STOCHASTIC BUBBLE

FORMATION AND BEHAVIOR

IN NON-NEWTONIAN FLUIDS

JESSICA REDMON

Submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Dissertation Adviser: Prof. Dr. Wojbor Woyczynski

Department of Mathematics, Applied Mathematics and Statistics

CASE WESTERN RESERVE UNIVERSITY

August 2019 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

Jessica Redmon

candidate for the degree of Doctor of Philosophy⇤

Committee Chair

Dr. Wojbor Woyczynski

Committee Member

Dr. David Gurarie

Committee Member

Dr. Longhua Zhao

Committee Member

Dr. Harsh Mathur

Defense Date

June 3rd, 2019

⇤We also certify that written approval has been obtained for any proprietary ma- terial contained therein. Contents

List of Figures ix

Abstract x

1 Introduction 1

2 Fractional Calculus 5

2.1 The Caputo Derivative ...... 10

2.2 Finite Di↵erence Approximations for Fractional Operators ...... 11

2.2.1 Finite Di↵erence Approximation for the Fractional Derivative 11

2.2.2 Finite Di↵erenceApproximationError ...... 12

3 Burgers Equation 19

3.1 Burger’sequationfromNavier-Stokes ...... 20

3.2 Generalized Burger’s Equation ...... 22

3.3 PowerLawFluids...... 24

3.4 Fractional General Burger’s Equation ...... 27

i 3.5 Time Fractional Generalized Burger’s Equation for non-Newtonian Fluids 31

4 The Rayleigh-Plesset Equation 33

4.1 InNewtonianFluids ...... 33

4.2 InNon-NewtonianFluids...... 38

4.3 Simulations ...... 40

4.3.1 Shear-ThinningFluids ...... 40

4.3.2 Shear-ThickeningFluids ...... 42

5 Bubble Cavitation: Formation and Collapse 43

5.1 Bubble Formation ...... 44

5.1.1 Di↵usionofSeedNulcei ...... 44

5.1.2 InitialRadii...... 51

5.2 Bubble Collapse ...... 52

6 Non-Spherical Bubbles 56

6.1 The Equator-Pole Model ...... 57

6.2 Uncoupling the Di↵erentialEquations...... 61

6.3 Nondimensionalizing the Di↵erentialEquations ...... 63

6.4 Numericalexperiments ...... 66

7 Finite Di↵erence Scheme 77

7.1 Finite Di↵erenceSchemesforBurger’sEquation ...... 77

7.2 Deriving the Finite Di↵erenceScheme...... 78

ii 7.3 NumericalExperiments...... 81

8 Future Work 90

iii List of Figures

3.1 Abdou and Soliman’s Fig. 1: The behavior of u(x,t) evaluated by

variational iteration method versus x for di↵erent values of time with

fixed values d=1, = .125, = .6, ↵ = .4...... 22

3.2 Wei and Borden’s Figure 1: The profiles of the transition layers with

u = 1, u = 1. The thick solid line represents a Newtonian fluid 1 2 (n = 1), the thin solid line is a shear-thickening fluid with n =2and

1 the dashed line represents a shear-thinning fluid with n = 2 ...... 27

3.3 Li, Zhang, and Ran’s numerical solution for 3.25 with p =2,↵ = .25

and d = .05...... 30

3.4 Li, Zhang, and Ran’s numerical solution for 3.25 with p =2,↵ = .75

and d = .05...... 31

4.1 Bubble stretched to 1.1R0 and governed by the Rayleigh-Plesset equa-

tion with no outside forcing. Note the damped oscillations as the bub-

ble returns to equilibrium...... 37

iv 4.2 Bubble stretched to 1.1R0 and governed by the Rayleigh-Plesset equa-

tion with sinusoidal outside forcing. The early behavior, from t =0to

t =1e 4, is dominated by the system’s tendency toward equilibrium with the driving force determining the radius after t =1.5e 4seconds, with a transitional region from t =1e 4tot =1.5e 4...... 38 4.3 Bubble behavior in a power law fluid with n =1.1withinitialradius

