Hydrostatic Shapes
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Capillary Surface Interfaces, Volume 46, Number 7
fea-finn.qxp 7/1/99 9:56 AM Page 770 Capillary Surface Interfaces Robert Finn nyone who has seen or felt a raindrop, under physical conditions, and failure of unique- or who has written with a pen, observed ness under conditions for which solutions exist. a spiderweb, dined by candlelight, or in- The predicted behavior is in some cases in strik- teracted in any of myriad other ways ing variance with predictions that come from lin- with the surrounding world, has en- earizations and formal expansions, sufficiently so Acountered capillarity phenomena. Most such oc- that it led to initial doubts as to the physical va- currences are so familiar as to escape special no- lidity of the theory. In part for that reason, exper- tice; others, such as the rise of liquid in a narrow iments were devised to determine what actually oc- tube, have dramatic impact and became scientific curs. Some of the experiments required challenges. Recorded observations of liquid rise in microgravity conditions and were conducted on thin tubes can be traced at least to medieval times; NASA Space Shuttle flights and in the Russian Mir the phenomenon initially defied explanation and Space Station. In what follows we outline the his- came to be described by the Latin word capillus, tory of the problems and describe some of the meaning hair. current theory and relevant experimental results. It became clearly understood during recent cen- The original attempts to explain liquid rise in a turies that many phenomena share a unifying fea- capillary tube were based on the notion that the ture of being something that happens whenever portion of the tube above the liquid was exerting two materials are situated adjacent to each other a pull on the liquid surface. -
The Shape of a Liquid Surface in a Uniformly Rotating Cylinder in the Presence of Surface Tension
Acta Mech DOI 10.1007/s00707-013-0813-6 Vlado A. Lubarda The shape of a liquid surface in a uniformly rotating cylinder in the presence of surface tension Received: 25 September 2012 / Revised: 24 December 2012 © Springer-Verlag Wien 2013 Abstract The free-surface shape of a liquid in a uniformly rotating cylinder in the presence of surface tension is determined before and after the onset of dewetting at the bottom of the cylinder. Two scenarios of liquid withdrawal from the bottom are considered, with and without deposition of thin film behind the liquid. The governing non-linear differential equations for the axisymmetric liquid shapes are solved numerically by an iterative procedure similar to that used to determine the equilibrium shape of a liquid drop deposited on a solid substrate. The numerical results presented are for cylinders with comparable radii to the capillary length of liquid in the gravitational or reduced gravitational fields. The capillary effects are particularly pronounced for hydrophobic surfaces, which oppose the rotation-induced lifting of the liquid and intensify dewetting at the bottom surface of the cylinder. The free-surface shape is then analyzed under zero gravity conditions. A closed-form solution is obtained in the rotation range before the onset of dewetting, while an iterative scheme is applied to determine the liquid shape after the onset of dewetting. A variety of shapes, corresponding to different contact angles and speeds of rotation, are calculated and discussed. 1 Introduction The mechanics of rotating fluids is an important part of the analysis of numerous scientific and engineering problems. -
Artificial Viscosity Model to Mitigate Numerical Artefacts at Fluid Interfaces with Surface Tension
