively in the present study we use a standard sphere pre-heat temperature,

TS = 250 °C, for all spheres investigated as representing cases of the Leidenfrost regime.

For simplicity we will refer to the room temperature spheres experiment, TS = 21 °C as the

“no-vapor-sphere case”, and to the experiments conducted with spheres heated to

TS = 250 °C as the “Leidenfrost sphere case”.

Experiments with smaller diameters steel (ST) and tungsten carbide (TC) spheres;

2R =10 mm to 25 mm were conducted in the taller, 3 meter tank. The velocity progression of both steel and tungsten carbide spheres falling in either the 3 m tall (L = 3 m) or the 2 m tall (L = 2 m) tanks. Each figure compares the velocity of identical spheres for room temperature (no vapor layer) case sphere and the heated Leidenfrost spheres

62 a

64 e

65 g

Figure 11 - Comparison of spheres fall velocity progression with time, during the fall of identical steel spheres in FC-72, of room temperature, TS = 21 ⁰C (blue circles) or under Leidenfrost regime,

TS = 250 ⁰C (red squares) for diameters: (a) 2R = 10 mm (b) 13 mm (c) 15 mm (d) 20 mm (e) 25 mm (f) 30 mm (g)40 mm (h) 45 mm (i) 50 mm and (j) 60 mm .

68 m

Figure 12 - Comparison of spheres fall velocity progression with time during the fall of identical tungsten carbide spheres in FC-72, for sphere of room temperature, TS = 21 ⁰C (blue circles) or

Leidenfrost spheres, TS = 250 ⁰C (red squares): for diameters (k) 2R = 10 mm (l) 15 mm (m) 20 mm (n) 25 mm (o) 40 mm and (p) 45 mm .

The data in Figure 11 and Figure 12 demonstrates the magnitude of the Leidenfrost sphere velocity increase due to the vapor layer drag reduction effect over the range of ST and TC sphere sizes investigated. The dramatic difference in the fall velocities between the no-vapor-layer and Leidenfrost vapor layer lubricated spheres is best demonstrated by the free-fall of the 20 mm steel spheres toward the lower part of the 3 meter tall tank. More importantly the Figure 11 and Figure 12 show that except for the smallest 10 mm steel sphere studied here, all other Leidenfrost spheres are still accelerating when reaching the bottom of the tank, even though in this study we use taller and wider tanks than in our previous study. For most of sphere sizes investigated the no vapor layer spheres reach terminal velocity before hitting the bottom of the tank.

However, the largest sphere sizes studied (Figure 11j and Figure 12p) are also accelerating when reaching the tank bottom, indicating a transition to drag crisis regime in the no-vapor-case. The transition to the drag crisis regime during the free-fall is indicated by the sudden change in the slope of the velocity curves, as is readily noticeable for the 2R

= 15 mm Leidenfrost sphere in Fig. 11c and for the 2R = 10 mm TC Leidenfrost case in

Fig. 12k.

An additional set of experiments was conducted in the 2 m tank with the heating device using FC-72 liquid heated to 55 °C. In this case we used a range of steel spheres pre-heated to the liquid temperature, TS = 55 °C and identical steel spheres which were heated before release to the Leidenfrost sphere temperature, TS = 160 °C

8.2 Falling Sphere Equation of Motion

In this section we highlight the equations of motion of the sphere during its free- fall to derive the simplified expressions for the interdependence of the sphere velocity, acceleration and the drag. In general the drag on the sphere as mentioned in the Chapter 1

Section 1.1, was characterized being a function of the Reynolds Number. The coefficient of drag being calculated from the drag force which in turn was found from the force balance

2퐹 between buoyancy, gravity and drag force. The drag coefficient was 퐶 = 퐷 , being a 퐷 휋푅2휌푈2

2휌푅푈 function of the Reynolds number 푅푒 = . Here FD is the drag force on the sphere, ρ the  fluid density, R the sphere radius and  the dynamic viscosity of the fluid.

The dynamic equation of motion of the sphere, during the free-fall in still

Newtonian fluids is given by the Basset-Boussinesq-Oseen (BBO) equation, which for the

Reynolds number range in our experiment can be written as [16,79,80]

푑푈 퐶 휋퐷2 푑푈 3 푡 푑푈⁄푑휏 푚 = (푚 − 푚 )𝑔 − 퐷 휌 ( ) 푈2 − 푘푚 − 퐷2√휋휌휇 ∫ 푑휏 (8.1) 푆 푑푡 푆 푓 2 4 푓 푑푡 2 0 √푡−휏

Where g is the acceleration of the gravity, D = 2R, ms is the sphere mass, m the mass of the displaced fluid, k is the added mass coefficient (for subcritical Re spheres, k ≈

0.5). The left hand side of the equation is the sphere inertia force. The right-hand side first, second, third and fourth terms are respectively the submerged weight, drag, added mass and the Basset history force. If the dependence of the drag coefficient on the velocity is known for the entire fall, this equation can be solved numerically to obtain the sphere fall

72 trajectory and velocity time dependencies [16]. For the high Reynolds number conditions of our experiment and relevantly high density of the spheres compared to that of the surrounding fluid, the Basset history force has been shown to be negligible (less than 1%)

[16] and can be therefore omitted to a good approximation, to simplify Eq. (8.1) to:

푑푈 3휌퐶 (휌 + 푘휌) = (휌 − 휌)𝑔 − 퐷 푈2 (8.2) 푆 푑푡 푆 4퐷

Where the expression was evaluated using the sphere and displaced fluid masses using the sphere and fluid densities, ρs and ρ respectively. In order to solve this equation analytically using the partial fractions method the drag coefficient was assumed to be constant. Initializing the initial condition as U = 0 at t = 0 we find the solution:

푡 1−푒−2푡⁄휏 푈 = 푈 푡푎푛ℎ ( ) = 푈 ( ) , (8.3) 푇 휏 푇 1+푒−2푡⁄휏 where:

4퐷 1⁄2 푈푇 = ( (휌푆 − 휌)𝑔) (8.4) 3휌퐶퐷 and

1⁄2 3휌퐶퐷 1 ( (휌푆−휌)푔) = 4퐷 . (8.5) 휏 (휌푆+푘휌)

The solution described above is true for any segment of the free-fall of the sphere during the period wherein the drag coefficient is constant. The time is offset at the

73 beginning of the segment to be a match with the velocity value. This leads us to matching the velocity value to U0 at t = t0:

1 푡−(푡0−푡0) 푈 = 푈 푡푎푛ℎ ( ), for t > t0 and CD = constant (8.6) 푇 휏

1 where t0 satisfies:

푡1 푈 = 푈 푡푎푛ℎ ( 0) (8.7) 0 푇 휏

It is known that in the subcritical Reynolds number regime, Re = 103 to 2×105 the drag coefficient has a plateau value of about 0.45 to 0.55. We have further noticed that for the range of ST and TC spheres sizes used in our experiment, the velocities corresponding to Re > 103 is less than 1/10th of the spheres terminal velocities, and is reached shortly after the sphere is released in the tank. Respectively, in Figure 13 we demonstrate that for the no vapor spheres falling at subcritical Reynolds numbers, the U vs. t experimental dependences closely follow the Eq. (8.3) given solution. The Figure 13 example is for the

2R = 25 mm ST and 2R = 25 mm TC no vapor spheres falling in FC-72 and Figure 13b for is the same spheres falling in room temperature water. In the case of Figure 13b we used data collected in our previous study [77] for sphere falling in a water-filled tank of 2.5 meter height and 40×40 cm cross-section, which results in less sideway velocity fluctuation due to the tank wall effects.

In some of the previous studies [2, 15, 82] on drag on free-falling spheres the U vs. t data were extrapolated using the empirical exponential relation:

−푡⁄휏1 푈 = 푈푇(1 − 푒 ) , (8.8)

where UT and τ1 are two independent fitting parameters. However, the examples in

Figure 13 show very good agreement with the Eq. (8.3) solution of the simplified BBO equation, using only one independent fitting parameter, which could be the drag coefficient, CD that determines both the terminal velocity, UT and the characteristic time τ according to Eq. (8.4) and Eq. (8.5) respectively. This is the same equation as used in Part

I of the thesis Eq. (3.4) .Alternatively, one can use the experimentally determined UT value to calculate CD and τ. The two empirical parameters Eq. (8.8) can still be useful in some cases, as they were shown to be effective in predicting the terminal velocities of spheres including cases when the drag varies during the fall [15].

(a) 3 FC - 72

TC, 25 mm

2 Steel, 25 mm

Velocity [m/s] Velocity 1

0 0.00 0.50 1.00 1.50 Time [s]

(b) 4 Water TC, 25 mm 3

Steel, 25 mm

2 Velocity [m/s] Velocity 1

0 0.00 0.50 1.00 1.50 Time [s]

Figure 13- Comparison of the experimental sphere free-fall velocity progression with time with the Eq. (8.3) analytical solution of the BBO equation assuming a constant drag coefficient fall. (a) Fall of 2R = 25 mm steel sphere (squares) and 2R = 25 mm tungsten carbide (triangles) in FC-72 measured in the 3 meters tall, 14 cm in diameter tank. (b) Fall of the same spheres in water measured in a 2.5 meter tall, 40 × 40 cm cross-section tank (experimental data from Jetly et al. 2018 [77]).

The red lines are Eq. (8.3) fits, where CD and τ were determined from the experimental value of the terminal velocity, UT using Eq. (8.4) and Eq. (8.5) respectively.

8.3 Drag Coefficients vs Reynold Number

The drag coefficient (CD) for spheres falling at terminal velocity was determined using the below relation which arised from the balance of buoyancy, gravity, and drag force:

4퐷 퐶퐷 = 2 (휌푆 − 휌)𝑔 (8.9) 3휌푈푇

The above equation was same as the one used in the previous Chapter 3, Section

3.5 Eq. (3.3). It must be noted that in the case of superhydrophobic spheres, the sphere would reach its terminal velocity wherein for the case of the Leidenfrost spheres and the large no-vapor-layer spheres, the spheres still tended to be accelerating as they were at the bottom of the tank as shown in the results above. For these cases, wherein the spheres were still accelerating the equation (8.10) could be used to calculate the instantaneous value of the drag coefficients. This instantaneous value was calculated accounting for the sphere acceleration as

4퐷 푑푈 퐶 = ((휌 − 휌)𝑔 − (휌 + 0.5휌) ) (8.10) 퐷 3휌푈2 푆 푆 푑푡

This equation converges to Eq. (8.9) for spheres falling at terminal velocity, e.g. for which dU/dt=0. Here we remind that in Eq. (8.2) and respectively Eq. (8.10) we ignore the history force which is expected to be negligible in our experiments, and we use the approximate value of the added mass coefficient, k = 0.5. The standard added mass coefficient, k = 0.5 value for a sphere has been experimentally verified for Re up to about

2.5 × 105 [99-100]. There is no well-established added mass coefficient correction at supercritical Reynolds number [80]. However as in our system the sphere density is

77 substantially higher than the liquid density (4 to 8 times) the drag coefficient estimate is not very sensitive to possible deviation from the k = 0.5 value.

Figure 14 shows an example for the evaluation of the instantaneous drag coefficient using Eq. (8.10). Figure 14a shows the sphere depth vs. time data, x(t), for the

2R = 25 mm ST and 2R = 25 mm TC Leidenfrost spheres falling in the 3 m tank. Figure

14b is the corresponding velocity vs time dependence, U (t) = dx(t)/dt, calculated as moving median from the x(t) data. Figure 14c is the corresponding drag coefficient vs time calculated using Eq. (8.10) in which the acceleration, dU(t)/dt is calculated as moving median from the U(t) data. The most significant result illustrated in Figure 14 is that, while the sphere velocity is far from the terminal value at the end of the fall (Figure 14b), the drag coefficient on the sphere reaches a plateau value (Figure 14c), at a time corresponding to a depth of about 0.5 meters. This result is important from practical point of view as we can now evaluate the drag coefficients using tanks that are substantially shorter than the one that will be needed for the spheres to reach a velocity close to the terminal velocity value.