90% of the equilibrium radius...... 41

4.4 Bubble behavior in a power law fluid with n =1.2withinitialradius

90% of the equilibrium radius...... 41

4.5 Bubble behavior in a power law fluid with n =1.5withinitialradius

90% of the equilibrium radius...... 41

4.6 Bubble behavior in a power law fluid with n =0.5withinitialradius

90% of the equilibrium radius...... 42

5.1 The figure above from ”The Rheology Handbook” shows viscosity against

shearing stress for Newtonian, shear-thinning and shear-thickening flu-

ids. The solid line is a Newtonian fluid, the dotted line is a shear-

thickening fluid, and the dashed line is a shear-thinning fluid...... 48

5.2 Non-Newtonian Di↵usionProcesses ...... 49

5.3 StandardBrownianmotionforaNewtonianfluid...... 49

5.4 Di↵usion for a shear-thickening fluid on the left, and its two-dimensional

projection on the right...... 50

v 5.5 The left figure is di↵usion for a shear-thinning fluid next to its two-

dimensional projection on the right...... 50

5.6 Figure 4.2 from Brennan’s ”Fundamentals of Multiphase Flows”. ”The

nucleus of radius, R0,entersalow-pressureregionatadimensionless

time of 0 and is convected back to the original pressure at a dimension-

less time of 500. The low-pressure region is sinusoidal and symmetric

about 250.” [12] ...... 54

6.1 Evolution of bubble radius and size of the deformation with initial

radius equal to R0, the equilibrium radius, and initial deformation size

2 10 R0. Thisfluidhasthesamepropertiesaswater...... 67

6.2 Evolution of bubble radius and size of the deformation with initial

radius equal to R0, the equilibrium radius, and initial deformation size

1 10 R0. Thisfluidhasthesamepropertiesaswater...... 68

6.3 Evolution of bubble radius and size of the deformation with initial ra-

dius 1.05R0 where R0 is the equilibrium radius and initial deformation

3 size 10 R0. The theoretical fluid used is 100 times more viscous than

water,withthesamesurfacetension...... 69

6.4 Evolution of bubble radius and size of the deformation with initial ra-

dius 1.05R0 where R0 is the equilibrium radius and initial deformation

2 size 10 R0. The theoretical fluid used is 100 times more viscous than

water,withthesamesurfacetension...... 70

vi 6.5 Evolution of bubble radius and size of the deformation with initial

radius 1.1R0 where R0 is the equilibrium radius and initial deformation

3 size 10 R0.Thetheoreticalfluidusedis5timesmoreviscousthan

water,withthesamesurfacetension...... 71

6.6 Evolution of bubble radius and size of the deformation with initial

radius 1.1R0 where R0 is the equilibrium radius and initial deformation

1 size 10 R0.Thetheoreticalfluidusedis5timesmoreviscousthan

water,withthesamesurfacetension...... 72

6.7 Evolution of bubble radius and size of the deformation with initial ra-

dius 1.05R0 where R0 is the equilibrium radius and initial deformation

2 size 10 R0.Thetheoreticalfluidusedis10timesmoreviscousthan

water,withthesamesurfacetension...... 73

6.8 Evolution of bubble radius and size of the deformation with initial

radius 1.2R0 where R0 is the equilibrium radius and initial deformation

2 size 10 R0.Thetheoreticalfluidusedis10timesmoreviscousthan

water,withthesamesurfacetension...... 74

6.9 Evolution of bubble radius and size of the deformation with initial

radius 1.1R0 where R0 is the equilibrium radius and initial deformation

3 size 10 R0.Thetheoreticalfluidusedis5timesmoreviscousthan

water,withthesamesurfacetension...... 75

vii 6.10 Evolution of bubble radius and size of the deformation with initial

radius 1.1R0 where R0 is the equilibrium radius and initial deformation

1 size 10 R0.Thetheoreticalfluidusedis5timesmoreviscousthan

water,withthesamesurfacetension...... 76

7.1 This is a shear-thinning fluid. For smooth initial conditions, the fluid

velocity propagates smoothly toward an equilibrium solution. Note the

bowing of the steady state for choices of ↵ further from 1...... 82

7.2 For smooth initial conditions, the shear-thickening fluid velocity prop-

agates smoothly toward an equilibrium solution. Note that the bowing

of the steady state for choices of ↵ further from 1 does not seem to

depend on fluid type, only ↵...... 83

7.3 With step function initial conditions, we see bubbles occur under cer-

tain conditions. For larger values of ↵,theshear-thickeningfluidresists

breaking to form bubbles, but as ↵ decreases, the sharp changes in ve-

locity indicative of bubble formation begin to occur...... 84

7.4 For larger values of ↵,thereisagainresistancetofluidrupture,and

the graph only begins to feature the sharp changes in u indicative of

bubble formation at ↵ =0.5...... 85

7.5 For smaller ⌘,thepowerlawindex,weseethefluidbreakingforall

values of ↵ rather than only for small alpha...... 86

viii 7.6 For this shear-thinning fluid, we see both increased resistance to fluid

rupture and a slower decay to a steady state...... 87

7.7 Here we see clear fluid rupture for all values of ↵ and for ↵>1, we see

the break continue indicating the bubble’s continued existance. . . . . 88

7.8 Finally, we see as with ⌘ = .5, bubble formation for all values of ↵.We

also see more bubbles staying formed...... 89

ix Stochastic Bubble Formation and Behavior

in Non-Newtonian Fluids by

JESSICA REDMON

Abstract

Conditions for bubble cavitation and behavior in non-Newtonian fluids have numerous applications in physical sciences, engineering and medicine. Non-Newtonian fluids are a rich, but relatively undeveloped area of fluid dynamics, with phenomena from di↵u- sion to bubble growth just beginning to receive attention. In the course of examining bubble cavitation, it became apparent that the random particle motion responsible for determining potential bubble formation had not been researched. As cavitation bub- bles collapse, they deform into a variety of non-spherical shapes. Due to the complex dynamics and the radial focus of current equations on bubble behavior, no accepted model has yet emerged. This work explores the behavior using numerical methods on both fluid and bubble models to examine this system from di↵erent prospectives, culminating in a time-fractional, power-law Burger’s type equation showing bubble formation under these conditions.

x Chapter 1

Introduction

For my doctoral research, I have been most interested in the work pertaining to non-Newtonian and bubbling fluids, though I have also spent time examining more general properties of non-Newtonian fluids and the behavior of non-spherical bubbles in Newtonian fluids. I have primarily been examining bubble formation and collapse in non-Newtonian fluids by way of a time-fractional Burger’s equation. The models developed for examining these phenomena have been studied individually, but have not previously been combined to describe this system.

Over the past couple of decades, research in nonlinear dynamics has taken an interest in fractional di↵erential equations. The foundation of fractional calculus has created an entire class of di↵erential equations that can be used to describe phenom- ena whose current behavior depends on time or space in a longer or shorter reaching way. This class of equation is becoming popular in a variety of applications in fluid

1 mechanics, biology, finance and physics, to name a few [1, 5, 7]. Increasing attention has gone into researching methods to approximate or solve these equations, from trial equations, to homotopy perturbation, and a plethora of numerical methods including

finite di↵erence methods [6]. I will be focusing on using a finite di↵erence approach to examine bubble formation under a modified fractional Burger’s equation applying to non-Newtonian fluids.

The generalized Burger’s equation without a fractional derivative has been studied intently for several decades, giving rise to considerable literature on many interesting applications [2, 7, 8]. The development of the time- and space-fractional derivative opened up a new type of Burger’s equation that include force through the boundary

[1, 4, 5, 6], creating a more accurate system for various Newtonian fluid flows. For non-Newtonian fluids, the generalized power-law Burger’s equation has been devel- oped, and traveling wave solutions discovered and studied [2]. However, this leaves many more complex systems unexplored.

Non-Newtonian fluids are a rich, but relatively undeveloped area of fluid dynamics, with phenomena from di↵usion to bubble growth just beginning to receive attention.

In the course of examining bubble cavitation, it became apparent that the random particle motion of the fluid responsible for determining potential bubble formation had not been researched. I built a model for this process from basic principles based

2 on fundamental particle movement in a similar manner to a derivation for the di↵u- sion equation. While it has not been rigorously tested, initial results follow expected behavior.

Recent interest in bubble growth in non-Newtonian fluids has yielded a new model similar to the Rayleigh-Plesset equation for shear thickening fluids, but derived using more readily accessible fluid properties like coecient of consistency and shearing stress so that it may be implemented for engineering and manufacturing applications without much, if any, modification [2]. As there were no assumptions in the deriva- tion process specific to shear-thickening fluids rather than any non-Newtonian fluid,

Iappliedthemodeltoshear-thinningfluidswithgreatsuccess.

Another area that has been of longstanding interest is the behavior of non- spherical bubbles. As cavitation bubbles collapse, they deform into a variety of non-spherical shapes that determine the shape and direction of any resulting shock waves [15]. However, because of the complex dynamics and the radial focus of current equations on bubble behavior, no accepted model has yet emerged. As the model for this is new and this is the first successful implementation, and because of the complex dynamics, it is currently intended for fixed viscosity, or Newtonian, fluids.

Numerical methods are often employed in the study of fractional Burger’s equation

3 and generalized Burger’s equation [1, 4, 5, 9] due to the nonlinear nature of the equa- tion. Interesting cases have been found for non-Newtonian fluids, bubbling fluids and parameter choices for the time-fractional Burger’s equation separately [1, 2, 4, 6, 8, 9].

Finite di↵erence approximations have become increasingly popular for solving both fractional di↵erential equations and wave and Burger’s type equations [1, 3, 6]. Re- cently Li, Zhange and Ran developed a linear finite di↵erence scheme for a generalized time fractional Burger’s equation, sometimes used to model bubbling fluids, that is globally stable, providing a groundwork for my own research into non-Newtonian bubbling systems.

4 Chapter 2

Fractional Calculus

This chapter will focus on laying the groundwork for the fractional derivative defini- tions I am using. I will show why it is applicable here, a finite di↵erence approximation for the Caputo derivative and end with a proof to show viability.

Fractional calculus is a generalization of classical di↵erentiation and integration developed to study di↵erential equations of fractional order [24]. There are many es- tablished ways to define a fractional derivative of order ↵ that are di↵erent, but all are considered correct as long as they approach the classical derivatives as ↵ approaches an integer [24, 11]. The first was derived by Liouville in 1832 for the exponential function, followed by a fractional derivative for a power function by Riemann in 1847

[24]. Some of the more common definitions include [11]:

5 1 dn t x(⌧) Riemann-Liouville D↵x(t)= d⌧ t (n ↵) dtn (t ⌧)↵ n+1 Z0 1 t dn x(⌧) Caputo D↵x(t)= d⌧ t (1 ↵) dtn (t ⌧)↵ Z0 n 1 (k) k ↵ t (n) ↵ x (0)t 1 x (⌧) Gr¨unwald-Letnikov Dt x(t)= + ↵+1 n d⌧ (k +1 ↵) (n ↵) 0 (t ⌧) Xk=0 Z 1 Riesz D↵x(t)= t 2cos ⇡↵ (n ↵) 2 n t d x(⌧) n 1 x(⌧) n ↵ n+1 d⌧ +( 1) ↵ n+1 d⌧ dt (t ⌧) t (t ⌧) Z1 Z

where n is the ceiling of ↵.