Computers and Fluids 143 (2017) 59–72 Contents lists available at ScienceDirect Computers and Fluids journal homepage: www.elsevier.com/locate/compfluid Artificial viscosity model to mitigate numerical artefacts at fluid interfaces with surface tension ∗ Fabian Denner a, , Fabien Evrard a, Ricardo Serfaty b, Berend G.M. van Wachem a a Thermofluids Division, Department of Mechanical Engineering, Imperial College London, Exhibition Road, London, SW7 2AZ, United Kingdom b PetroBras, CENPES, Cidade Universitária, Avenida 1, Quadra 7, Sala 2118, Ilha do Fundão, Rio de Janeiro, Brazil a r t i c l e i n f o a b s t r a c t Article history: The numerical onset of parasitic and spurious artefacts in the vicinity of fluid interfaces with surface ten- Received 21 April 2016 sion is an important and well-recognised problem with respect to the accuracy and numerical stability of Revised 21 September 2016 interfacial flow simulations. Issues of particular interest are spurious capillary waves, which are spatially Accepted 14 November 2016 underresolved by the computational mesh yet impose very restrictive time-step requirements, as well as Available online 15 November 2016 parasitic currents, typically the result of a numerically unbalanced curvature evaluation. We present an Keywords: artificial viscosity model to mitigate numerical artefacts at surface-tension-dominated interfaces without Capillary waves adversely affecting the accuracy of the physical solution. The proposed methodology computes an addi- Parasitic currents tional interfacial shear stress term, including an interface viscosity, based on the local flow data and fluid Surface tension properties that reduces the impact of numerical artefacts and dissipates underresolved small scale inter- Curvature face movements. -
Astronomy 201 Review 2 Answers What Is Hydrostatic Equilibrium? How Does Hydrostatic Equilibrium Maintain the Su
Astronomy 201 Review 2 Answers What is hydrostatic equilibrium? How does hydrostatic equilibrium maintain the Sun©s stable size? Hydrostatic equilibrium, also known as gravitational equilibrium, describes a balance between gravity and pressure. Gravity works to contract while pressure works to expand. Hydrostatic equilibrium is the state where the force of gravity pulling inward is balanced by pressure pushing outward. In the core of the Sun, hydrogen is being fused into helium via nuclear fusion. This creates a large amount of energy flowing from the core which effectively creates an outward-pushing pressure. Gravity, on the other hand, is working to contract the Sun towards its center. The outward pressure of hot gas is balanced by the inward force of gravity, and not just in the core, but at every point within the Sun. What is the Sun composed of? Explain how the Sun formed from a cloud of gas. Why wasn©t the contracting cloud of gas in hydrostatic equilibrium until fusion began? The Sun is primarily composed of hydrogen (70%) and helium (28%) with the remaining mass in the form of heavier elements (2%). The Sun was formed from a collapsing cloud of interstellar gas. Gravity contracted the cloud of gas and in doing so the interior temperature of the cloud increased because the contraction converted gravitational potential energy into thermal energy (contraction leads to heating). The cloud of gas was not in hydrostatic equilibrium because although the contraction produced heat, it did not produce enough heat (pressure) to counter the gravitational collapse and the cloud continued to collapse. -
STARS in HYDROSTATIC EQUILIBRIUM Gravitational Energy
STARS IN HYDROSTATIC EQUILIBRIUM Gravitational energy and hydrostatic equilibrium We shall consider stars in a hydrostatic equilibrium, but not necessarily in a thermal equilibrium. Let us define some terms: U = kinetic, or in general internal energy density [ erg cm −3], (eql.1a) U u ≡ erg g −1 , (eql.1b) ρ R M 2 Eth ≡ U4πr dr = u dMr = thermal energy of a star, [erg], (eql.1c) Z Z 0 0 M GM dM Ω= − r r = gravitational energy of a star, [erg], (eql.1d) Z r 0 Etot = Eth +Ω = total energy of a star , [erg] . (eql.1e) We shall use the equation of hydrostatic equilibrium dP GM = − r ρ, (eql.2) dr r and the relation between the mass and radius dM r =4πr2ρ, (eql.3) dr to find a relations between thermal and gravitational energy of a star. As we shall be changing variables many times we shall adopt a convention of using ”c” as a symbol of a stellar center and the lower limit of an integral, and ”s” as a symbol of a stellar surface and the upper limit of an integral. We shall be transforming an integral formula (eql.1d) so, as to relate it to (eql.1c) : s s s GM dM GM GM ρ Ω= − r r = − r 4πr2ρdr = − r 4πr3dr = (eql.4) Z r Z r Z r2 c c c s s s dP s 4πr3dr = 4πr3dP =4πr3P − 12πr2P dr = Z dr Z c Z c c c s −3 P 4πr2dr =Ω. Z c Our final result: gravitational energy of a star in a hydrostatic equilibrium is equal to three times the integral of pressure within the star over its entire volume. -