3 (a) TC, 25 mm

2 ST, 25 mm

Depth [m]Depth 1

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Time [s]

(b) 6

5 TC, 25 mm

4 ST, 25 mm 3

2 Velocity [m/s] Velocity 1

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Time [s]

0.7

0.6 D

C 0.5 TC, 25 mm 0.4 ST, 25 mm 0.3 TC, 10 mm ST, 40 mm

0.2 ST, 15 mm Drag Coefficient , , Coefficient Drag

0.1 TC, 15 mm

0 10000104 2 3 4 6 8100000 105 2 3 4 6 1000000 8 106 Reynolds Number, Re

Figure 15- Drag coefficient vs Reynold number for selected steel (ST, 2R = 15, 25, 40 mm) Leidenfrost spheres and tungsten carbide (TC, 2R = 10, 15, 25 mm) Leidenfrost spheres. Each data set is evaluated from the sphere fall trajectory data, as done in the Figure 14, and then plotted vs. the sphere instantaneous Reynolds number.

In Fig 15 we show CD vs Re dependences evaluated from the fall trajectories of selected sizes of steel and tungsten carbide Leidenfrost spheres. For all spheres shown the drag coefficient in the beginning of the free-fall is close to the drag of the no-vapor-layer sphere, starting from the values in subcritical regime, i. e. CD ~ 0.5 – 0.6. We assume that it takes some time from the start of the free-fall for the flow separation around the

Leidenfrost sphere to settle to the rear of the sphere, corresponding to the low-drag-value plateau. It seems that the plateau values dependence on the Reynolds number approaches a universal dependence for the range of sphere sizes and densities investigated.

Each data point is calculated using Eq. (8.10), where the sphere velocity, U(t) and acceleration dU(t)/dt values are measured close to the bottom of the 3 meter or the 2 meter tank. Figure 16a uses a linear vertical axis for the drag coefficient data and Figure 16b shows the same data using a logarithmic vertical axis for the drag coefficient values to more clearly present the spread in the Leidenfrost spheres drag coefficient data.

The CD data for the no-vapor cases closely follows the literature values, with subcritical range of CD ~ 0.45 – 0.55 followed by drag crisis transition at Re ~ 2.5 to 3.5 ×

5 10 and supercritical values, CD ~ 0.15 – 0.20 [14, 15, 82, 87]. There was good agreement between the literature drag coefficient values calculated from the terminal velocity of the sphere falling in a very large (~ 6.5 meter tall) tank [14] and our 2 meter tank data obtained using Eq. (8.10).

As can be expected the results for the Leidenfrost sphere follow the trend outlined in our prior study of an early drag crisis transition starting from Re ~ 4 × 104. However, here by using a larger tank and accounting for the sphere acceleration we measure drag coefficient values, CD ~ 0.04 ± 0.01, which are about half than those measured in the

82 smaller tank. We as well show, that the vapor-layer drag reduction effect persists for supercritical Reynolds numbers, Re > 3.5 × 105. Thus, the vapor layer reduces the drag by an order of magnitude in the subcritical Reynolds number range, and several fold for the supercritical Reynold number range investigated here (Figure 16b). We note that similar range of drag reduction at supercritical Reynolds number was hinted for Leidenfrost spheres falling in heated 95 ⁰C water [82].However in this case the estimation was based on drag coefficients calculated using extrapolated values of the sphere fall velocities whereas in the present study the drag coefficients are calculated using the measured sphere velocity and acceleration. The vapor layer drag reduction compares favourably to any alternative drag reduction approach that induces an early drag crisis, as the use of surface modification with roughness feature [3], addition of flexible polymers [14 - 16], or mass transfer [2].

The magnitude of the drag reduction by the gas layer will depend on the properties of the gas layer, most importantly the gas layer thickness and the ratio between the fluid and gas viscosities. This is demonstrated in Fig. 16 by the data collected for FC-72 heated to the 55 °C, at which temperature FC-72 has a lower viscosity than room temperature FC-

72, resulting in a shift to higher Reynolds numbers of the drag reduction transition. This is consistent with the results for the high viscosity liquids, for which the drag reduction effect is shifted to lower Reynolds numbers [84]. The extended data presented here for the

Leidenfrost spheres could help to further develop analytical and simulation models [45,77 and 84] that can be used to optimize the gas layer drag reduction effect.

Figure 16- Drag coefficient, CD vs Reynold number, Re dependence for the room temperature no vapor spheres, TS = 21 °C and Leidenfrost sphere, TS = 250 °C falling in FC-72. Panel (a) uses a linear vertical axis and (b) uses a logarithmic vertical axis for the CD values. Each data point is calculated using Eq. (8.10) with the sphere velocity and acceleration taken close to the end of the 3 m or 2 m tank fall. Open diamonds (blue) are for the no-vapor steel spheres, open triangles (blue) for the no-vapor tungsten carbide sphere. Solid diamonds (red) are for Leidenfrost steel spheres and solid triangles (red) for Leidenfrost tungsten carbide spheres. Open squares (blue) are for the no-vapor-spheres, TS = 55 °C and solid squares (green) are data for Leidenfrost steel sphere,

TS = 160 °C falling in FC-72 heated to 55 °C. Literature values for no vapor steel spheres falling in an “infinite” water tank measured by White [14] are shown as crosses.

Chapter 9: Conclusions for Part II

In this part of the thesis we have investigated the drag reduction induced by

Leidenfrost vapor layers sustained on the surface of heated metallic spheres falling in the perfluorocarbon liquid FC-72, using liquid tanks which are substantially larger than the ones used in prior studies. Nevertheless, we found that for most of the cases of Leidenfrost spheres investigated, the spheres were still accelerating when they reached the tank bottom.

Analyzing the equation of the sphere motion we derive simplified expressions that correctly predict the sphere fall trajectory in the case of a known drag coefficient.

Subsequently, we use these equations to estimate the sphere drag coefficient accounting for the experimentally measured sphere velocity and acceleration.

A practically important result from this analysis is that for the range of Leidenfrost spheres investigated, during the free-fall the drag coefficient reaches a plateau value, well before the sphere reaches terminal velocity. The methodology for analyzing the free-fall trajectories demonstrated here can be used in any related study of free-falling spheres. The drag coefficient value for the Leidenfrost spheres of about 0.04 are arguably the lowest CD measured for a free-falling sphere and the closest practical realization of a non-slip sphere falling in the super-critical Reynolds number range.

Part III

Stable-streamlined cavities following the impact of non- superhydrophobic spheres on water

Published in the SoftMatter (Vakarelski et al. 2019, see reference [155])

Chapter 10: Introduction to Part III

10.1 Sphere impact on liquid

The impact resulting from a solid sphere onto a liquid leads to formation of an air cavity, which is best exemplified by the classical water-entry problems [101]. The study is relevant to several military, industrial and sports application including the entrance of the ballistic missiles [102] slamming of boats [103] sea plane landing [104 - 105] dip coating procedures [106] and drag on rowing oars [107].The first study on this subject was by

Worthington and Cole [108 - 109] and the study by Worthington [110] exemplified the splashing and ensuing cavity formation with the use of single-spark photography.

Further, studies [111] helped to describe the cavity shapes and dynamics associated with its formation. The first major set of results which assisted in quantifying the evolution of the formation of the cavity arose with the improvement of the high speed cine- photography. These results were showcased by the works [102, 112 -114] who studied the impact of solid spheres onto water to study the dependence of atmospheric pressure, surface characteristics of splash and cavity formation and sphere density.

Prior studies showcased the dependence of sphere wettability and liquid properties on the critical impact velocity (U*) for the sphere. It was found that for hydrophilic sphere

* * impacts 휃푂 < 90 °; U implicitly had no dependence on the static contact angle (θ0). U

휎퐿푉 was linearly dependent on the capillary velocity . The σLV = liquid – vapor surface 휇퐿

* tension and µL = liquid viscosity. In the case of hydrophobic spheres where 휃푂 ≥ 90 ° ; U

휎퐿푉 3 showed dependence proportional to × (휋 − 휃표) [115]. 9휇퐿

The next major study on the dynamics of the transient cavities formed at velocities above the U*, showcased the close comparison between the experimental and theoretical

퐻 findings. The pinch off was located at 푃 which was the deep seal location. The pinch of 퐻 scaled with the pinch of depth location (퐻/푅표) and time for pinch off (τ). The pinch of

퐻 1/2 time followed the scaling 퐻푝/퐻 = ½ , ~퐹푟 . Uo was the sphere impact velocity; the 푅0

2 푅표 relaxation time (τ) τ ≈ 2.06 √ [116]. Here 퐹푟 is the Froude’s number which is the ratio 푔 of the inertia to the gravitational force.

An analytical model predicting the amount of air entrapped by a cavity for the impacts of low density spheres onto water were further developed. This model concluded the volume of air entrapped on impact lowered for spheres of lower densities [117]. The forces experienced by cavity forming and non-cavity forming spheres on impact was studied exhibiting that the coefficient of forces for the cavity forming was higher than the

3 former [118]. Experiments found that for low density Teflon spheres (ρs = 2.2 g/cm ) the teardrop cavity collapsed after the first cavity pinch-off following the impact [119].

Studies to see the influence of two-layer stratified fluid (oil-water) on the cavity formation showed the presence of ripple like patterns on the cavity walls. The pattern was attributed to the shear forces acting between the oil-layer thin film and the surrounding water. The ripples were found to be more eminent shortly after passing the oil film and during the later stages of the cavity formation these were found to be less prominent [120].

Recently, a new phenomenon based on the experimental investigation for the water entry for spheres through highly viscous liquid layer on bulk water pointed the difference in the splash and cavity formation in comparison to a single liquid. The viscous layer on top of

89 the bulk liquid substantially modified the splash dynamics. It was concluded that the viscous layer on the top of the bulk liquid played a major role in comparison to the pressure imbalance inside and outside the cavity in determining the pinch off mechanism for these cavities [121]. These ripple like pattern were not observed and a stable cavity was found to exist in the case of a single-layer system [110].

The high velocity impacts of polypropylene spheres onto viscous Newtonian fluid’s was found to make transient holes which were different from the impacts onto a water surface. The cavity shape was independent of the gravity but largely depended on the density ratio and the Reynolds Number. The depth of the cavity could be predicted by using a model based on the skin friction assuming the isotropic energy dissipation [123].

Numerous, other studies focused on the water entry into glycerine-water mixtures [124], viscoelastic fluids [123,125,126], high-viscosity fluids [127], ethanol [128] and

Perfluorohexane [107]. Studies on the cavitation phenomenon induced by penetrable fabrics in hydrophilic spheres concluded that fabric tunnel present on the fabric floated on the surface of water, which increased drag at the time of entry.

This was in comparison to the unattainable cavities which were observed following the impact of hydrophilic spheres [129]. Further investigations both experimentally and theoretically helped to predict the exact location of the contact line pinned on the sphere following impact onto liquid surface [130]. The hydrodynamic numerical simulation for the water entry of free-falling spheres using coupled Eulerian-Lagrangian (CEL formulation) was performed using Abaqus. The study focused on the water entry for different density spheres impacting the surface of water [131].

10.2 Prior works on stable cavities

The formation of air cavity upon the sphere impact on water occurs above a critical impacts velocity which is a function of the sphere-water contact angle, as well as the sphere surface micro-roughness [101, 115,132-134]. For smooth hydrophilic spheres the threshold velocity is about 7.5 m/s corresponding to a drop height of more than 3 m. For rough or hydrophobic sphere impact the critical height decreases rapidly, with superhydrophobic sphere making an air cavity practically at any impact velocity [115].

Recently a novel phenomenon of complete encapsulation of a solid metallic sphere in a stable streamlined gas cavity fallowing the impact of the sphere on liquid held in a deep tank was exhibited. The phenomenon was observed following the impact of heated

Leidenfrost sphere on perfluorocarbon liquid, PP1 (perfluoro-2-methylpentane, F2

Chemicals, Ltd) or on water heated to 95 °C, as well as by the impact of a superhydrophobic sphere on room temperature water [79, 135]. At sufficiently high impact velocity controlled by the release height of the sphere above the liquid surface, the sphere entrains a cylinder of air during the impact which afterword’s pinches off to finally pacify and form a stable teardrop shaped gas cavity. These sphere-in-cavity formations exhibit very low hydrodynamic drag, less than 1/10th of the drag measured on similar streamlined shape solid projectile. Nevertheless the near-zero drag formation falls at a discreet terminal velocity obeying the Bernoulli equation of potential flow applied on the cavity interface, with the larger cavity falling with higher characteristic velocity [135].