While no geometric interpretation of the fractional derivative has been verified, un- like integer order integrals and derivatives, it is generally accepted that the fractional derivative acts as ”memory” for the system [24, 32]. Fractional di↵erential equations have been increasingly employed to model complex systems in di↵erent research ar- eas and engineering applications with nonlinear behavior or long term memory. It is particularly useful for fatiguing or viscoelastic systems [10, 11, 24]. Many problems in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science are also better described by fractional order di↵erential equations [4]. Fractional dif- ferential equations have already been applied to a variety of solids and fluids [24, 23], but work in elastic solids suggests suitability to non-Newtonian fluids.

6 Fractional di↵erential equations (FDE) have been appearing more frequently in both research and engineering applications. The fractional derivative has applications to many physical problems such as ?frequency dependent damping behavior of mate- rials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PIDµ controller for the control of dynamical systems, etc. [4] Di↵erential equations of fractional order also describe phenomena in fields like electromagnetics, acoustics, viscoelasticity, electrochemistry, and material science. In

Bagley and Torvik?s review of work done in this field before 1980, they demonstrate that the frequency dependence of the damping materials is well described by models of half-order fractional di↵erential equations.

Mainardi [41] and Rossikhin and Shitikova [42] published an overview of frac- tional derivatives generally for solid mechanics, and particularly for the modeling of viscoelastic damping. Magin [43] published a critical review in three parts for the ap- plications of fractional calculus to bioengineering. Further applications of fractional derivatives in other fields, as well as related mathematical tools and techniques can be found in [44, 45, 46, 47, 48], while applications of fractional derivatives in elec- trochemical processes [36, 37], dielectric polarization [38], colored noise [39],..., and chaos [40] have been demonstrated by other authors.

Fractional calculus has been used to model both physical and engineering pro-

7 cesses that are best described by FDEs, so a reliable and ecient technique for solv- ing FDEs is necessary. Atanackovic and Stankovic [49] analyzed lateral motion of an elastic column this is fixed at one end and loaded at the other in terms of a sys- tem of FDE. Shawagfeh [50] used an Adomian decomposition method for non-linear

FDE. Daftardar-Gejji and Babakhani [52] also used Adomian decomposition method to obtain solutions of a system of FDEs, and explored the convergence of the method.

Daftardar-Gejji and Babakhani [51] studied the existence, uniqueness and stability of solutions to a system of FDE, and presented an analysis of the system. They also later showed an iterative method of solving nonlinear functional equations. Finally,

Momani [54] used Adomian decomposition method to present nonperturbative ana- lytical solutions of both space and time-fractional Burger?s equations. [4]

More recently, the fractional calculus was successfully applied to non-di↵erentiable problems arising in the areas of solid mechanics [68], heat transfer and wave prop- agation [69], di↵usion [70], hydrodynamics [71], vehicular trac flow [72] and other topics [6]. A fractional calculus approach has recently been used to describe wave propagation in non-local media, sub-and super- di↵usion and in various generalized

Burger’s equations. Sapora, Cornetti and Carpinteri modeled wave propagation in an elastic media where disturbances e↵ect non-local points. They considered displace- ment of a one dimensional, Eringen bar with length L and endpoints a and b [23].

Reimann-Liouville fractional derivates of order 1 <↵<2describetheinteractions

8 between nonadjacent points. Using this model, they were able to analyze wave prop- agation in this one dimensional elastic media and found the non-local interactions resulted in deformations of the wave shape which were more pronounced if only one end of the bar was fixed[23].

D. del-Castillo-Negrete, B. A. Carreras and V. E. Lynch used a fractional oper- ator in a reaction di↵usion system to model the front dynamics of a superdi↵usive media like plasma and other sheer-thinning non-Newtonian fluids. They examined the system at the particle motion level to determine that a fractional operator in the space was appropriate for super di↵usion. Previous studies in reaction-di↵usion systems exhibiting both subdi↵usion and superdi↵usion including [27] ,[28] and [5] used a temporal fractional derivative. At the particle level, superdi↵usive fluid par- ticles did not move according to a stochastic Gaussian process, as is necessary for the use of Laplacian operators, but instead were better described by ↵ stable L´evy distributions with 1 <↵<2. In particular, they analyzed the Fisher-Kolmogorov equation

F ()=(1 )

with a finite domain: 0

9 2.1 The Caputo Derivative

The most commonly used fractional derivatives are the Riemann-Liouville and Ca- puto definitions. The di↵erence between the two definitions lies in the placement of the derivative with respect to the integral. Both of these definitions have advantages and drawbacks. The Caputo derivative preserves the initial condition, but does not coincide with the classical derivative for integer ↵ from above, except in it’s smooth- ness requirement. The Riemann-Liouville is the opposite, with the initial condition involving a fractional derivative or integral, but gives the classical derivative for in- teger values of ↵ [11].

For 0 <↵<1, the Riemann-Liouville fraction derivative is defined as

1 d t u(s) D↵u(x, t)= ds t (1 ↵) dt (t s)↵ Z0

And the Caputo definition takes the form:

1 t @u(x, s) 1 D↵u(x, t)= ds t (1 ↵) @s (t s)↵ Z0

[10]

Clearly, the Caputo definition will apply to fewer functions than the Riemann-

Liouville, as all functions must be di↵erentiable to have the Caputo fractional deriva- tive applied to them. It is worth noting that the Caputo definition, unlike the

Liouville-Riemann, of a fractional order derivative of a constant gives zero. Also

10 ↵ (n) ↵+n notice that for this definition Dt u (x, t)=Dt u(x, t). This, together with preser- vation of the initial conditions, makes the Caputo definition preferable for our system.

2.2 FiniteDi↵erence Approximations for Fractional

Operators

Finding a numerical solution to a fractional di↵erential equation requires some del- icacy. Most classical methods like Euler method, both implicit and explicit, are unstable when used directly on the Gr¨unwald approximation [29]. Meerschaert and

Tadjeran had some success approximating the Riemann-Liouville derivative using a shifted Gr¨unwald formula, the implicit Euler method, and the Crank-Nicholson method[29]. However, R. Scherer et al. found a finite di↵erence approximation to the

Caputo derivative using the Gr¨unwald-Letnikov approach.

2.2.1 FiniteDi↵erence Approximation for the Fractional Deriva-

tive

Consider the time discrete ⌧ = t t .Forthefractionaloperator,wehave n+1 n

1 t @u(x, s) 1 D↵u(x, t)= ds t (1 ↵) @s (t s)↵ Z0

Since this cannot be taken directly with all potential functions u(x, t), we use a

11 quadrature type approximation, with an based on the definition of the fractional derivative, to fit into the finite di↵erence scheme

n 1 ↵ n k 0 Dt u(x, t) uj (an 1 k an k)uj an 1uj (2.1) ⇡ Xk=1 where

tn+1 1 a = dt n t↵ Ztn 1 1 ↵ 1 ↵ = (t ) (t ) 1 ↵ n+1 n 1 ↵ ⇥ ⇤ ⌧ 1 ↵ 1 ↵ = (n +1) (n) 1 ↵ ⇥ ⇤

So we define

1 ↵ ⌧ 1 ↵ 1 ↵ a = (n +1) (n) (2.2) n 1 ↵ ⇥ ⇤

2.2.2 Finite Di↵erence Approximation Error

Sun and Wu [31] used this type of approximation for a numerical simulation of a di↵usion wave equation. They developed a version of the following Lemma for 1 <

↵<2 which was applied to 0 <↵<1 by Li, Zhang, and Ran [1].