Energy Dissipation Due to Viscosity During Deformation of a Capillary Surface Subject to Contact Angle Hysteresis
Physica B 435 (2014) 28–30 Contents lists available at ScienceDirect Physica B journal homepage: www.elsevier.com/locate/physb Energy dissipation due to viscosity during deformation of a capillary surface subject to contact angle hysteresis Bhagya Athukorallage n, Ram Iyer Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA article info abstract Available online 25 October 2013 A capillary surface is the boundary between two immiscible fluids. When the two fluids are in contact Keywords: with a solid surface, there is a contact line. The physical phenomena that cause dissipation of energy Capillary surfaces during a motion of the contact line are hysteresis in the contact angle dynamics, and viscosity of the Contact angle hysteresis fluids involved. Viscous dissipation In this paper, we consider a simplified problem where a liquid and a gas are bounded between two Calculus of variations parallel plane surfaces with a capillary surface between the liquid–gas interface. The liquid–plane Two-point boundary value problem interface is considered to be non-ideal, which implies that the contact angle of the capillary surface at – Navier Stokes equation the interface is set-valued, and change in the contact angle exhibits hysteresis. We analyze a two-point boundary value problem for the fluid flow described by the Navier–Stokes and continuity equations, wherein a capillary surface with one contact angle is deformed to another with a different contact angle. The main contribution of this paper is that we show the existence of non-unique classical solutions to this problem, and numerically compute the dissipation. -
1. A) the Sun Is in Hydrostatic Equilibrium. What Does
AST 301 Test #3 Friday Nov. 12 Name:___________________________________ 1. a) The Sun is in hydrostatic equilibrium. What does this mean? What is the definition of hydrostatic equilibrium as we apply it to the Sun? Pressure inside the star pushing it apart balances gravity pulling it together. So it doesn’t change its size. 1. a) The Sun is in thermal equilibrium. What does this mean? What is the definition of thermal equilibrium as we apply it to the Sun? Energy generation by nuclear fusion inside the star balances energy radiated from the surface of the star. So it doesn’t change its temperature. b) Give an example of an equilibrium (not necessarily an astronomical example) different from hydrostatic and thermal equilibrium. Water flowing into a bucket balances water flowing out through a hole. When supply and demand are equal the price is stable. The floor pushes me up to balance gravity pulling me down. 2. If I want to use the Doppler technique to search for a planet orbiting around a star, what measurements or observations do I have to make? I would measure the Doppler shift in the spectrum of the star caused by the pull of the planet’s gravity on the star. 2. If I want to use the transit (or eclipse) technique to search for a planet orbiting around a star, what measurements or observations do I have to make? I would observe the dimming of the star’s light when the planet passes in front of the star. If I find evidence of a planet this way, describe some information my measurements give me about the planet. -
Arxiv:1808.01401V1 [Math.NA] 4 Aug 2018 Failure in Manufacturing Microelectromechanical Systems Devices, Is Caused by an Interfacial Tension [53]