A precondition for the formation of the sphere in cavity in our previous studies was the non-wetting sphere impact, either by the means of the Leidenfrost effect or by the use of superhydrophobic spheres in water. 91

Chapter 11: Experimental Setup and Methods for Part III

11.1 Experimental Setup

The liquid in this experiment was room temperature water at 21 °C, having density 998 kg/m3 and kinematic viscosity of 1 mPa s. The experiments were performed in two tanks:

- 12×12 cm cross section tank with a depth of 2 m. The other was a 40 × 40 cm cross section tank with a depth of 2.5 m. Both the tanks were manufactured in the Central workshop at KAUST. The tanks were equipped with glass windows on both the front and back side. The practical limitation of the drop heights for the experiments was the height of the ceilings in the lab. Most of the experiments were done in the lab with a ceiling height of 4 m, which limited the drop heights to around 2.0 m. The other experiments were performed by moving the 2.0 m deep tank to the lab with ceiling heights of 5 m thereby having an effective drop height of over 5.0 m above the surface of the liquid in the 2.0 m deep water column.

The spheres were purchased from FRITSCH GmbH (FRITSCH Grinding balls) or

Simply Bearing, Ltd. The properties of the spheres are depicted in the table 1. The experiments were performed using the same setup as in Chapter 3, Section 3.1 using the same high speed camera and analyzing technique as described earlier. The frame rate was in these experiments was set at 2000 frames per second and having a shutter speed at

1/20000.

11.2 Sphere Preparation

The spheres used in the experiments were mainly of three different densities

3  Polished zirconium oxide (zirconia, ZO), ρs = 5.6 g/cm of diameters DS = 10, 15, 20

3  Polished steel (ST) spheres, ρs = 7.8 g/cm of diameters DS = 10, 15, 20 mm

3  Polished tungsten carbide (TC) spheres, ρs = 14.8 g/cm of diameters DS = 10 and 15

In these sets of experiments performed the spheres were classified based on their wettability and the water contact angles (θ). The ethanol and water washed as-received spheres were hydrophilic in nature and exhibited contact angles θ ~ 60°. Once the spheres were retrieved back from the tank the contact angle exhibited by them increased to θ ~ 90°.

In these experiments we will refer these spheres as unmodified. The spheres which were cleaned using the plasma cleaner devices (Harrick PDC-002) became fully wetted and these fully wetted spheres exhibited contact angles θ<30°. The spheres could be made hydrophobic using the commercially available coating Ultra Glaco (Soft 99, http://www.soft99.co.jp). The application of this coating increased the water contact angle to around θ ~ 120°. In comparison, the sphere-in-cavity experiments were performed with either the Leidenfrost spheres or the superhydrophobic spheres. In the previous studies the sphere was coated with the Glaco Mirror Coat Zero (Soft 99, http://www.soft99.co.jp) using the procedure explained in Chapter 3, Section 3.1 of this thesis. The presence of hydrophobic silica nano particles within the Glaco Mirror Coat Zero further increased the water contact angle θ > 160°.

Table 1- Physical parameters for zirconium oxide (ZO), steel (ST) or tungsten carbide (TC) spheres with attached cavity formation falling at constant velocity in room temperature water, TW = 21 °C.

All data are collected using unmodified spheres of θ  90° which were released from about 2.0 meter height above the water level in the tank for a tank filled with pure water.

11.3 Surfactant Solutions Preparation

To evaluate the effect of the interface mobility on the drag of the sphere-cavity formations we conducted excrement using surfactant additives which are well known to affect the interface mobility and related momentum transfer through liquid-vapor interface.

Following foam rheology studies we use surfactant formulation of low and high surface modules, characterizing the interface ability to recover following dilation.

In the surfactant-solution experiments we used the anionic surfactant sodium dodecylsulfate (SDS) or a mixture of the anionic surfactant sodium lauryl-dioxyethylene sulfate (SLES), zwitterionic surfactant cocoamidopropyl betaine (CAPB) and myristic acid

(MAc). SDS and MAc were purchase from Aldrich, SLES from AK ChemTech Co., Ltd., and CAPB from Mystic Moments, UK. We also used a commercial shampoo (Johnson’s®

Baby Shampoo) which is soap free, and a toilet soap (Coast® soap) that contains various sodium fatty acids.

(MAc). SDS and MAc were purchase from Aldrich, SLES from AK ChemTech Co., Ltd., and CAPB from Mystic Moments, UK. We also used a commercial shampoo (Johnson’s®

Baby Shampoo) which is soap free, and a toilet soap (Coast® soap) that contains various sodium fatty acids.

The SDS concentration used was of 10 mM slightly above the SDS critical micelle concentration (CMC) of 8 mM. In preparing the surfactant solution mixture we followed

[136] by first preparing a stock solution of 10 wt % (6.6 wt % SLES + 3.4 wt % CAPB) in which 0.4 % MAc was dissolved by heating the mixture to 60 °C at mild stirring, until a clear solution was obtained. The stock solution was diluted 2.5 times with DI water when in the water tank. The Johnson’s® Baby Shampoo was used as ~ 1 wt % in water solution.

The Coast® soap bar was first grounded into small pieces and dissolved as 5 wt % stock solution by heating at about 60 °C while stirring. The stock solution was then dissolved in

25 liters of water in the tank to about 0.04 wt % final concentration.

For simplicity we will refer to the above surfactant solutions at the specified concentrations used in the tank, as SDS solution, shampoo solution, SLES + CAPB + MAc solution and soap solution. The summary of the surfactant solutions composition, surface tension and surface dilation modulus is shown in table 2.

Table 2 - Short names, composition, surface tension and surface-dilation modulus of the surfactant solutions used. The surface tensions were measure with a Kruss tensiometer. Data for the surface dilation modulus ES are taken from Denkov et al. 2005 [136]

Chapter 12: Experimental Results for Part III

12.1. Formation of the cavity

For smooth hydrophilic spheres the threshold velocity is about 7.5 m/s corresponding to a drop height of more than 3 m. For rough or hydrophobic sphere impact the critical height decreases rapidly, with superhydrophobic sphere making an air cavity practically at any impact velocity [112]. For the smooth unmodified sphere used in most of the experiments here (θ  90°) we were able to form a cavity using an impact velocity of about 6.0 m/s, that correspond to a sphere release height of about 2 meter above the water surface.

If the steel spheres were washed with ethanol and water before use (θ  60°), the release height needed to be increased to above 3 m, and for plasma cleaned spheres (θ<

30°) to above 4 m above the water surface, which are in good agreement with prior experimental and theoretical studies [115,117,119,122,132-134]. All results present in this section are for spheres impact on pure water, however we notice that when surfactant is added the critical cavity velocity can be affected due to dynamic surface tension effects

[137].

When the impact velocity of the sphere was below the threshold the sphere will cross the air-water interface without forming a cavity as in the example shown in Fig. 17a, for an ethanol-washed 10 mm steel sphere impacting from a 2 meter release height.

However when an unmodified 10 mm steel sphere was released from the same height the sphere forms a cavity upon water impact which eventually results in the sphere with attached cavity formation. The entire free-fall of the stable streamlined attached cavity is

97 shown in in Fig. 17b-k. For the figures the camera is fixed at a different positions below the water surface, to track the cavity formation vs depth. The sphere depth is indicated on each of the Fig. 17 snapshots.

The snapshots given in Figs. 17b-d capture the initial cavity forming upon the unmodified sphere impact on the water surface. A seal-off cavity that is typical for such impact velocities is observed (Figs. 17c, d) [132,134]. Figures 17e-g track the gradual shedding of bubble behind the cavity that is assisted by the acoustic ripples propagating along the cavity surface [79]. Further on the acoustic ripples gradually pacify with only smaller bubble been shed, as seen in Figs. 17g, h. Finally, in Fig. 17k the ripples are fully pacified and the steady state sphere with attached cavity fall can be observed through the depths 115 to 150 mm below the water surface.

Most of the experiments in our study were conducted using unmodified spheres.

However, as illustrated in Fig. 18 for the case of 10 mm steel spheres nearly identical attached cavities were produced for a variety of surface treatments, i.e. for plasma cleaned spheres (θ< 30°, Fig. 18a), ethanol washed spheres (θ  60°, Fig. 18b), unmodified sphere

(θ 90°, Fig. 18c), or hydrophobic spheres (θ  120°, Fig. 18d), when they are released from the appropriate height above the water surface. The cavity-formation release height threshold was about 4 m for the plasma-cleaned sphere, 3.5 m for the ethanol cleaned sphere, and about 2 m for the unmodified sphere.

For hydrophobic spheres the threshold height was about 0.5 meters, which is close to the release height used in our prior study with superhydrophobic spheres [132]. A

Leidenfrost or superhydrophobic sphere will form an initial cavity upon impact even when dropped from much lower heights. However, formation of an initial cavity is not a

98 sufficient condition. As detailed in our prior studies[79,135] even for non-wetted impact conditions one needs a larger threshold release height as the volume of the teardrop-cavity formed after the first cavity pinch-off should be large enough to secure approximate neutral buoyancy of the sphere-in-cavity formation.

3 We note that for lower-density Teflon spheres (ρs = 2.2 g/cm ) used in a recent study

[119] the teardrop cavity is found to collapse following the first cavity pinch-off after the impact. The present study demonstrates a stable cavity formation for much larger sphere/fluid density ratios of ρS/ρ between 5.0 and 15. There may therefore be a critical minimum density ratio needed to achieve stable cavity. The comparison in Fig. 18 demonstrates that the sphere with attached cavity formation is nearly identical for the entire range of smooth spheres hydrophobicity investigated herein. The contact-lines in all cases appear to be pinned at the same location, slightly above the equator ( 110° from the sphere bottom).

The depth at which the stabilization of sphere with cavity formation was observed to increases with the sphere density and size, as well as with the sphere release height. For the 10 mm zirconia sphere it was about 1.0 meter depth and for and 20 mm steel sphere about 2.0 meter depth. Figure 19 gives typical examples for the depth trajectory vs time and decent velocity vs time progression. For most of the spheres investigated here the sphere with attached cavity formation were still stable, when reaching the bottom of the tank. However, we notice that if a deep enough tank is used the cavity will eventually collapse due to the buildup of hydrostatic pressure [79].

A detailed analysis of the physics of the stable-streamlined cavity was done in our initial investigation for Leidenfrost-sphere impact on fluorocarbon liquid [79]. There it was

99 suggested that a pre-condition for the pacification of the acoustic ripples along the side of the cavity is the lack of physical contact between the sphere and the cavity. However present experiments demonstrated that in the case of water and a smooth sphere of a water contact angle between < 30° and 120°, the ripples do eventually completely pacify even when a physical contact line between the cavity and the sphere is present. By extending the phenomenon of stable-streamline cavity to wetting impacts we demonstrate that this is a far more general phenomenon than anticipated before.

Simplified, if one releases a sphere from a sufficient height above the water surface, at a certain depth a stable-streamlined cavity attached to the falling sphere can be produced.

We note that although there have been numerous prior investigation [101,112,115-

117,122,132-133] of water-impact by steel spheres similar to the ones used in the present study, most of them were limited to the observation of the initial sphere impact until the cavity seal or shortly thereafter. Using deeper water tanks than in prior investigations allows us to track the sphere-with-cavity formation until the constant fall velocity steady- streamlined cavity state is observed.

The Figure 20 showcased the pinning of the contact line for all different spheres used at almost the same location which was at θ~110 °. The pinning was assumed to be assisted by the corner-flow separation upstream of the contact line. The use of Zirconium oxide sphere helped to visualize the contact line clearly due to its appearance. The figure also shows the difference in the sphere-in-cavity (Figure 20a) with the sphere-with-cavity

(Figures 20 b-d)

100

Figure 17 - Snapshots of the various stages in the formation of a stable cavity from the time of impact onto a liquid surface. The number below the frame indicates the depth of the sphere below the surface of water.