Lemma 2.2.1. Let f(t) C2 [0,t ], a as in equation 2.2, and let 2 n n

↵ 1 tn f 0(s) ⌧ n 1 R(f(tn)) ds a0f(tn) (an 1 j an j) f(tj) an 1f(t0)  (1 ↵) 0 (t s)↵ (2 ↵) j=1 n R h P i 12 be the remainder between the fractional derivative and the approximation. Then,

2 ↵ 1 1 ↵ 2 ↵ 2 ↵ R(f(tn)) = + (1 + 2 ) max f 00(t) ⌧ (2 ↵) 12 2 ↵ 0 t tn    Proof. Let

tn n tk dt f(tk) f(tk 1) dt 0 A = f (t) ↵ ↵ 0 (tn t) ⌧ tk 1 (tn t) Z Xk=1 Z n tk f(tk) f(tk 1) dt 0 = f (t) ↵ tk 1 ⌧ (tn t) Xk=1 Z 

Recall ⌧ = t t . Using a Taylor expansion with an integral remainder, we n+1 n write

t tk 1 g(tk) g(tk 1) 1 f 0(t) = f 00(s)(s tk 1)ds f 00(s)(tk s)ds ⌧ ⌧ " tk 1 t # Z Z

Putting this back into A yields

n tk f(tk) f(tk 1) dt 0 A = f (t) ↵ tk 1 ⌧ (tn t) Xk=1 Z  n tk t tk 1 1 dt = f 00(s)(s tk 1)ds f 00(s)(tk s)ds ⌧ (t t)↵ tk 1 " tk 1 t # n Xk=1 Z Z Z

Changing the order of integration and integrating against t,wehave

13 1 n tk 1 ↵ s tk 1 1 ↵ tk s 1 ↵ A = (tn s) (tn tk) + (tn tk 1) f 00(s)ds k=1 tk 1 1 ↵ ⌧ ⌧ ✓  ◆ P R

Analyzing this, we can see

1 n tk 1 ↵ s tk 1 1 ↵ tk s 1 ↵ A (tn s) (tn tk) + (tn tk 1) f 00(s) ds k=1 tk 1 | |1 ↵ ⌧ ⌧ | | ✓  ◆ P R 1 n tk 1 ↵ s tk 1 1 ↵ tk s 1 ↵ maxt t t f 00(t) (tn s) (tn tk) + (tn tk 1) ds 0 n k=1 tk 1  1 ↵   | | ⌧ ⌧  ✓ ◆ P R

Now we need to show that

n tk 1 ↵ s tk 1 1 ↵ tk s 1 ↵ 0 (tn s) (tn tk) + (tn tk 1) ds  tk 1 ⌧ ⌧ Xk=1 Z ✓  ◆ 2 ↵ 1 ↵ 2 + (1 + 2↵)  12 2 ↵ 

1 ↵ Let g(s)=(tn s) and ⇠k (tk 1,tk), then we can rewrite the inside of the 2 integral so that

14 1 ↵ s tk 1 1 ↵ tk s 1 ↵ (tn s) (tn tk) + (tn tk 1) = ⌧ ⌧  s tk 1 tk s 1 g(s) g(tk)+ g(tk 1) = g00(⇠k)(t tk)(t tk 1) ⌧ ⌧ 2  1 ↵ 1 = (1 ↵)(↵)(tn ⇠k) (tk t)(t tk 1) 2 0

using this expression in the integral, we see

n 2 tk s tk 1 tk s g(s) g(tk)+ g(tk 1) ds tk 1 ⌧ ⌧ k=1 Z ✓  ◆ X n 2 tk 1 ↵ 1 = (1 ↵)(↵)(tn ⇠k) (tk s)(s tk 1)ds tk 1 2 k=1 Z X n 2 tk 1 ↵ 1 (1 ↵)(↵) (tn tk) (tk s)(s tk 1)ds  2 tk 1 Xk=1 Z

Note that tk ⌧ 3 (tk s)(s tk 1)ds = tk 1 6 Z

15 This allows us to take the integral out of the summation, so we have

n 2 tk s tk 1 tk s g(s) g(tk)+ g(tk 1) ds tk 1 ⌧ ⌧ k=1 Z ✓  ◆ X n 2 3 ⌧ ↵ 1 (1 ↵)(↵) (t t )  12 n k Xk=1 2 tn 1 ⌧ ↵ 1 (1 ↵)(↵) (t t) dt  12 n Zt1 2 ⌧ ↵ ↵ = (1 ↵) (tn tn 1) (tn t1) 12

1 ↵ 2 ↵ ⌧  12

Now to account for the last two terms, we can see

n tk s tk 1 tk s g(s) g(tk)+ g(tk 1) ds tk 1 ⌧ ⌧ k=n 1 ✓  ◆ X Z tn 1 = g(s)ds ⌧ g(tn 1)+g(tn 1) tn 2 2 Z  tn 1 ↵ 1 1 ↵ 1 ↵ = (g(s))tn s) ds ⌧ (tn tn 2) +(tn tn 1) tn 2 2 Z  2 ↵ 2 ↵ 2 ↵ = (1 + 2 ) ⌧ 2 ↵ ✓ ◆

Putting these together, we get

16 n tk s tk 1 tk s g(s) g(tk)+ g(tk 1) ds tk 1 ⌧ ⌧ Xk=1 Z ✓  ◆ n 2 n tk s tk 1 tk s = g(s) g(tk)+ g(tk 1) ds ⌧ ⌧ ! tk 1 k=1 k=n 1 ✓  ◆ X X Z 2 ↵ 2 ↵ 1 ↵ 2 ↵ = ⌧ + (1 + 2 ) 12 2 ↵ 

Now we can say

1 n tk 1 ↵ s tk 1 1 ↵ tk s 1 ↵ A maxt t t f 00(t) (tn s) (tn tk) + (tn tk 1) ds 0 n k=1 tk 1  1 ↵   | | ⌧ ⌧ | |  ✓ ◆ P R 2 ↵ 1 2 ↵ 1 ↵ 2 ↵ max f 00(t) ⌧ + (1 + 2 )  1 ↵ t0 t tn | | 12 2 ↵   

From this, we only need to show

n f(tl) f(tl 1) tl ds 1 n 1 = a0f(tn) (an 1 j an j) f(tj) an 1f(t0) tl 1 ↵ j=1 ⌧ (tn s) ⌧ l=1 X R h P i (2.3)

which comes directly from the integral. Recall that ↵ =0. 6

17 n n f(tk) f(tk 1) tk ds f(tk) f(tk 1) 1 1 ↵ 1 ↵ = [(tn+1 k) (tn k) ] tk 1 ↵ ⌧ (tn s) ⌧ 1 ↵ k=1 k=1 X R X 1 n = an k (f(tk) f(tk 1)) ⌧ k=1 X n 1 1 = a0f(tn) (an 1 k an k) f(tk)an 1f(t0) ⌧ " # Xk=1

This verifies the accuracy of the finite di↵erence approximation for the fractional derivative.

18 Chapter 3

Burgers Equation

Here I will discuss the history, derivation and applications of Burger’s equation. This will culminate in the equation I am using to model bubble creation in non-Newtonian

fluids.