A CONTINUATION METHOD FOR COMPUTING CONSTANT MEAN CURVATURE SURFACES WITH BOUNDARY N. D. BRUBAKER∗ Abstract. Defined mathematically as critical points of surface area subject to a volume constraint, con- stant mean curvatures (CMC) surfaces are idealizations of interfaces occurring between two immiscible fluids. Their behavior elucidates phenomena seen in many microscale systems of applied science and engineering; however, explicitly computing the shapes of CMC surfaces is often impossible, especially when the boundary of the interface is fixed and parameters, such as the volume enclosed by the surface, vary. In this work, we propose a novel method for computing discrete versions of CMC surfaces based on solving a quasilinear, elliptic partial differential equation that is derived from writing the unknown surface as a normal graph over another known CMC surface. The partial differential equation is then solved using an arc-length continuation algorithm, and the resulting algorithm produces a continuous family of CMC surfaces for varying volume whose physical stability is known. In addition to providing details of the algorithm, various test examples are presented to highlight the efficacy, accuracy and robustness of the proposed approach. Keywords. constant mean curvature, interface, capillary surface, symmetry-breaking bifurcation, arc- length continuation AMS subject classifications. 49K20, 53A05, 53A10, 76B45 1. Introduction Determining the behavior of an interface between nonmixing phases is crucial for under- standing the onset of phenomena in many microscale systems. For example, interfaces induce capillary action in tubules [23], produce beading in microfluidics [24], and change the wetting properties of patterned substrates [34]. Additionally, stiction, the leading cause of arXiv:1808.01401v1 [math.NA] 4 Aug 2018 failure in manufacturing microelectromechanical systems devices, is caused by an interfacial tension [53]. -
Statics and Dynamics of Capillary Bridges
Colloids and Surfaces A: Physicochem. Eng. Aspects 460 (2014) 18–27 Contents lists available at ScienceDirect Colloids and Surfaces A: Physicochemical and Engineering Aspects j ournal homepage: www.elsevier.com/locate/colsurfa Statics and dynamics of capillary bridges a,∗ b Plamen V. Petkov , Boryan P. Radoev a Sofia University “St. Kliment Ohridski”, Faculty of Chemistry and Pharmacy, Department of Physical Chemistry, 1 James Bourchier Boulevard, 1164 Sofia, Bulgaria b Sofia University “St. Kliment Ohridski”, Faculty of Chemistry and Pharmacy, Department of Chemical Engineering, 1 James Bourchier Boulevard, 1164 Sofia, Bulgaria h i g h l i g h t s g r a p h i c a l a b s t r a c t • The study pertains both static and dynamic CB. • The analysis of static CB emphasis on the ‘definition domain’. • Capillary attraction velocity of CB flattening (thinning) is measured. • The thinning is governed by capillary and viscous forces. a r t i c l e i n f o a b s t r a c t Article history: The present theoretical and experimental investigations concern static and dynamic properties of cap- Received 16 January 2014 illary bridges (CB) without gravity deformations. Central to their theoretical treatment is the capillary Received in revised form 6 March 2014 bridge definition domain, i.e. the determination of the permitted limits of the bridge parameters. Concave Accepted 10 March 2014 and convex bridges exhibit significant differences in these limits. The numerical calculations, presented Available online 22 March 2014 as isogones (lines connecting points, characterizing constant contact angle) reveal some unexpected features in the behavior of the bridges. -
Georgia Institute of Technology in Partial Fulfillment Of
M INVESTIGATION OF CAPILLARY SPREADING POWER OF EISJLSI01IS J A THESIS Presentad to the Faculty of the Division of Graduate Student: Georgia Institute of Technology In Partial Fulfillment of the Requirements for the Degree Master of Science in Chemical Engineering fey Boiling Gay Braviley September 19/49 0752fl rt 9 AN INVESTIGATION OF CAPILLARY SPREADING POWER OP EMULSIONS Approved: ^i S'-"—y<c\-— /. ./ J Date Approved by Chairman &, \j^e^^cZt^u^ <C~*Si. ^g*i #q t AC XBQRLgDGSWTS The author wishes to express his appreciation to the personnel of Southern Sizing Company for making this study possible. To Dr. J» M« DallaValle, my advisor, I am most deeply indebted for his suggestion of the study and his valuable guidance and interest in the problem. SABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION AMD SCOPE OF THE PROBLEM 1 II. REVIEW OF THE LITERATURE 3 Capillary Flow 3 Surface Tension 5 Wotting, Spreading, and Penetration 6 III. EQUIPMT 7 Selection 7 Description and Operation 8 IV. SELECTION, STABILITY, AND PHYSICAL PROPERTIES OF EMULSIONS . , 11 V. CAPILLARY SPREADING POWER 19 Development of Equations* •••••••• • 19 Effect of Temperature •• ••••• 26 Effect of Emulsion Concentration • 26 VI. SUT«RY £SB CONCLUSIONS 29 Summary 29 Conclusions •••••••••.