101

Figure 18- Snapshots of the sphere attached-streamlined cavity for a 10 mm steel sphere for: (a)

Plasma cleaned sphere with water contact angle, θ < 30°, released from h0 = 4.2 meters above the water surface; (b) Ethanol cleaned sphere θ  60°, h0 = 3.2 m; (c) Unmodified sphere θ  90°, h0 =

2.0 m; (d) Ultra Glaco coated sphere, θ  120°, h0 = 0.5 m

102

Figure 19 - Examples of the depth trajectory vs time (a) and decent velocity vs time (b) for the case of a 10 mm unmodified steel sphere with attached cavity impacting from about 2 m height above the water surface (red squares and 10 mm unmodified tungsten carbide sphere impacting from about the same height of about 2 meter above the water surface (blue circles). Arrows mark the establishment of the steady streamlined cavity regime, with no more bubble shedding

103

a b c d

Figure 20 - High-speed camera snapshots of sphere-in-cavity formation for a 10 mm Leidenfrost steel sphere, TS = 250 °C free-falling in PP1 (a) and for spheres with attached streamlined air-cavity formation free-falling in water using: (b) 20 mm zirconia sphere; (c) 15 mm steel sphere; (d) 10 mm tungsten carbide sphere. All spheres in (b-d) have water contact angle of about 90° and were released from about 2.0 m above the water surface. The cavity – sphere contact line is marked with white arrow in each image.

104

12.2 Cavity Shape, Fall Velocity & Drag

Figure 21 shows snapshots which compares the free-fall of a sphere with attached cavity formed by the impact of unmodified DS = 15 mm steel sphere (Fig. 21a) and that formed by DS = 15 mm superhydrophobic steel sphere (Fig. 21b). For reference we also show the sphere-in-cavity shape for DS = 15 mm Leidenfrost steel sphere, TS = 400 °C in

95 °C water (Fig. 21c). As seen in Fig. 21 the sphere-with-attached cavity and sphere-in- cavity formations have similar cavity shape and close free-fall velocities. This suggests that similar relations for the cavity volume, velocity and drag might hold for both formations. Therefore herein, we follow the same characterization process for sphere-with- attached cavity as done earlier for the enclosed sphere-in-cavity formations.

As in the case of a sphere-in-cavity formation, we observe that the sphere with attached cavities have self-similar shapes which depend only on the ratio between the sphere and liquid densities. The aspect ratio of the sphere with cavity length to maximum diameter was found to be L/D ~ 4.0 for the zirconia spheres; L/D ~4.8 for the steel spheres, and L/D ~ 6.3 for tungsten carbide spheres (see Fig. 21a for L and D definitions). These ratios are slightly higher than for the sphere-in-cavity formations, reflecting the fact that the cavity is now attached above the sphere equator, giving a slimmer streamlined profile

(Fig. 21).

The total volume of the cavity with sphere formation, VSC (including the sphere and the cavity volume), was estimated from video snapshots using a profile tracking MATLAB image processing code. It can be shown that the formation shape can be also fitted using a

105 three-piece algebraic curve that comprises of a spherical front section, an elliptical mid- section and a parabolic tail [135].

As in the sphere-in-cavity case we find that the ratio of the volume of the sphere- with-attached cavity formation, to the volume of the sphere VSC/VS was always only slightly less than the ratio of the sphere density to fluid density, ρS/ρ. This is demonstrated by the

Fig. 22 data showing the VSC/VS vs ρS/ρ dependence. This means that the sphere-with- attached cavity formation is nearly neutrally buoyant. Furthermore, the volume of the sphere-cavity formation was found to be empirically related to L and D dimensions as VSC

 0.47LD2 for all combinations of spheres density and sizes studied, as shown in Fig. 23.

The drag coefficient of the sphere-with-cavity, CD, was estimated from the formation terminal velocity U and sphere with cavity formation volume VSC, using the following relation from the balance of the buoyancy and drag force:

8푔(푚 −휌푉 ) 퐶 = 푠 푆퐶 , (12.1) 퐷 휋휌퐷2푈2

where ms is the mass of the sphere. Because the effective density of the sphere- with-cavity formation is always close to the liquid density, the estimation of the drag coefficient is most sensitive to the accuracy with which VSC is determined.

In the figure 24, it can be clearly seen how the profile of the cavity and the volume of the cavity including the sphere can be calculated using MATLAB. The profile in red was the profile which was derived from MATLAB and it showcased the drag coefficient

CD = 0.028. The blue curve in contrast represented the neutral buoyancy curve which would be exhibited when the drag coefficient had a value; CD = 0.000 or for the curve represented

106 by the green profile having CD = 0.060. It was found as in case of the sphere-in-cavity the ratio of the volume of the sphere with attached cavity formation to the volume of the sphere

VSC/VS was always slightly less than the ratio of the sphere density to the fluid density ρS/ρ.

The dependence of these was showcased in figure 22. The results presented made sure that that the cavity formation was nearly neutrally buoyant. The volume of the sphere-in-cavity

2 was found to be empirically related as VSC  0.47LD for all combinations of spheres density and sizes as shown in figure 22.

Figure 26 shows the dependence of the sphere-with-cavity drag coefficient, CD, on the Reynolds number, Re = ρDU/µ, using data for zirconia, steel and tungsten carbide spheres. For comparison we also show data for drag of enclosed sphere-in-cavity formation using superhydrophobic steel spheres, and the drag of similar streamlined-shape solid projectiles measured in our recent study [135]. The average values of the drag coefficients,

CD = 0.025 ± 0.010 of the sphere with attached cavity formation is similar to the value found for sphere-in-cavity formation (e.g. superhydrophobic or Leidenfrost spheres in cavity). This is close to an order of magnitude lower than the drag on the similar shape streamlined solid projectile, CD  0.2, or sphere without cavity, CD  0.45 [87]. Drag measured on Leidenfrost spheres (without cavities) in our prior studies can also be comparatively low reaching CD = 0.04 ±0.01 values, but is manifested as an early drag crisis phenomenon, triggered above a certain Reynolds number [81-82,84,138] In contrast the drag on sphere with cavity seems to be equally pronounced in the entire range of

Reynolds numbers investigated.

107

Leidenfrost steel sphere, TS = 400 °C in 95 °C water

108

20 10 mm 15 mm 15 20 mm TC

S ZO V ST

/ 10 ZO SC

V TC

ST 0 0 5 10 15 20 ρ /ρ S

Figure 22 - Dependence of the ratio of the total sphere-with-cavity over the sphere volume, VSC/VS on the sphere to liquid density ratio, ρS/ρ. Data shown are for 10 mm (green triangles), 15 mm (red squares) and 20 mm (blue circles) unmodified zirconia (ZO), steel (ST) or tungsten carbide (TC) spheres. The dotted line corresponds to neutral buoyancy of the sphere-with-cavity formation.

109

30 ]

3 20

[cm SC

V 10

0 0 20 40 60 LD2 [cm3]

Figure 23 - Cavity volume, as a function of (퐿 퐷2), cavity diameter D and length L, for the same sphere sizes as in Figure 22. The dotted line is best linear fit to the data, that gives the relation VSC = 0.47LD2.

110

Figure 24 - An example of a high-speed video-camera snapshot used to determine the volume of the sphere-with-cavity formation, VSC. b) Photograph of a 3D-printed solid projectile used to estimate the drag on a similar streamlined-shape solid body

111

The low drag on the sphere-with-cavity is due to the streamlined shape of the formation and importantly to the exchange of the no-slip boundary condition on the solid- water interface with the free-slip or stress-free boundary condition along the cavity gas – liquid interface. Here we show similar drag value for the sphere-in-cavity formation in which case the sphere is completely isolated from the liquid and for the sphere-with-cavity formation in which case there is direct contact between the front part of the sphere and the liquid. However this is perhaps, not surprising, having in mind that the air layer between the front of the sphere and the cavity before the cavity separation from the sphere is very thin and provides only partial effective slip [29,37,77] The direct contact between the sphere and the liquid could explain the residual drag on the cavity formation.

An important property of the sphere-in-cavity formation, found in our prior study, was that although the formation drag is near-zero, the cavity fall-velocity has a discrete value [135]. The velocity value is determined by matching the hydrostatic pressure gradient in the pool with the dynamic pressure from the Bernoulli equation along the free surface, while the pressure inside the gas-cavity is constant:

1 2 푃퐶 = 푃0 − 2 휌 푢 − 휌𝑔푧 (12.2)

Here PC is the pressure inside the cavity, z is the vertical coordinate measured from the tail of the cavity, u the fluid velocity on the cavity surface and P0 the reference pressure in the fluid at u = 0 and z = 0. By numerically solving the Laplace equation for the velocity potential,  at the cavity surface, ∇2휙 = 0, it was confirmed that the resulting velocity according to potential flow theory u, is in agreement with equation (12.2) for all cavities

112 shapes and sphere-in-cavity fall velocity investigated [135]. Based on these finding, and following the semi-empirical relation between the free-fall velocity of the sphere-in-cavity formation, U, the sphere dimeter DS and the ratio between the sphere and the liquid densities ρS/ρ, was deduced:

2 푈 ≈ 퐶 (𝑔 퐷푆) [(휌푆/휌) − 1] (12.3)

With the empirical coefficient, C = 3.3, found from the best fit of the experimental data for Leidenfrost sphere-in-cavity formations falling both in 95 °C water and in PP1. In

2 Fig. 25 we plot the same dependence of U /gDS vs [(휌푆/휌) − 1] using the present velocities data for sphere-with-cavity formations. A similar linear dependence is obtained, with a slightly different coefficient of C = 4.7. The variation in the coefficient reflects the earlier noted difference in the cavity shape, where the attached cavity is more slender than that the cavity that fully encapsulates the sphere.

The close values of drag coefficient and fall velocity between the sphere-in-cavity formation and the sphere-with-attached cavity formation shown here demonstrate that both are near-zero drag formations for which the fall velocity complies with the potential flow theory and Bernoulli equation along the cavity interface. We also note that the physical model used to describe fall velocity of the streamlined cavity is reminiscent of the Davies-

Taylor study of the rise velocity of a large spherical cap gas bubble in liquid [139].

113

) S

gD 40

(

2 U 20

0 0 4 8 12 16

ρS/ρ - 1

Figure 25- Variation of the sphere-in-cavity terminal velocity 푈 with sphere diameter 퐷푆 and

sphere to liquid density ratio, ρS/ρ -1. The same sphere symbols are used as in Fig. 22. The dotted line is a linear best fit to the data, giving a coefficient C = 4.7 in Eq. (12.3).

114

Solid sphere

D C

Solid projectile 0.1

Sphere with cavity Drag Coefficient, Drag

0.01 10000 100000 Reynolds number, Re

Figure 26 - Variation of the drag coefficient CD with the Reynolds number Re for sphere-with-

cavity for unmodified zirconia spheres of diameter DS = 10, 15 and 20 mm, L/R ~4.0 (solid red

diamonds), unmodified steel spheres of diameter DS = 10, 15 and 20 mm, L/R ~ 4.7 (solid red

triangles) and unmodified tungsten carbide spheres, DS = 10 and 15, L/R ~ 6.3 (solid red circles). Data for solid spheres without a cavity (empty green squares) are taken from Jetly et al.[73] Data

for superhydrophobic steel sphere-in-cavity formation, DS = 15, 20 and 25 mm, L/R ~4.5 (empty

red triangles) and solid streamlined projectiles of similar shape, DP = 25 mm, LP/RP = 4.5 (solid blue squares), are taken from Vakarelski et al. [135].

115

12.3 Effect of Surfactant

The near-zero drag on the sphere with streamlined cavity is a result of the streamlined shape and the free-slip boundary condition along the cavity interface, as confirmed by comparing the drag on sphere-with-cavity and sphere-in-cavity formations with the drag on similar shape solid projectiles (Fig. 26). It is well known that surface active additives can effectively immobilize the air-liquid interface. For small bubbles, even trace amounts of surfactant can lead to interface immobilization. However for larger gas cavities and high interface shear rates in our experiments the interface sensitivity to surfactant additives will depend on the surfactant types and concentration.