The classic Burger’s equation has been well researched since its development in

1915 [19]. It was originally used to study turbulence in 1948 [17]. It was generalized in 1985 by Sugimoto and Kakutani to look at nonlinear viscoelastic waves and shock solutions [9]. Since then, Burger’s equation has been used to model weakly nonlinear waves [9], waves in viscous fluids [18], gas dynamics [19], trac flow [19, 6], and plane waves[6].

The fractional Burgers’ equation is an increasingly active area of interest in non- linear dynamics research. It can be applied to a wide range of phenomena in fluid

19 dynamics, as the fractional derivative can model the cumulative e↵ect of outside force through the boundary layer, as well as the influence of prior states of the sys- tem [1, 4]. This opens up models for nonlinear media, from weakly nonlinear acoustic waves through a gas filled pipe to bubbly fluids and shallow water waves [1, 9].

3.1 Burger’s equation from Navier-Stokes

Burger’s equation was first developed as a simplification of the Navier-Stokes equation[17].

However, the relationship between the two equations is more nuanced than the orig- inal derivation would suggest. There are several similarities in behavior, as one @u would expect, most notably the u term, which conserves both u(x, t)dx and @x R (u(x, t))2dx,sothesystemtendstoyieldareaswithsteepvelocitygradients[21]. R The di↵erences, however, are just as interesting, and perhaps more surprising. The most notable di↵erence is the tendency for shocks to appear and propagate nicely in

Burger’s equation, reducing initial chaos, instead of increasing as one sees with the

Navier-Stokes equation[20, 21]. These behaviors coupled with its friendliness toward numerical solvers makes Burger’s equation a hot area of research for several decades now [2, 5, 6, 9, 17, 18, 19, 20, 21].

To derive Burger’s equation, we begin with the one dimensional Navier-Stokes

20 equation for a viscous fluid is

@u @u @⌧ @p ⇢ + ⇢ u = + ⇢ g (3.1) L @t L @x @x @x L

where ⇢L is the fluid density, u is the velocity, ⌧ is the shearing stress, p is pressure and g is the external force, usually gravity.

Using the rheological relationship between the shear and strain tensors, ⌧ = n n 1 2 K ✏ ✏, and noting that for Newtonian fluids, n =1and2K = µ where µ k k is the viscosity[2], we get:

@u @u @2u(x, t) @p ⇢ + ⇢ u = µ + ⇢ g (3.2) L @t L @x @x2 @x L which is a Burger’s equation with outside forces, in this case, pressure and gravity. If @p we set = ⇢ g, we get the classic Burger’s equation for Newtonian flows. @x L

@u @u @2u(x, t) ⇢ + ⇢ u = µ (3.3) L @t L @x @x2

or equivalently @u @u @2u(x, t) + u = d (3.4) @t @x @x2

where d is a constant, typically d = µ . ⇢L

21 Even propagating the classical Burger’s equation is not a trivial exercise, due to its nonlinear nature. Abdou and Soliman modeled Burger’s equation with initial ↵ + +( ↵)e ↵ condition u(x, 0) = ,for = (x )[22]. 1+e d

Figure 3.1: Abdou and Soliman’s Fig. 1: The behavior of u(x,t) evaluated by variational iteration method versus x for di↵erent values of time with fixed values d=1, = .125, = .6, ↵ = .4.

3.2 Generalized Burger’s Equation

There is no one definition for generalized Burger’s equation, but rather it is usually formed when the advection or di↵usion term is adjusted, or an outside forcing term is added to encompass broader applications then the traditional Burger’s equation.

This yields 3 basic classes that are usually used, often individually, but sometimes combined, when discussing a generalized Burger’s equation.

22 @u @2u @u = d (u) (3.5) @t @x2 @x

@u @ @u @u = u (3.6) @t @x @x @x ✓ ◆

@u @2u @u = d u + f(u)(3.7) @t @x2 @x

Note that 3.7 is simply the traditional Burger’s equation with a force term added, and does not require further discussion to understand its purpose or derivation.

To arrive at 3.5 we follow Dr Woyczynski’s work on Burger’s-KPZ Turbulence, assuming that the fluid flow of u(t, x)issubjecttotheconservationlaw

@ x1 u(t, x)dx + (t, x ) (t, x )=0 @t 1 0 Zx0

Where the flux (t, x)=(u(t, x)) only depends on the local velocity, then as x 1 ! x0, the above equation becomes a Riemann type, quasilinear equation

@u @u 0(u(t, x)) =0 (3.8) @t @x

However, if the flux additionally depends on the gradient of the velocity of u(t, x), @u so that =(u(t, x)) d ,thentheaboveconservationequationyields[20] @x

23 @u @u @2u 0(u(t, x)) = d (3.9) @t @x @x2

In the case that 0(u)= (u), this leads to 3.5, completing its derivation.

3.3 Power Law Fluids

To derive a generalized Burger’s equation for power law, non-Newtonian fluids, we follow the work done by Wei and Borden [2], and begin with the general Navier-Stokes equation for incompressible, viscous flows in one-dimension

@u @u @⌧ @p ⇢ + u = + f (3.10) @t @x @x @x  where f is the external force.

⌘ (n 1) It is well-known, rheologically, that ⌧ =2K ✏ ✏,where⌘ is the power-law | | index, ✏ is the strain, and K is related to the viscosity by 2⌘K = µ [2]. This gives

@u @u @ (⌘ 1) @p ⇢ + u = µ ✏ ✏ + f (3.11) @t @x @x | | @x  Du @u @u Recall that Du = ✏ is the strain, and that = + u . Now, we write Dt @t @x

24 ⌘ 1 @u @u @ @u @u @p ⇢ + u = µ + f (3.12) @t @x @x @x @x @x  ! @p µ Let f = and ⌫ = , then we have the generalized Burger’s equation for @x ⇢ power-law fluids. [2]

⌘ 1 @u @u @ @u @u + u = ⌫ (3.13) @t @x @x @x @x ! ⌘ 1 It is worth noting that this is equivalent to eqution 3.6 if we let (u)= u u. | |

Wei and Borden found a traveling wave solution for 3.13. First, they applied the standard transformation u(x, t) u(⇠)where⇠ = x t,andthus3.13becomes !

du du d du + u = ⌫ (3.14) d⇠ d⇠ d⇠ d⇠ ✓ ◆

which infers

d 1 du u + u2 ⌫ =0 d⇠ 2 d⇠  ✓ ◆ or, for some constant A

1 du u + u2 ⌫ = A 2 d⇠ ✓ ◆

They then applied upstream and downstream boundary conditions

25 lim u(⇠)=u1 lim u(⇠)=u2 lim u0(⇠)=0 ⇠ ⇠ ⇠ !1 !1 | |!1

which, for = 1 (u + u )andA = 1 u u ,gavethem 2 1 2 2 1 2

du 1 1 = 2u + u2 2A = (u u )(u u )(3.15) d⇠ 2⌫ 2⌫ 1 2 ✓ ◆ 1 ⌘ 1 1 1 Since (t)= t t, (t)= t ⌘ t,thisyields | | | |

1 n du 1 1 1 1 = (u u )(u u ) = ((u u )(u u )) d⇠ 2⌫ 1 2 2⌫ 1 2 ✓ ◆

Using separation of variables, this can be written as

1 1 1 n du = d⇠ 1 ((u u )(u u )) 2⌫ Z 1 2 Z u u For n =1and 2 < 1, they used Mathematica to find the following power 6 u2 u1 series solution for the generalized power-law Burger’s equation

1 1 1 k n n 1 1 1 n n (u2 u) (1 n ,k)( n ,k) u2 u ⇠ = 1 (3.16) 2⌫ n 1 s u2 u1 (2 n ,k)k! u2 u1 Xk=0 ✓ ◆

They gave the profiles of the transition layers for a shear-thickening (n>1) fluid, a Newtonian fluid (n =1),andashear-thinningfluid(n<1).