••••••••••••• 30 BIBLIOGRAPHY 3J APPMDIX „ 36 * LIST OF TABLES TABLES PAGE I. Physical Properties of Test Emulsion 13 II. Time of Capillary Plow vs. Viscosity-Surface Tension Ratio •••• • • 21 III. Time of Capillary Plow vs. Concentration and Temperature #••.••••»••••••••••••*• ho IV. Variation of Physical Properties of Test Emulsion with Temperature ••••• •• hi V. Variation of Physical Properties of Test Emulsion with Concentration •••••••••••• •• i\Z ' LIST OP FIGURES FIGURES PAGE 1. -
Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations
Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations Dissertation zur Erlangung des Doktorgrades vorgelegt von Johannes Daube an der Fakult¨at f¨ur Mathematik und Physik der Albert-Ludwigs-Universit¨at Freiburg Februar 2017 Dekan: Prof. Dr. Gregor Herten 1. Gutachter: Prof. Dr. Dietmar Kr¨oner 2. Gutachter: Prof. Dr. Helmut Abels Datum der m¨undlichen Pr¨ufung: 09.11.2016 Contents Abstract v Acknowledgements vii List of Symbols ix 1 Introduction 1 1.1 PhaseTransitions................................ 1 1.2 CapillaryEffects ................................. 2 1.3 Sharp- and Diffuse-Interface Models and the Sharp-Interface Limit . 3 1.4 The Navier–Stokes–Korteweg Model . .... 4 1.5 TheStaticCase.................................. 10 1.6 ExistingResults ................................. 13 1.7 NewContributions ................................ 16 1.8 Outline ...................................... 16 2 Mathematical Background 19 2.1 Notation...................................... 19 2.2 Measures ..................................... 26 2.3 Functions of Bounded Variation . 30 3 The Diffuse-Interface Model 35 3.1 The Double-Well Potential . 35 3.2 TheNotionofWeakSolutions. 37 3.3 APrioriEstimates ................................ 43 3.4 CompactnessofWeakSolutions. 56 3.5 LimitingInterfaces .............................. 60 3.6 Remarks...................................... 63 4 The Sharp-Interface Model 67 4.1 Two-Phase Incompressible Navier–Stokes Equations with Surface Tension . 67 4.2 Hypersurfaces................................... 70 -
Astronomy 241: Review Questions #1 Distributed: September 20, 2012 Review the Questions Below, and Be Prepared to Discuss Them I
Astronomy 241: Review Questions #1 Distributed: September 20, 2012 Review the questions below, and be prepared to discuss them in class on Sep. 25. Modified versions of some of these questions will be used in the midterm exam on Sep. 27. 1. State Kepler’s three laws, and give an example of how each is used. 2. You are in a rocket in a circular low Earth orbit (orbital radius aleo =6600km).Whatve- 1 locity change ∆v do you need to escape Earth’s gravity with a final velocity v =2.8kms− ? ∞ Assume your rocket delivers a brief burst of thrust, and ignore the gravity of the Moon or other solar system objects. 3. Consider a airless, rapidly-spinning planet whose axis of rotation is perpendicular to the plane of its orbit around the Sun. How will the local equilibrium surface temperature temperature T depend on latitude ! (measured, as always, in degrees from the planet’s equator)? You may neglect heat conduction and internal heat sources. 3 4. Given that the density of seawater is 1025 kg m− ,howdeepdoyouneedtodivebefore the total pressure (ocean plus atmosphere) is 2P0,wherethetypicalpressureattheEarth’s 5 1 2 surface is P =1.013 10 kg m− s− ? 0 × 5. An asteroid of mass m M approaches the Earth on a hyperbolic trajectory as shown " E in Fig. 1. This figure also defines the initial velocity v0 and “impact parameter” b.Ignore the gravity of the Moon or other solar system objects. (a) What are the energy E and angular momentum L of this orbit? (Hint: one is easily evaluated when the asteroid is far away, the other when it makes its closest approach.) (b) Find the orbital eccentricity e in terms of ME, E, L,andm (note that e>1).