Studies by Rushton and Davies [160] showcased the limits for the reduction of the frictional drag force and showed how for a sphere with a single fluid of radius a (i.e. ε →

0) had a drag correction compared to Stokes Law. This equation was the same as the

Hadamard-Rybczinski results and thus for an encapsulating phase 1 fluid is extremely thin

(i.e. ε → 1) the compound fluid would experience the same drag as a solid particle of radius a, irrespective of the core being a fluid. This is because the encapsulating fluid shell becomes suppressed and thus the so called phase 1 of the fluid becomes immobile and the equivalent effect of the adsorption of surfactants on the bubble rigidifying on the gas-liquid interface renders the interface to be immobile. This rigidifying of the gas-liquid interface prevents any momentum transfer and results in higher drag, which is equivalent to the drag predicted by the Stokes law [159]

116

In our choice of test surfactants we follow foam rheology investigations [136, 140-

143] that have clarified two types of surfactant additives with respect to their effect on the foam-bubble mobility. The first are typically synthetic surfactants (for example SDS) which show low-surface-modulus and fast relaxation of the surface tension following surface dilation. The second type, exemplified by the sodium and potassium salts of fatty acids (also known as soaps), have high-surface-modulus and slow surface tension relaxation. Foam shearing experiments indicate that for the low-surface-modulus surfactants the bubble interface behave as mobile, and in contrast for the high-surface- modulus surfactant the foam bubbles interface behaves as immobile [140-143]

As examples of the low-modulus surfactants here we use 10 mM SDS water solution and a commercial shampoo (Johnson’s® Baby Shampoo) water solution which is soap free. As example of high-modulus surfactants we use a mixture of SLES + CAPB surfactants + Mystic acid (MAc) and also a high concentration solution of a regular toiled soap (Coast© soap) that contains various sodium fatty acids. The advantage of the SLES

+ CAPB + MAc mixture is that following the preparation protocol the solution stays clear and transparent, whereas the toilet soap solution is fast becoming turbid [136]. Details on all surfactant solutions composition, concentration and preparation are given in the experimental section and summarized in Table 2 in Section 11.2.

The comparison of the free-fall of 10 mm unmodified steel sphere with attached cavity in pure water, SDS solution, shampoo solution, SLES + CAPB+ MAc solution and soap solution is shown in Figure 27. It can be seen in the snapshots in the cases of SDS solution (Fig. 27b) and shampoo solution (Fig. 27c) we observed stable attached cavities with nearly identical shape and fall velocity to the case of cavities formed in pure water

117

(Fig. 27a). This result indicates that the low-surface-modulus surfactants are not affecting the mobility of the cavity interface and the cavity behaves as having a free-slip interface.

It also shows that the strength of the air-liquid surface tension does not seems to affect substantially the cavity formation. This is in agreement with previous results for sphere-in- cavity formation using low surface tension PP1 liquid [79].

The observation that even very high concentrations of surfactant above the CMC that is readily lowering the surface tension of water do not affect the mobility and the shape of the cavity formation might seem surprising. However this result is in good agreement with the foam rheology investigations that implicate free-slip on bubble for foams formed using low-surface-modulus surfactants [140-142]. At the same time it is in sharp contrast with the behavior of micron sizes bubbles in water, in which case even trace amounts of contamination are found to immobilize the interface as evaluated in rising bubble terminal velocity experiment [144-147] and in bubble interaction in atomic force microscopy

(AFM) experiments [148-149]. This behavior clearly demonstrates that the mobility of the air – water interface strongly depends on the flow regime and related tangential stress applied on the bubbles or cavity interface. For small bubble and low shear stress close to the Stokes regime even trace amount of surfactant that are not changing the surface tension will fully immobilize the interface and the bubbles behave like rigid particles [144-150].

For larger deformable bubbles their rise velocity is in agreement with the mobile-surface rise velocity predicted by the Moore’s theory [151] of stress-free interface bubbles, unless a higher concentration of surfactant is added [152-154]. Finally for the larger foam bubbles and higher shear rates, as well as in our falling cavity experiments, even for concentration of synthetic surfactants above the CMC, the interface remains free-slip.

118

In contrast to the low modulus surfactants, in the case of the SLES + CAPB + MAc solution it was not possible to form a stable cavity. Shortly after the sphere impacts the water surface the cavity was either completely detached from the sphere, or as in the Fig.

27d, was constantly shedding. In the case of soap solution (Fig. 27e), we observed an intermediate case of a “cut-tail” cavity formation. The fall velocity of the sphere with cut- tail cavity is close to that of the sphere with streamlined cavities .However for the case of cut-tail the cavity volume, VSC is much smaller, i.e. the formation is less buoyant and therefore the drag is higher, estimate to be CD ~ 0.1, for the example given in the Fig. 27e, and even higher, CD ~ 0.3 for the example given in Fig. 27d. On the other hand, when the cavity is completely removed the drag will increase to the limiting case of a sphere without cavity CD ~ 0.45. Thus we show that viscous stress on the cavity interface eventually leads to partial, in the case of soap solution, or complete as in the case of the SLES + CAPB +

MAc solution cavity destruction.

The surfactant-mixture and soap-solution results are in agreement with the foam shearing experiments confirming, the trend that high-elastic-modulus surfactants are effective in immobilizing the interface. We note however that the SLES + CAPB + MAc concentration needed in our experiment to destabilize the cavity was about 8 times higher

(~ 4 wt. %) than the concentration used in the foam rheology experiments [140-142]. Using a lower ~ 0.5 wt. % concentration, which was shown to immobilize bubbles in foam shear experiments we were still able to produce stable cavity formations. This reflects the fact that the cavity formation is a more dynamic process than foam shearing, which follows the general trend of higher shear stress resulting in higher interface mobility. A related

119 experimental detail was that when a sphere was impacting a foam covered water interface, instead of a foam-free interface as in the rest of the experiments, a stable cavity formation was occasionally observed even in the case of the higher concentration SLES + CAPB +

MAc solution. The effect of foam on top of water on the initial cavity formation has been recently investigated [137].

We finally note that the surfactant mixtures solution experiments showed that the immobilization of the interface did not result in a stable-streamlined cavity of higher drag close to that of the similar shape solid projectile (CD ~ 0.2) as one might assume. Instead the cavity was partially formed as in the cut-tail formation (Fig. 27e) or completely destroyed (Fig. 27d). This indicates that the teardrop streamlined cavity is a stable formation only as long as its interface is stress-free. One way to estimate if the cavity mobility will be affected by the presence of surfactant additive is to compare the magnitude of the surface dilation modules, Es. [140-142]

120

Figure 27 - Snapshots comparing 10 mm unmodified steel spheres with cavity formation free- falling in (A) pure water with different surfactants: (B) SDS solution; (C) Shampoo solution; (D) SLES + CAPB + MAc solution; (E) Soap solution

121

Chapter 13: Conclusions for Part III

A practical realization of an object moving in a fluid which approached the near zero drag was the free-falling sphere inside a liquid and forming the stable-streamlined gas cavity. The zero drag has been well known and predicted by the D’Alemberts paradox

[135]. In the previous studies performed the preconditions for the formation of a stable cavity were known to occur for the non-wetting sphere impacts.

In this chapter we showcased that the similar stable-streamlined cavity could be formed for the wide range of water contact angles from the fully wetted hydrophilic to the hydrophobic spheres. These stable streamlined cavities were formed as long as the drop height was high enough and the tank was deep enough to allow the cavity to stabilize. The only difference between the wetting and non-wetting sphere impacts was that for the wetted case the spheres streamlined cavity was attached right above the sphere equator rather than enclosing the sphere completely. In spite of this difference both the spheres-in-cavity and sphere-with-cavity exhibited low drags and same dependence of the free-fall velocity on the sphere size and density.

The last part of this chapter was concentrating on the effect of stable-streamlined cavity formation for the water surfactants solution. It was found that even for high concentrations of low-surface-modulus the synthetic surfactants mixtures did not affect the shape of the cavity nor the fall velocity which indicated that the interface remained free- slip. When high concentration of high-surface-modulus surfactant mixtures, e.g. soap solutions were used the cavity was partial or completely destroyed indicating interface immobilization. These results were in good agreement with foam rheology experiments.

122

However, a higher surfactant concentration was needed to immobilize the interface than in the case of foam shearing experiments, confirming the trend of higher liquid-air interface tangential mobility at higher interface shear rates.

123

Chapter 14: General Conclusions

Drag reduction by a lubricating air layer introduced between a solid body and the surrounding liquid is a promising approach to save energy and improve efficiency for a range of practical applications, from microfluidics to marine vessels. The pursued research is to advance this approach by conducting free-falling sphere experiments that build on the high-speed imaging laboratory expertise.

Part I of the thesis focused on the capabilities of drag reduction due to superhydrophobic coatings. The study with superhydrophobic spheres has produced the most systematic and convincing experimental demonstration for drag reduction induced by an air plastron on bluff body at high Reynolds numbers. Experiments have showcased that the drag on the bigger superhydrophobic spheres was dramatically reduced by up to 50–80% by triggering an early drag crisis at high Re ∼ 1.2 × 105, which was substantially lower than the corresponding drag crisis occurring for smooth solid spheres. This drag reduction was a result of the very thin air layers (∼1–2 μm) which encapsulated the superhydrophobic spheres.

Part II of the thesis focused on the capabilities of drag reduction due to Vapor Layer

Cases (Leidenfrost layers). The investigation of Leidenfrost drag reduction extended previous results and demonstrated the lowest drag on a free-falling sphere ever measured.

The drag coefficient value for the Leidenfrost spheres was about 0.04 which was the lowest

CD measured for a free-falling sphere and the closest practical realization of a non-slip sphere falling in the super-critical Reynolds number range.

124

Part III of the thesis complimented the recently discovered phenomenon of near- zero-drag streamlined cavity, showing it to be a far more general phenomenon than initially anticipated. The contact line for the attached cavity was clearly located at  110° from the sphere bottom. The similar stable-streamlined-cavity formation could be observed following the water impact of a smooth sphere over a wide range of water contact angles, from fully wetted hydrophilic spheres to hydrophobic spheres provided the impact is from a high enough height and into a deep enough tank.

Nevertheless, the sphere with an attached cavity showed the same low drag and same dependence of the free-fall velocity on the sphere size and density as the sphere in the enclosed cavity, investigated before. Finally, we study the effect of added surfactants to the water on these streamlined cavities. It was found that even high concentration low- surface-modulus synthetic surfactant mixtures did not affect the cavity shape or fall- velocity, indicating that the interface remains free-slip. When high concentration high- surface-modulus surfactant mixtures, e.g. soap solutions, were used the cavity was partially or completely destroyed, indicating interface immobilization. These results were in good agreement with foam rheology experiments. However, a higher surfactant concentration was needed to immobilize the interface than in the case of foam-shearing experiments, confirming the trend of higher liquid–air interface tangential mobility at higher interface shear rates.

125

Publications by the Defendant

1. Self-determined shapes and velocities of giant near-zero drag gas cavities Vakarelski, I. U., Klaseboer, E., Jetly, A., Mansoor, M. M., Aguirre-Pablo, A. A., Chan, D. Y. C. & Thoroddsen, S. T. Science Advances, 3: e1701558 (2017).

2. Drag crisis moderation by thin air layers sustained on super hydrophobic spheres falling in water Jetly, A., Vakarelski, I. U. & Thoroddsen, S. T. Soft Matter, 14, 1608-1613 (2018).

3. Experiments on the breakup of drop-impact crowns by Marangoni holes Aljedaani, A. B., Wang, C., Jetly, A. & Thoroddsen, S. T. Journal of Fluid Mechanics, 844: 162-186 (2018).

4. Giant drag reduction on Leidenfrost spheres evaluated from extended free-fall trajectory Jetly, A., Vakarelski, I. U., Yang, Z. & Thoroddsen, S. T. Experimental Thermal and Fluid Science 102, 181–188 (2019).

5. Stable-streamlined cavities following the impact of non-superhydrophobic spheres on water Vakarelski, I. U., Jetly, A., & Thoroddsen, S. T. Soft Matter, 2019, 15, 6278-6287 (2019).