26 Figure 3.2: Wei and Borden’s Figure 1: The profiles of the transition layers with u1 = 1, u2 = 1. The thick solid line represents a Newtonian fluid (n =1),the thin solid line is a shear-thickening fluid with n =2andthedashedlinerepresentsa 1 shear-thinning fluid with n = 2 .

3.4 Fractional General Burger’s Equation

The space fractional Burger’s equation was originally derived from the Lagrangian equations of motion by Sugimoto and Kakutani as a “Generalized Burger’s equation” as follows.

@2u @L @ @u ✏⇢ x = xx = 1+✏ x K (3.17) L @t2 @x @x @x xx ✓ ◆

where uxx is the non-vanishing component of the displacement vector, Lxx the

Lagrangian stress tensor, Kxx the Kirchho↵’s stress tensor and ✏ is a parameter that measures the order of strain and thus the nonlinearity. Each term is nondimension-

27 alized by the following transformations

x Lx, t Tt, u Uu ,L SL ,K SK ,⇢ ⇢⇢ ! ! x ! x xx ! xx xx ! xx L ! L

@u where L, T, U, S, ⇢ are the characteristic values for the system. We also let x = @x v,notingthev is the displacement gradient. If we assume the viscoelastic forces are weak, we can express Kxx in terms of the Lagrangian strain Exx

t @E K =( 2µ)E + lE2 + O(E )+ K(t s) xx ds + O(E2 )(3.18) xx xx xx xx @s xx Z1

with µ and as the equilibrium Lam´econstants, l is the nonlinear elastic constant,

K(t)istherelaxationfunctionand

@u ✏2 @u E = ✏ x + x (3.19) xx @x 2 @x ✓ ◆

Now, expressing K(t)intermsoftherelaxationspectrumH(⇤) with ⇤as a relaxation time

1 t/⇤ d⇤ K(t)= H(⇤)e (3.20) ⇤ Z0

28 where H(⇤) can be approximated as

⇤ ⌫ H(⇤) = , 0 <⌫<1(3.21) (⌫)(1 ⌫)

which gives us

t ⌫ K(t)= 0

2 2 2 2 2 t @ v 2 @ v @ v 1 @ 1 @v = c 2 + ✏a 2 = 2 ⌫ ds (3.23) @t @x @x ⇢L (1 ⌫) @x (t s) @s Z1 +2µ where c = , a = 3 c2 + l ⇢ 2 ⇢L r L Now we introduce new coordinates

x ⇠ = t , and ⌧ = ✏t c

Expressing 3.23 in terms of ⇠ and ⌧ to first order ✏ and ,weget

@v @1+⌫v @v = d + v (3.24) @⌧ @⇠1+⌫ @⇠

The time fractional generalized Burger’s equation does not seem to have been derived in such a manner as the space fractional version, but rather fit to problems

29 on the basis of the memory property of the fractional derivative [24, 32]. Fractional

Burger’s equation is particularly useful in viscoelastic or fatiguing systems [10, 11, 24].

For propagating nonlinear acoustic waves through a gas filled pipe, shallow water waves and waves in bubbling liquids the following time fractional generalized Burger’s equation has been used [1, 3, 4]

@2u(x, t) @u(x, t) D↵u(x, y)=d u(x, t)p (3.25) t @x2 @x

Li, Zhang, and Ran developed a linear, implicit finite di↵erence scheme to solve the above time fractional generalized Burger’s equation, and found the following results

Figure 3.3: Li, Zhang, and Ran’s numerical solution for 3.25 with p =2,↵ = .25 and d = .05.

30 Figure 3.4: Li, Zhang, and Ran’s numerical solution for 3.25 with p =2,↵ = .75 and d = .05.

3.5 Time Fractional Generalized Burger’s Equa-

tion for non-Newtonian Fluids

To develop my own Burger’s type equation suitable for bubbling, non-Newtonian

fluids, I combined previously studied models with the necessary properties. I began with the time fractional generalized Burger’s equation above, as it has been used to model propagating waves in bubbling liquids.

Recall that the Burger’s equation for non-Newtonian fluids takes the form

⌘ 1 @u @u µ @ @u @u + u = @t @x ⇢ @x @x @x !

31 Note that only the di↵usion term di↵ers from the classical Burger’s equation, sug- gesting that this adjustment would make any generalized Burger’s equation suitable ⌘ 1 @ @u @u @2u(x, t) for power law fluids. Now, substituting for in 3.25, we @x @x @x @x2 ! arrive at

⌘ 1 µ @ @u(x, t) @u(x, t) @u(x, t) D↵u(x, y)= u(x, t)p (3.26) t ⇢ @x @x @x @x " # a generalized Burger’s type equation suitable for non-Newtonian, bubbling fluids.

Note that this equation is more nonlinear than the time fractional generalized Burger’s equation, and potentially more nonlinear than even the existing Burger’s equation for non-Newtonian fluids, making its implementation a delicate exercise.

32 Chapter 4

The Rayleigh-Plesset Equation

4.1 In Newtonian Fluids

In this chapter, I will explore] ] the Rayleigh-Plesset equation and how it must be modified to work with non-Newtonian fluids. I will derive it from conversation equa- tions and use it to discuss bubble growth.

The classic Rayleigh-Plesset equation describes the changing radius of a pulsating, spherical vapor bubble in a Newtonian, incompressible liquid, under a time varying pressure field [14]. It can be derived from the Navier-Stokes equation, which describes the motion of viscous fluids, using kinematic and dynamic boundary conditions and conservation of mass at the bubble’s surface.

Several authors have provided an analytical analysis for the growth of a single bubble including Plesset and Zwick [56], Prosperetti and Plesset [57], and Hesih

33 [58], while Mohammadein et al [55, 59] and others have studied problems around the growth of a single bubble in Newtonian fluids. Yang and Yeh [?]pioneeredthe theoretical work with bubble dynamics in viscous fluids. Arefmanesh et al [61] built on potential theory to develop an accurate numerical technique for studying di↵usion- induced bubble formation in viscous fluids. Chhabra and Dhyinga [62] worked o↵

Carreau fluid models to study single fluid spheres, and Chhabra et al [63, 64, 65, 66, 67] studied drops and bubbles in CMC solutions based on power-law models, while others explored problems surrounding the growth of a single bubble in non-Newtonian Fluids.