126

Bibliography

1. Schiller, L., & Naumann, A. (1933). Fundamental Calculations In Gravitational

Processing. Zeitschrift Des Vereines Deutscher Ingenieure, 77, 318-320.

2. Vakarelski, I. U., Chan, D. Y., & Thoroddsen, S. T. (2015). Drag Moderation By The

Melting Of An Ice Surface In Contact With Water. Physical Review Letters, 115(4),

044501.

3. Choi, J., Jeon, W. P., & Choi, H. (2006). Mechanism Of Drag Reduction By Dimples

On A Sphere. Physics Of Fluids, 18(4), 041702.

4. Isaev, S. A., Leontiev, A. I., Kudryavtsev, N. A. E., & Pyshnyi, I. A. (2003). The Effect

Of Rearrangement Of The Vortex Structure On Heat Transfer Under Conditions Of

Increasing Depth Of A Spherical Dimple On The Wall Of A Narrow Channel. High

Temperature, 41(2), 229-232.

5. Alekseev, V. V., Gachechiladze, I. A., Kiknadze, G. I., & Oleinikov, V. G. (1998).

Tornado-Like Energy Transfer On Three-Dimensional Concavities Of Reliefs-

Structure Of Self-Organizing Flow, Their Visualisation, And Surface Streamlining

Mechanisms. In Transactions Of The 2nd Russian Nat. Conf. Of Heat Transfer, Heat

Transfer Intensification Radiation And Complex Heat Transfer (Vol. 6, Pp. 33-42).

6. Bhushan, B. (2011). Biomimetics Inspired Surfaces For Drag Reduction And

Oleophobicity/Philicity. Beilstein Journal Of Nanotechnology, 2(1), 66-84.

7. Wilkinson, S. P., & Lazos, B. S. (1988). Direct Drag And Hot-Wire Measurements On

Thin-Element Riblet Arrays. In Turbulence Management And Relaminarisation (Pp.

121-131). Springer, Berlin, Heidelberg

127

8. Choi, K. S. (1989). Near-Wall Structure Of A Turbulent Boundary Layer With

Riblets. Journal Of Fluid Mechanics, 208, 417-458.

9. Karniadakis, G. E., & Choi, K. S. (2003). Mechanisms On Transverse Motions In

Turbulent Wall Flows. Annual Review Of Fluid Mechanics, 35(1), 45-62.

10. Webb, R. L., & Kim, N. Y. (2005). Enhanced Heat Transfer. Taylor And Francis, Ny.

11. Weigand, B., Semmler, K., & Von Wolfersdorf, J. (2001). Heat Transfer Technology

For Internal Passages Of Air‐Cooled Blades For Heavy‐Duty Gas Turbines. Annals Of

The New York Academy Of Sciences, 934(1), 179-193.

12. Kays, W. M., & Whitelaw, J. H. (1967). Convective Heat And Mass Transfer.

13. Toms, B. A. (1977). On The Early Experiments On Drag Reduction By Polymers. The

Physics Of Fluids, 20(10), S3-S5.

14. A.White, Drag On Spheres In Diluted High Polymer Solutions, Nature 216 (1967) 994-

995.

15. N. Lyotard, W.L. Shew, L. Bocquet, J.-F. Pinton, Polymer And Surface Roughness

Effects On The Drag Crisis For Falling Spheres, Eur. Phys. J. B 60 (2017) 469-476.

16. K. Watanabe, H. Kui, I. Motosu, Drag Of A Sphere In Dilute Polymer Solutions In

High Reynolds Number Range, Rheol. Acta 37 (1998) 328-335.

17. Alyanak, E., Grandhi, R., & Penmetsa, R. (2006). Optimum Design Of A

Supercavitating Torpedo Considering Overall Size, Shape, And Structural

Configuration. International Journal Of Solids And Structures, 43(3-4), 642-657.

18. Yang, D., Xiong, Y. L., & Guo, X. F. (2017). Drag Reduction Of A Rapid Vehicle In

Supercavitating Flow. International Journal Of Naval Architecture And Ocean

Engineering, 9(1), 35-44.

128

19. Ceccio, S. L. (2010). Friction Drag Reduction Of External Flows With Bubble And

Gas Injection. Annual Review Of Fluid Mechanics, 42, 183-203

20. Verschoof, R. A., Van Der Veen, R. C., Sun, C., & Lohse, D. (2016). Bubble Drag

Reduction Requires Large Bubbles. Physical Review Letters, 117(10), 104502

21. Lay, K. A., Yakushiji, R., Mäkiharju, S., Perlin, M., & Ceccio, S. L. (2010). Partial

Cavity Drag Reduction At High Reynolds Numbers. Journal Of Ship Research, 54(2),

109-119.

22. Mizokami, S.H., Kawakita, C.H., Kodan, Y.O., Takano S.H., Higasa, S.E., Shigenaga,

R.Y. Experimental Study Of Air Lubrication Method And Verification Of Effects On

Actual Hull By Means Of Sea Trial. Mitsubishi Heavy Industries Technical Review.

2010 Sep;47(3):41-7.

23. Qin, S., Chu, N., Yao, Y., Liu, J., Huang, B., & Wu, D. (2017). Stream-Wise

Distribution Of Skin-Friction Drag Reduction On A Flat Plate With Bubble

Injection. Physics Of Fluids, 29(3), 037103.

24. Sanders, W. C., Winkel, E. S., Dowling, D. R., Perlin, M., & Ceccio, S. L. (2006).

Bubble Friction Drag Reduction In A High-Reynolds-Number Flat-Plate Turbulent

Boundary Layer. Journal Of Fluid Mechanics, 552, 353-380.

25. Elbing, B. R., Winkel, E. S., Lay, K. A., Ceccio, S. L., Dowling, D. R., & Perlin, M.

(2008). Bubble-Induced Skin-Friction Drag Reduction And The Abrupt Transition To

Air-Layer Drag Reduction. Journal Of Fluid Mechanics, 612, 201-236.

26. Mccormick, M. E., & Bhattacharyya, R. (1973). Drag Reduction Of A Submersible

Hull By Electrolysis. Naval Engineers Journal, 85(2), 11-16.

129

27. Rothstein, J. P. (2010). Slip On Superhydrophobic Surfaces. Annual Review Of Fluid

Mechanics, 42, 89-109.

28. Mchale, G., Newton, M. I., & Shirtcliffe, N. J. (2010). Immersed Superhydrophobic

Surfaces: Gas Exchange, Slip And Drag Reduction Properties. Soft Matter, 6(4), 714-

719.

29. Mchale, G., Shirtcliffe, N. J., Evans, C. R., & Newton, M. I. (2009). Terminal Velocity

And Drag Reduction Measurements On Superhydrophobic Spheres. Applied Physics

Letters, 94(6), 064104.

30. Dong, H., Cheng, M., Zhang, Y., Wei, H., & Shi, F. (2013). Extraordinary Drag-

Reducing Effect Of A Superhydrophobic Coating On A Macroscopic Model Ship At

High Speed. Journal Of Materials Chemistry A, 1(19), 5886-5891.

31. Adamson, A. W., & Gast, A. P. (1967). Physical Chemistry Of Surfaces (Vol. 15). New

York: Interscience

32. Israelachvili, J. N. (2015). Intermolecular And Surface Forces. Academic Press.

33. Bhushan, B. (1999). Principles And Applications Of Tribology. John Wiley & Sons.

34. Bhushan, B. (2002). Introduction To Tribology, John Wiley& Sons. New York.

35. Bhushan, B. (2005). Nanotribology And Nanomechanics. Wear, 259(7-12), 1507-

1531.

36. Lauga, E., & Stone, H. A. (2003). Effective Slip In Pressure-Driven Stokes

Flow. Journal Of Fluid Mechanics, 489, 55-77.

37. Busse, A., Sandham, N. D., Mchale, G., & Newton, M. I. (2013). Change In Drag,

Apparent Slip And Optimum Air Layer Thickness For Laminar Flow Over An

Idealised Superhydrophobic Surface. Journal Of Fluid Mechanics, 727, 488-508.

130

38. Ou, J., Perot, B., & Rothstein, J. P. (2004). Laminar Drag Reduction In Microchannels

Using Ultrahydrophobic Surfaces. Physics Of Fluids, 16(12), 4635-4643.

39. Ou, J., & Rothstein, J. P. (2005). Direct Velocity Measurements Of The Flow Past

Drag-Reducing Ultrahydrophobic Surfaces. Physics Of Fluids, 17(10), 103606.

40. Choi, C. H., & Kim, C. J. (2006). Large Slip Of Aqueous Liquid Flow Over A

Nanoengineered Superhydrophobic Surface. Physical Review Letters, 96(6), 066001.

41. Choi, C. H., Ulmanella, U., Kim, J., Ho, C. M., & Kim, C. J. (2006). Effective Slip

And Friction Reduction In Nanograted Superhydrophobic Microchannels. Physics Of

Fluids, 18(8), 087105.

42. Watanabe, K., Udagawa, Y., & Udagawa, H. (1999). Drag Reduction Of Newtonian

Fluid In A Circular Pipe With A Highly Water-Repellent Wall. Journal Of Fluid

Mechanics, 381, 225-238.

43. Gogte, S., Vorobieff, P., Truesdell, R., Mammoli, A., Van Swol, F., Shah, P., &

Brinker, C. J. (2005). Effective Slip On Textured Superhydrophobic Surfaces. Physics

Of Fluids, 17(5), 051701.

44. Daniello, R. J., & Rothstein, J. P. (2008, November). Turbulent Drag Reduction Using

Superhydrophobic Surfaces. In Aps Division Of Fluid Dynamics Meeting Abstracts.

45. Mchale, G., Flynn, M. R., & Newton, M. I. (2011). Plastron Induced Drag Reduction

And Increased Slip On A Superhydrophobic Sphere. Soft Matter, 7(21), 10100-10107.

46. Gruncell, B. R., Sandham, N. D., & Mchale, G. (2013). Simulations Of Laminar Flow

Past A Superhydrophobic Sphere With Drag Reduction And Separation Delay. Physics

Of Fluids, 25(4), 043601.

131

47. Brennan, J. C., Geraldi, N. R., Morris, R. H., Fairhurst, D. J., Mchale, G., & Newton,

M. I. (2015). Flexible Conformable Hydrophobized Surfaces For Turbulent Flow Drag

Reduction. Scientific Reports, 5, 10267.

48. Geraldi, N. R., Dodd, L. E., Xu, B. B., Wells, G. G., Wood, D., Newton, M. I., &

Mchale, G. (2017). Drag Reduction Properties Of Superhydrophobic Mesh

Pipes. Surface Topography: Metrology And Properties, 5(3), 034001.

49. Zhang, S., Ouyang, X., Li, J., Gao, S., Han, S., Liu, L., & Wei, H. (2014). Underwater

Drag-Reducing Effect Of Superhydrophobic Submarine Model. Langmuir, 31(1), 587-

593.

50. Lee, C., & Kim, C. J. (2011). Underwater Restoration And Retention Of Gases On

Superhydrophobic Surfaces For Drag Reduction. Physical Review Letters, 106(1),

014502.

51. Vakarelski, I. U., Chan, D. Y., Marston, J. O., & Thoroddsen, S. T. (2013). Dynamic

Air Layer On Textured Superhydrophobic Surfaces. Langmuir, 29(35), 11074-11081.

52. Srinivasan, S., Kleingartner, J. A., Gilbert, J. B., Cohen, R. E., Milne, A. J., &

Mckinley, G. H. (2015). Sustainable Drag Reduction In Turbulent Taylor-Couette

Flows By Depositing Sprayable Superhydrophobic Surfaces. Physical Review

Letters, 114(1), 014501.

53. Jones, P. R., Hao, X., Cruz-Chu, E. R., Rykaczewski, K., Nandy, K., Schutzius, T. M.,

& Espinosa, H. D. (2015). Sustaining Dry Surfaces Under Water. Scientific Reports, 5,

12311.

54. Li, Y., Quéré, D., Lv, C., & Zheng, Q. (2017). Monostable Superrepellent

Materials. Proceedings Of The National Academy Of Sciences, 114(13), 3387-3392.