[16]

The Navier-Stokes Equations are arguably the most fundamental set of equations in fluid dynamics, comparable to conservation of momentum. It describes an infinites- imal fluid volume in a viscous, compressible fluid, making it applicable to a broad range of fluid types [14]. For a Newtonian fluid in spherical coordinates, we use

1 @p @u @u 1 @ @u 2u = + u ⌫ r2 (4.1) ⇢ @r @t @r L r2 @r @r r2 L  ✓ ◆

where all terms are defined as in 3.1. We find that conservation of mass gives us

F (t) u(r, t)= (4.2) r2

where F (t) is some function in time. Applying a zero mass transport relation at the bubble’s surface

34 dR(t) dR(t) u(R, t)= = F (t)=R2(t) (4.3) dt ) dt

Note that:

@u(r, t) 2F (t) = @r r3 @u(r, t) F (t) = 0 @t r2

And

1 @ @u 2u 1 @ 2F (t) 2 F (t) r2 = r2 r2 @r @r r2 r2 @r r3 r2 r2 ✓ ◆ ✓ ◆ 1 @ 2F (t) 2F (t) = r2 @r r r4 ✓ ◆ 1 2F (t) 2F (t) = r2 r2 r4 ✓ ◆ =0

So, using this and putting 4.2 into the Navier-Stokes equation we get

1 @p F 0(t) F (t) 2F (t) = 2 + 2 3 (4.4) ⇢L @r r r r

35 Integrating with respect to r gives

2 p p F 0(t) F (t) 1 = 4 (4.5) ⇢L r 2r

Considering p,weknowthattheforceperunitareainther-direction is

2S rr + pB (4.6) r=R R Which must be zero due to the mass transport condition. And, as = p + rr @u µ ,weget L @r 2S @u p = pB p + µL (4.7) r=R R @r This yields 2 pB p F 0(t) F (t) 2S µL @u 1 = 4 + + (4.8) ⇢L r 2r ⇢LR ⇢L @r

And now, using 4.3, we get the classical Rayleigh-Plesset equation [12]:

2 2 pB(t) p (t) d R 3 dR 4⌫L dR 2 1 = R + + + (4.9) ⇢ dt2 2 dt R dt ⇢ R L ✓ ◆ L

For a bubble stretched to just past equilibrium with no outside pressure field, this gives the following simulation

36 10-5 8.8

8.6

8.4

8.2

8

7.8 Bubble Radius

7.6

7.4 0 0.5 1 1.5 2 2.5 3 3.5 4 Time 10-4

Figure 4.1: Bubble stretched to 1.1R0 and governed by the Rayleigh-Plesset equa- tion with no outside forcing. Note the damped oscillations as the bubble returns to equilibrium.

2(3 1) 3p 1 Adding a sinusoidal acoustic field with frequency 3 + 2 ,weget s ⇢LR0 ⇢LR0 the following.

37 10-5 8.8

8.6

8.4

8.2

8

7.8 Bubble Radius 7.6

7.4

7.2 0 0.5 1 1.5 2 2.5 3 3.5 4 Time 10-4

Figure 4.2: Bubble stretched to 1.1R0 and governed by the Rayleigh-Plesset equation with sinusoidal outside forcing. The early behavior, from t =0tot =1e 4, is dominated by the system’s tendency toward equilibrium with the driving force determining the radius after t =1.5e 4seconds,withatransitionalregionfrom t =1e 4tot =1.5e 4.

By manipulating the p term in 5.8, you can model bubble growth and collapse in 1 Newtonian fluids in acoustic fields. Fields can be chosen to show a variety of behavior including bubble stability and cavitation [12, 13, 14].

4.2 In Non-Newtonian Fluids

More recently, Mohammadein and Abu-Bakr built on this work to derive a similar equation to apply to shear-thickening, non-Newtonian fluid from conservation equa- tions. Assuming a single spherically symmetric gas bubble growing between two finite

38 radii, R0 and Rm, in an incompressible liquid, they combined mass, momentum and pressure-balance conservation equations to derive the di↵erential equation for the ra- dius. They also neglected gravity, gas pressure change, and any inconsistency in the pressure inside the bubble [16].

The mass equation u(r, t) = 0 has the following solution for a spherically r· symmetric gas bubble

R2(t)R˙ (t) u(r, t)=" r2

For momentum balance, Mohammadein and Abu-Bakr modified the classical equa- tion to account for power law, non-Newtonian shearing stress as follows

3 2 pL p 1 1 R(t)R¨(t)+ R˙ (t)= 1 ⌧dr (4.10) 2 ⇢ ⇢ r· L L ZR

and for pressure balance they used

2 pL = pB ⌧ R(t) r=R(t) Using the spherical symmetry for the divergence of ⌧,wecansay

1 1 2⌧ @⌧ ⌧dr = + dr r· r @r ZR ZR ✓ ◆

39 Combining the second two equations gives a non-Newtonian equivalent of the

Rayleigh-Plesset equation

3 2 1 2 3 1 ⌧rr RR¨ + R˙ = pB p + dr (4.11) 2 ⇢ 1 R ⇢ r L ✓ ◆ L ZR

where ⌧rr is the radial component of the shear stress. Note that all of the as- sumptions made to apply this equation to a shear-thickening fluid also apply to a shear-thinning fluid, indicating that this equation should hold for those cases.

4.3 Simulations

I used this di↵erential equation in MatLab to model bubble growth and collapse for a variety of power-law fluids in di↵erent acoustic fields. I was able to verify

Mohammadein and Abu-Bakr’s results for shear thickening fluids.

4.3.1 Shear-Thinning Fluids

Comparing growth in shear-thickening fluids to Newtonian fluids, it is possible to extrapolate how a shear-thinning fluid might behave in similar circumstances, and my

MatLab trials were consistent with this. In fact, the model derived by Mohammadein and Abu-Bakr ran more eciently for more values of 1

40 10-5 8.5

8

7.5

7 0 1 2 3 4 5 6 10-4

Figure2 4.3: Bubble behavior in a power law fluid with n =1.1withinitialradius

90%1.5 of the equilibrium radius.

1

0.5 10-5 8.5 0

-0.5 8 -1 0 1 2 3 4 5 6 -4 7.5 10

7 0 1 2 3 4 5 6 10-4

Figure1.5 4.4: Bubble behavior in a power law fluid with n =1.2withinitialradius 90%1 of the equilibrium radius.

0.5

10-5 8.20

-0.58

7.8-1 0 1 2 3 4 5 6 -4 7.6 10

7.4

7.2 0 1 2 3 4 5 6 10-4

Figure0.4 4.5: Bubble behavior in a power law fluid with n =1.5withinitialradius 90% of the equilibrium radius. 0.3

0.2 41

0.1

0 0 1 2 3 4 5 6 10-4 4.3.2 Shear-Thickening Fluids

10-5 9

8.5

8

7.5

7 0 1 2 3 4 5 6 10-4

Figure2 4.6: Bubble behavior in a power law fluid with n =0.5withinitialradius 90% of the equilibrium radius. 1

0

-1

-2 0 1 2 3 4 5 6 10-4

42 Chapter 5

Bubble Cavitation: Formation and

Collapse

This chapter will consider how bubbles are formed, the di↵usive processes involved, and the types of general conditions necessary for cavitation.

One of the more interesting problems in bubble dynamics is that of bubble cavita- tion, which can be defined as “the process of nucleation in a liquid when the pressure falls below the vapor pressure.” [13] This collapse can cause noise, shock waves, in- tense heat and, in special circumstances, flashes of light. It is an omnipresent problem in practical fluid mechanics, damaging all types of moving parts from propellers and pumps to dolphin’s tails. This damage is not limited to moving parts. Cavitation shock waves are known to cause surprising amounts of damage to solid surfaces near

43 the collapsing bubbles.[13]

5.1 Bubble Formation

Bubble formation is a two stage process, the first stage is to find weaknesses in the

fluid, called seed nuclei, where bubbles can potentially form. The number, size and distribution of the seed nuclei is determined by fluid properties like density, viscosity, and temperature. The second stage is the expansion of these seed nuclei by pressure change into full bubbles. The amount of pressure required to do this depends on the size of the seed nuclei as well as fluid properties like hydrostatic pressure and surface tension.