132

55. Domingues, E. M., Arunachalam, S., & Mishra, H. (2017). Doubly Reentrant Cavities

Prevent Catastrophic Wetting Transitions On Intrinsically Wetting Surfaces. Acs

Applied Materials & Interfaces, 9(25), 21532-21538.

56. Martínez-Gómez, A., López, S., García, T., De Francisco, R., Tiemblo, P., & García,

N. (2017). Long-Term Underwater Hydrophobicity: Exploring Topographic And

Chemical Requirements. Acs Omega, 2(12), 8928-8939.

57. Hof, B. (2017). Fluid Dynamics: Water Flows Out Of Touch. Nature, 541(7636), 161.

58. Ishikura, K., Yoshimi, J., Matsuda, A., Yamashita, K., Katsu, M., Otsuka, S., &

Tkakagi, H. (2014). Influence Of A Newly-Developed Triathlon Suit On The Passive

Drag And Body Position. Procedia Engineering, 72, 338-343.

59. Sheng, X., & Zhang, J. (2011). Air Layer On Superhydrophobic Surface

Underwater. Colloids And Surfaces A: Physicochemical And Engineering

Aspects, 377(1-3), 374-378.

60. Jung, Y. C., & Bhushan, B. (2008). Wetting Behaviour During Evaporation And

Condensation Of Water Microdroplets On Superhydrophobic Patterned

Surfaces. Journal Of Microscopy, 229(1), 127-140.

61. Lee, C., & Kim, C. J. C. (2009). Maximizing The Giant Liquid Slip On

Superhydrophobic Microstructures By Nanostructuring Their

Sidewalls. Langmuir, 25(21), 12812-12818.

62. Lee, D. G., & Kim, H. Y. (2009). The Role Of Superhydrophobicity In The Adhesion

Of A Floating Cylinder. Journal Of Fluid Mechanics, 624, 23-32.

133

63. Shibuichi, S., Onda, T., Satoh, N., & Tsujii, K. (1996). Super Water-Repellent Surfaces

Resulting From Fractal Structure. The Journal Of Physical Chemistry, 100(50), 19512-

19517.

64. Coulson, S. R., Woodward, I., Badyal, J. P. S., Brewer, S. A., & Willis, C. (2000).

Super-Repellent Composite Fluoropolymer Surfaces. The Journal Of Physical

Chemistry B, 104(37), 8836-8840

65. Miwa, M., Nakajima, A., Fujishima, A., Hashimoto, K., & Watanabe, T. (2000).

Effects Of The Surface Roughness On Sliding Angles Of Water Droplets On

Superhydrophobic Surfaces. Langmuir, 16(13), 5754-5760.

66. Öner, D., & Mccarthy, T. J. (2000). Ultrahydrophobic Surfaces. Effects Of Topography

Length Scales On Wettability. Langmuir, 16(20), 7777-7782

67. Feng, L., Li, S., Li, Y., Li, H., Zhang, L., Zhai, J., & Zhu, D. (2002). Super‐

Hydrophobic Surfaces: From Natural To Artificial. Advanced Materials, 14(24), 1857-

1860.

68. Erbil, H. Y., Demirel, A. L., Avcı, Y., & Mert, O. (2003). Transformation Of A Simple

Plastic Into A Superhydrophobic Surface. Science, 299(5611), 1377-1380.

69. Lau, K. K., Bico, J., Teo, K. B., Chhowalla, M., Amaratunga, G. A., Milne, W. I., ... &

Gleason, K. K. (2003). Superhydrophobic Carbon Nanotube Forests. Nano

Letters, 3(12), 1701-1705.

70. Burton, Z., & Bhushan, B. (2005). Hydrophobicity, Adhesion, And Friction Properties

Of Nanopatterned Polymers And Scale Dependence For Micro-And

Nanoelectromechanical Systems. Nano Letters, 5(8), 1607-1613

134

71. Li, W., & Amirfazli, A. (2005). A Thermodynamic Approach For Determining The

Contact Angle Hysteresis For Superhydrophobic Surfaces. Journal Of Colloid And

Interface Science, 292(1), 195-201.

72. Jung, Y. C., & Bhushan, B. (2006). Contact Angle, Adhesion And Friction Properties

Of Micro-And Nanopatterned Polymers For Superhydrophobicity. Nanotechnologyc

, 17(19), 4970

73. Golovin, K. B., Gose, J. W., Perlin, M., Ceccio, S. L., & Tuteja, A. (2016). Bioinspired

Surfaces For Turbulent Drag Reduction. Philosophical Transactions Of The Royal

Society A: Mathematical, Physical And Engineering Sciences, 374(2073), 20160189.

74. Jung, Y. C., & Bhushan, B. (2009). Biomimetic Structures For Fluid Drag Reduction

In Laminar And Turbulent Flows. Journal Of Physics: Condensed Matter, 22(3),

035104.

75. Zhou, Y., Li, M., Su, B., & Lu, Q. (2009). Superhydrophobic Surface Created By The

Silver Mirror Reaction And Its Drag-Reduction Effect On Water. Journal Of Materials

Chemistry, 19(20), 3301-3306.

76. Vakarelski, I. U., Patankar, N. A., Marston, J. O., Chan, D. Y., & Thoroddsen, S. T.

(2012). Stabilization Of Leidenfrost Vapour Layer By Textured Superhydrophobic

Surfaces. Nature, 489(7415), 274.

77. Jetly, A., Vakarelski, I. U., & Thoroddsen, S. T. (2018). Drag Crisis Moderation By

Thin Air Layers Sustained On Superhydrophobic Spheres Falling In Water. Soft

Matter, 14(9), 1608-1613.

135

78. Moosmann, M., Schimmel, T., Barthlott, W., & Mail, M. (2017). Air–Water Interface

Of Submerged Superhydrophobic Surfaces Imaged By Atomic Force

Microscopy. Beilstein Journal Of Nanotechnology, 8(1), 1671-1679.

79. Mansoor, M. M., Vakarelski, I. U., Marston, J. O., Truscott, T. T., & Thoroddsen, S.

T. (2017). Stable–Streamlined And Helical Cavities Following The Impact Of

Leidenfrost Spheres. Journal Of Fluid Mechanics, 823, 716-754.

80. Yin, Z., Wang, Z., Liang, B., & Zhang, L. (2017). Initial Velocity Effect On

Acceleration Fall Of A Spherical Particle Through Still Fluid. Mathematical Problems

In Engineering, 2017.

81. Vakarelski, I. U., Marston, J. O., Chan, D. Y., & Thoroddsen, S. T. (2011). Drag

Reduction By Leidenfrost Vapor Layers. Physical Review Letters, 106(21), 214501.

82. Vakarelski, I. U., Chan, D. Y., & Thoroddsen, S. T. (2014). Leidenfrost Vapour Layer

Moderation Of The Drag Crisis And Trajectories Of Superhydrophobic And

Hydrophilic Spheres Falling In Water. Soft Matter, 10(31), 5662-5668.

83. Quéré, D. (2013). Leidenfrost Dynamics. Annual Review Of Fluid Mechanics, 45,

197-215.

84. Vakarelski, I. U., Berry, J. D., Chan, D. Y., & Thoroddsen, S. T. (2016). Leidenfrost

Vapor Layers Reduce Drag Without The Crisis In High Viscosity Liquids. Physical

Review Letters, 117(11), 114503.

85. Berry, J. D., Vakarelski, I. U., Chan, D. Y., & Thoroddsen, S. T. (2017). Navier Slip

Model Of Drag Reduction By Leidenfrost Vapor Layers. Physics Of Fluids, 29(10),

107104.

136

86. Leidenfrost, J. G. (1756). De Aquae Communis Nonnullis Qualitatibus Tractatus.

Ovenius.

87. Achenbach, E. (1972). Experiments On The Flow Past Spheres At Very High Reynolds

Numbers. Journal Of Fluid Mechanics, 54(3), 565-575

88. Tomotika, S. (1936). Lamina Boundary Layer On The Surface Of A Sphere In A

Uniform Stream. Hm Stationery Office.

89. Sakamoto, H., & Haniu, H. (1990). A Study On Vortex Shedding From Spheres In A

Uniform Flow. Journal Of Fluids Engineering, 112(4), 386-392

90. Mittal, R. (1999). Planar Symmetry In The Unsteady Wake Of A Sphere. Aiaa

Journal, 37(3), 388-390

91. Constantinescu, G. S., & Squires, K. D. (2003). Les And Des Investigations Of

Turbulent Flow Over A Sphere At Re= 10,000. Flow, Turbulence And

Combustion, 70(1-4), 267-298.

92. Yun, G., Kim, D., & Choi, H. (2006). Vortical Structures Behind A Sphere At

Subcritical Reynolds Numbers. Physics Of Fluids, 18(1), 015102.

93. Hoffman, J. (2006). Adaptive Simulation Of The Subcritical Flow Past A

Sphere. Journal Of Fluid Mechanics, 568, 77-88.

94. Jones, D. A., & Clarke, D. B. (2008). Simulation Of Flow Past A Sphere Using The

Fluent Code (No. Dsto-Tr-2232). Defense Science And Technology Organization

Victoria (Australia) Maritime Platforms Div.

95. Lee, C., & Kim, C. J. (2012). Wetting And Active Dewetting Processes Of

Hierarchically Constructed Superhydrophobic Surfaces Fully Immersed In

Water. Journal Of Microelectromechanical Systems, 21(3), 712-720.

137

96. Wang, B., Wang, J., & Chen, D. (2014). Continual Automatic Generation Of Gases On

Hydrophobic Transverse Microgrooved Surface. Chemistry Letters, 43(5), 646-648

97. Bradfield, W. S., Barkdoll, R. O., & Byrne, J. T. (1962). Some Effects Of Boiling On

Hydrodynamic Drag. International Journal Of Heat And Mass Transfer, 5(7), 615-622.

98. Saranadhi, D., Chen, D., Kleingartner, J. A., Srinivasan, S., Cohen, R. E., & Mckinley,

G. H. (2016). Sustained Drag Reduction In A Turbulent Flow Using A Low-

Temperature Leidenfrost Surface. Science Advances, 2(10), E1600686.

99. Hamilton, W. S., & Lindell, J. E. (1971). Fluid Force Analysis And Accelerating

Sphere Tests. Journal Of The Hydraulics Division, 97(6), 805-817.

100. Pantaleone, J., & Messer, J. (2011). The Added Mass Of A Spherical

Projectile. American Journal Of Physics, 79(12), 1202-1210.

101. Truscott, T. T., Epps, B. P., & Belden, J. (2014). Water Entry Of

Projectiles. Annual Review Of Fluid Mechanics, 46, 355-378.

102. May, A. (1952). Vertical Entry Of Missiles Into Water. Journal Of Applied

Physics, 23(12), 1362-1372.

103. Zhao, R., & Faltinsen, O. (1993). Water Entry Of Two-Dimensional

Bodies. Journal Of Fluid Mechanics, 246, 593-612.

104. Von Karman, T. (1929). The Impact On Seaplane Floats During Landing.

105. Wagner, H. (1932). Phenomena Associated With Impacts And Sliding On Liquid

Surfaces. Z. Angew. Math. Mech, 12(4), 193-215.

106. Burley, R. (1992). Air Entrainment And The Limits Of Coatability. Surface

Coatings International, 75(5), 192-202.

138

107. Affeld, K., Schichl, K., & Ziemann, A. (1993). Assessment Of Rowing

Efficiency. International Journal Of Sports Medicine, 14(S 1), S39-S41.

108. Worthington, A. M., & Cole, R. S. (1897). V. Impact With A Liquid Surface,

Studied By The Aid Of Instantaneous Photography. Philosophical Transactions Of The

Royal Society Of London. Series A, Containing Papers Of A Mathematical Or Physical

Character, (189), 137-148.

109. Worthington, A. M., & Cole, R. S. (1899). Impact With A Liquid Surface, Studied

By The Aid Of Instantaneous Photography. Paper Ii. Proceedings Of The Royal Society

Of London Series I, 65, 153-154.

110. Worthington, A. M. (1908). A Study Of Splashes. Longmans, Green, And

Company

111. Ramsauer, C., & Dobke, G. (1927). Die Bewegungserscheinungen Des Wassers

Beim Durchgang Schnell Bewegter Kugeln. Annalen Der Physik, 389(22), 697-720.

112. Gilbarg, D., & Anderson, R. A. (1948). Influence Of Atmospheric Pressure On The

Phenomena Accompanying The Entry Of Spheres Into Water. Journal Of Applied

Physics, 19(2), 127-139

113. Richardson, E. G. (1948). The Impact Of A Solid On A Liquid

Surface. Proceedings Of The Physical Society, 61(4), 352.

114. May, A. (1951). Effect Of Surface Condition Of A Sphere On Its Water‐Entry

Cavity. Journal Of Applied Physics, 22(10), 1219-1222.

115. Duez, C., Ybert, C., Clanet, C., & Bocquet, L. (2007). Making A Splash With Water

Repellency. Nature Physics, 3(3), 180

139

116. Duclaux, V., Caillé, F., Duez, C., Ybert, C., Bocquet, L., & Clanet, C. (2007).

Dynamics Of Transient Cavities. Journal Of Fluid Mechanics, 591, 1-19

117. Aristoff, J. M., Truscott, T. T., Techet, A. H., & Bush, J. W. (2010). The Water

Entry Of Decelerating Spheres. Physics Of Fluids, 22(3), 032102

118. Truscott, T. T., Epps, B. P., & Techet, A. H. (2012). Unsteady Forces On Spheres

During Free-Surface Water Entry. Journal Of Fluid Mechanics, 704, 173-210.

119. Di Mundo, R., Bottiglione, F., Pascazio, G., & Carbone, G. (2018). Water Entry

And Fall Of Hydrophobic And Superhydrophobic Teflon Spheres. Journal Of Physics:

Condensed Matter, 30(44), 445001

120. Tan, B. C. W., & Thomas, P. J. (2018). Influence Of An Upper Layer Liquid On

The Phenomena And Cavity Formation Associated With The Entry Of Solid Spheres

Into A Stratified Two-Layer System Of Immiscible Liquids. Physics Of Fluids, 30(6),

064104

121. Sun, T., Wang, H., Zong, Z., Zhang, G., Wang, A., & Xu, C. (2019). Splash

Formation And Cavity Dynamics Of Sphere Entry Through A Viscous Liquid Resting

On The Water. Aip Advances, 9(7), 075211.

122. Mansoor, M. M., Marston, J. O., Vakarelski, I. U., & Thoroddsen, S. T. (2014).

Water Entry Without Surface Seal: Extended Cavity Formation. Journal Of Fluid

Mechanics, 743, 295-326.

123. Le Goff, A., Quéré, D., & Clanet, C. (2013). Viscous Cavities. Physics Of

Fluids, 25(4), 043101.

124. Bell, G. E. (1924). Lxxvi. On The Impact Of A Solid Sphere With A Fluid Surface

And The Influence Of Surface Tension, Surface Layers, And Viscosity On The

140

Phenomenon. The London, Edinburgh, And Dublin Philosophical Magazine And

Journal Of Science, 48(287), 753-764.

125. Podgorski, T., & Belmonte, A. (2002). Surface Folds During The Penetration Of A

Viscoelastic Fluid By A Sphere. Journal Of Fluid Mechanics, 460, 337-348.

126. Akers, B., & Belmonte, A. (2006). Impact Dynamics Of A Solid Sphere Falling

Into A Viscoelastic Micellar Fluid. Journal Of Non-Newtonian Fluid

Mechanics, 135(2-3), 97-108

127. Cheny, J. M., & Walters, K. (1996). Extravagant Viscoelastic Effects In The

Worthington Jet Experiment. Journal Of Non-Newtonian Fluid Mechanics, 67, 125-

135.

128. Grumstrup, T., Keller, J. B., & Belmonte, A. (2007). Cavity Ripples Observed

During The Impact Of Solid Objects Into Liquids. Physical Review Letters, 99(11),

114502

129. Watson, D. A., Stephen, J. L., & Dickerson, A. K. (2018). Jet Amplification And

Cavity Formation Induced By Penetrable Fabrics In Hydrophilic Sphere Entry. Physics

Of Fluids, 30(8), 082109.

130. Ding, H., Chen, B. Q., Liu, H. R., Zhang, C. Y., Gao, P., & Lu, X. Y. (2015). On

The Contact-Line Pinning In Cavity Formation During Solid–Liquid Impact. Journal

Of Fluid Mechanics, 783, 504-525.

131. Ahmadzadeh, M., Saranjam, B., Fard, A. H., & Binesh, A. R. (2014). Numerical

Simulation Of Sphere Water Entry Problem Using Eulerian–Lagrangian

Method. Applied Mathematical Modelling, 38(5-6), 1673-1684.

141

132. Aristoff, J. M., & Bush, J. W. (2009). Water Entry Of Small Hydrophobic

Spheres. Journal Of Fluid Mechanics, 619, 45-78.

133. Zhao, M. H., Chen, X. P., & Wang, Q. (2014). Wetting Failure Of Hydrophilic

Surfaces Promoted By Surface Roughness. Scientific Reports, 4, 5376.

134. Speirs, N. B., Mansoor, M. M., Belden, J., & Truscott, T. T. (2019). Water Entry

Of Spheres With Various Contact Angles. Journal Of Fluid Mechanics, 862.

135. Vakarelski, I. U., Klaseboer, E., Jetly, A., Mansoor, M. M., Aguirre-Pablo, A. A.,

Chan, D. Y., & Thoroddsen, S. T. (2017). Self-Determined Shapes And Velocities Of

Giant Near-Zero Drag Gas Cavities. Science Advances, 3(9), E1701558.

136. Golemanov, K., Denkov, N. D., Tcholakova, S., Vethamuthu, M., & Lips, A.

(2008). Surfactant Mixtures For Control Of Bubble Surface Mobility In Foam

Studies. Langmuir, 24(18), 9956-9961.

137. Speirs, N. B., Mansoor, M. M., Hurd, R. C., Sharker, S. I., Robinson, W. G.,

Williams, B. J., & Truscott, T. T. (2018). Entry Of A Sphere Into A Water-Surfactant

Mixture And The Effect Of A Bubble Layer. Physical Review Fluids, 3(10), 104004.

138. Jetly, A., Vakarelski, I. U., Yang, Z., & Thoroddsen, S. T. (2019). Giant Drag

Reduction On Leidenfrost Spheres Evaluated From Extended Free-Fall

Trajectories. Experimental Thermal And Fluid Science, 102, 181-188.

139. Davies, R. M., & Taylor, G. I. (1950). The Mechanics Of Large Bubbles Rising

Through Extended Liquids And Through Liquids In Tubes. Proceedings Of The Royal

Society Of London. Series A. Mathematical And Physical Sciences, 200(1062), 375-

390.

142

140. Denkov, N. D., Subramanian, V., Gurovich, D., & Lips, A. (2005). Wall Slip And

Viscous Dissipation In Sheared Foams: Effect Of Surface Mobility. Colloids And

Surfaces A: Physicochemical And Engineering Aspects, 263(1-3), 129-145.

141. Denkov, N. D., Tcholakova, S., Golemanov, K., Ananthapadmanabhan, K. P., &

Lips, A. (2008). Viscous Friction In Foams And Concentrated Emulsions Under Steady

Shear. Physical Review Letters, 100(13), 138301.

142. Denkov, N. D., Tcholakova, S., Golemanov, K., Ananthpadmanabhan, K. P., &

Lips, A. (2009). The Role Of Surfactant Type And Bubble Surface Mobility In Foam

Rheology. Soft Matter, 5(18), 3389-3408.

143. Marinova, K. G., Naydenova, K. T., Basheva, E. S., Bauer, F., Tropsch, J., &

Franke, J. (2017). New Surfactant Mixtures For Fine Foams With Slowed

Drainage. Colloids And Surfaces A: Physicochemical And Engineering Aspects, 523,

54-61.

144. Kelsall, G. H., Tang, S., Smith, A. L., & Yurdakul, S. (1996). Measurement Of Rise

And Electrophoretic Velocities Of Gas Bubbles. Journal Of The Chemical Society,

Faraday Transactions, 92(20), 3879-3885.

145. Parkinson, L., Sedev, R., Fornasiero, D., & Ralston, J. (2008). The Terminal Rise

Velocity Of 10–100 Μm Diameter Bubbles In Water. Journal Of Colloid And Interface

Science, 322(1), 168-172.

146. Vakarelski, I. U., Manica, R., Li, E. Q., Basheva, E. S., Chan, D. Y., & Thoroddsen,

S. T. (2018). Coalescence Dynamics Of Mobile And Immobile Fluid

Interfaces. Langmuir, 34(5), 2096-2108.

143

147. Pawliszak, P., Ulaganathan, V., Bradshaw-Hajek, B. H., Manica, R., Beattie, D. A.,

& Krasowska, M. (2019). Mobile Or Immobile? Rise Velocity Of Air Bubbles In High

Purity Water. The Journal Of Physical Chemistry C.

148. Manor, O., Vakarelski, I. U., Tang, X., O’shea, S. J., Stevens, G. W., Grieser, F.,

... & Chan, D. Y. (2008). Hydrodynamic Boundary Conditions And Dynamic Forces

Between Bubbles And Surfaces. Physical Review Letters, 101(2), 024501.

149. Vakarelski, I. U., Manica, R., Tang, X., O’shea, S. J., Stevens, G. W., Grieser, F.,

... & Chan, D. Y. (2010). Dynamic Interactions Between Microbubbles In

Water. Proceedings Of The National Academy Of Sciences, 107(25), 11177-11182.

150. Liu, B., Manica, R., Liu, Q., Klaseboer, E., Xu, Z., & Xie, G. (2019). Coalescence

Of Bubbles With Mobile Interfaces In Water. Physical Review Letters, 122(19),

194501.

151. Moore, D. W. (1965). The Velocity Of Rise Of Distorted Gas Bubbles In A Liquid

Of Small Viscosity. Journal Of Fluid Mechanics, 23(4), 749-766.

152. Okazaki, S. (1964). The Velocity Of Ascending Air Bubbles In Aqueous Solutions

Of A Surface Active Substance And The Life Of The Bubble On The Same

Solution. Bulletin Of The Chemical Society Of Japan, 37(2), 144-150.

153. Malysa, K., Krasowska, M., & Krzan, M. (2005). Influence Of Surface Active

Substances On Bubble Motion And Collision With Various Interfaces. Advances In

Colloid And Interface Science, 114, 205-225.

154. Manica, R., Klaseboer, E., & Chan, D. Y. (2016). The Hydrodynamics Of Bubble

Rise And Impact With Solid Surfaces. Advances In Colloid And Interface

Science, 235, 214-232.

144

155. Vakarelski, I. U., Jetly, A., & Thoroddsen, S. T. (2019). Stable-Streamlined

Cavities Following The Impact Of Non-Superhydrophobic Spheres On Water. Soft

Matter, 15(31), 6278-6287.

156. Mchale, G. S. N. J., Shirtcliffe, N. J., & Newton, M. I. (2004). Super-Hydrophobic

And Super-Wetting Surfaces: Analytical Potential?. Analyst, 129(4), 284-287.

157. Shirtcliffe, N. J., Mchale, G., Newton, M. I., Perry, C. C., & Pyatt, F. B. (2006).

Plastron Properties Of A Superhydrophobic Surface. Applied Physics Letters, 89(10),

104106.

158. Shirtcliffe, N. J., Mchale, G., Newton, M. I., & Zhang, Y. (2009).

Superhydrophobic Copper Tubes With Possible Flow Enhancement And Drag

Reduction. Acs Applied Materials & Interfaces, 1(6), 1316-1323.

159. Stokes, G. G. (1851). Trans. Camb. Phil. Soc.

160. Rushton, E., & Davies, G. A. (1973). The Slow Unsteady Settling Of Two Fluid

Spheres Along Their Line Of Centres. Applied Scientific Research, 28(1), 37-61.

161. Shirtcliffe, N. J., Mchale, G., Atherton, S., & Newton, M. I. (2010). An Introduction

To Superhydrophobicity. Advances In Colloid And Interface Science, 161(1-2), 124-

138

162. Mchale, G. (2007). Cassie And Wenzel: Were They Really So Wrong?. Langmuir,

23(15), 8200-8205.

145