5.1.1 Di↵usion of Seed Nulcei

The first step of bubble formation is predicting the distribution of seed nuclei in the fluid. In a perfectly pure liquid seed nuclei can be thought of as microscopic voids or gas pockets, pre-existing in the liquid [14, 13], that appear as a result of the

Brownian-type motion of the liquid particles. The stochastic nature of the particle motion creates natural density changes on an atomic scale, and a rapidly changing pressure field can then rupture the liquid at this location of low tensile strength, re- sulting in a vapor bubble. This process is called homogeneous nucleation.

Asecondtypeofseedingthatcanleadtobubbleformationismicron-sizedpock-

44 ets, or microbubbles, of contaminate gas. This can be more practical to produce bubbles in an experimental setting, but the bubble behavior will depend on the type and quantity of the contaminate gas. While this method is fascinating in its own right, the treatment of the contaminate gas in the resulting bubble dynamics places it outside the scope of my current research. I will be focusing on the first type of liquid weakness.

The dispersion of microscopic voids is more nuanced in the non-Newtonian case than in Newtonian fluids. As the fluid particles are no longer governed by a true

Brownian motion, the seed nuclei cannot accurately be modeled by a Gaussian pro- cess and thus classical di↵usion does not apply. Brownian motion is based on the principle of a random walk with equal step sizes. Since step size is dependent on vis- cosity of the fluid, and the apparent viscosity of a non-Newtonian fluid is dependent on the shearing stress and thus the acceleration of each particle, resulting in varying step sizes.

To examine di↵usion in a non-Newtonian fluid, we first consider the one-dimensional case: a single particle on a straight line can move either left or right, with equal prob- ability, so it came to a position x at a time t from the previous time step, from either the left or right, giving us the following

1 1 f(x, t)= f(x dx, t t)+ f(x + dx, t t)(5.1) 2 2

45 where d is the step size and can vary from time step to time step.

Taking the Taylor series expansion that is second order in space and first order in time of f(x dx, t t)andf(x + dx, t t), we get

1 1 f(x, t)= f(x dx, t t)+ f(x + dx, t t) 2 2 1 @f(x, t) 1 @2f(x, t) @f(x, t) = f(x, t) dx + (dx)2 t 2 @x 2 @x2 @t  1 @f(x, t) 1 @2f(x, t) @f(x, t) + f(x, t)+ dx + (dx)2 t 2 @x 2 @x2 @t  1 @2f(x, t) @f(x, t) = f(x, t)+ (dx)2 t 2 @x2 @t

So we now have

1 @2f(x, t) @f(x, t) 0= (dx)2 t 2 @x2 @t 1 @2f(x, t) d2x2 @f(x, t) 0= 2 @x2 t @t

by using the same parabolic scaling seen in the derivation of classical di↵usion, and passing to the limit as x2 0andt 0, we get ! !

@f @2f = D (5.2) @t @t2

46 where D = d2 is the drag coecient and is not always a constant, but may instead be a function of acceleration.

Following Peskir’s work on the di↵usion coecient [81], to find D we must first consider a system with N particles in a cylindrical vessel of fluid. The force on each particle is given by

Fp = mv where v is the particle’s velocity, is the frictional coecient of the fluid, and m is its mass. So the pressure p(x, t)isgivenby

p(x, t)=⌫kT where k is Boltzmann’s constant, T is temperature, and ⌫ is the average number of particles at position x at time t.sowesee

@p @⌫ = kT = ⌫F = m⌫v @x @x p using that @⌫ v⌫ = D @x we get that @⌫ @⌫ kT = m⌫v = mD @x @x

47 So this tells us kT kT = mD or D = m

Using the Stokes formula: 6⇡r = ⌘ m where r is the particles radius and ⌘ is the non-constant viscosity we get that

kT D = (5.3) 6⇡r⌘

Recall that ⌘ will be a function of shearing stress, which can be expressed as a function of acceleration of the particle. Work done in rheology shows that the rela- tionship between shearing stress and viscosity is approximately exponential for the magnitude of shearing stress seen in this application

Figure 5.1: The figure above from ”The Rheology Handbook” shows viscosity against shearing stress for Newtonian, shear-thinning and shear-thickening fluids. The solid line is a Newtonian fluid, the dotted line is a shear-thickening fluid, and the dashed line is a shear-thinning fluid.

48 Using this approximation, we were able to model di↵usion iteratively in both shear-thickening and shear-thinning fluids by calculating a new step size from the drag at each time step and compare the results with Brownian motion.

(a) Di↵usion process for shear-thickening (b) Di↵usion process for shear-thinning non-Newtonian fluid non-Newtonian fluid

Figure 5.2: Non-Newtonian Di↵usion Processes

Figure 5.3: Standard Brownian motion for a Newtonian fluid.

While some di↵erences in these di↵usion processes are readily apparent, projecting

49 these to a 2 dimensional plane more clearly illustrates the di↵erences in step size for the non-Newtonian fluids

Figure 5.4: Di↵usion for a shear-thickening fluid on the left, and its two-dimensional projection on the right.

The increased apparent viscosity with increased application of shearing stress that defines shear-thickening fluids results in easier movement for slow-moving particles, making the di↵usion process even less prone towards clustering than Brownian mo- tion, but resistant to fast-accelerating particles.

Figure 5.5: The left figure is di↵usion for a shear-thinning fluid next to its two- dimensional projection on the right.

50 Here, we see the opposite pattern, with clustering as slower particles experience higher apparent viscosity and large jumps for fast accelerating particles. This clearly demonstrates the lessening apparent viscosity and shearing stress increase that defines ashear-thinningfluid.

5.1.2 Initial Radii

For homogeneous nucleation, the intermolecular forces that tend to prevent the for- mation of large voids in the fluid are overcome by swift pressure changes and bubbles will form governed by [13]

2 pB p = (5.4) 1 R

where pB is the pressure inside the bubble, p is the surrounding liquid, is 1 the surface tension and R is the bubble radius. For a vapor bubble, the interior pressure will be given by the saturated vapor pressure, pV ,whichisafunctionof temperature and intrinsic to the liquid used [13]. When the outside pressure falls below p 2 ,theradiusofthemicroscopicvoidswillincrease,rupturingtheliquid V R to form a macroscopic bubble with equilibrium radius R0

2 R = (5.5) 0 p p V L where pL is the pressure of the liquid immediately outside the bubble [14]. However, not every void can necessarily become a bubble, and certainly not every void will

51 become a stable bubble [14]. The seed nucleus must be larger than a critical radius,

Rc, in order to experience the sharp growth required for cavitation. For isothermal conditions, and hydrostatic pressure po,thecriticalradiusbecomes

3R3 2 R = 0 p p + (5.6) c 2 o V R s ✓ 0 ◆

This critical radius was originally found by Blake in 1949, and later by Neppiras and Noltingk while working on the the relationship of radius and pressure required to hold a bubble in a static equilibrium and is sometimes called the Blake critical radius[14, 13]. All bubbles with radius R0 >Rc must be unstable, while bubbles with radius R0

5.2 Bubble Collapse

More attention has been paid to researching the collapsing bubble than the formation of a cavitation bubble. In certain conditions which will be discussed in this section, cavitation bubbles collapse so violently that they produce shock waves intense heat, and even occasionally bursts of light. It is an omnipresent problem in practical fluid mechanics, damaging all types of moving parts from propellers and pumps to dol-

52 phin’s tails. This damage is not limited to moving parts. Cavitation shock waves are known to cause surprising amounts of damage to solid surfaces near the collapsing bubbles [13].

The dynamics of an established bubble under a pressure field in a Newtonian fluid are modeled quite well by the Rayleigh-Plesset equation

2 2 pB(t) p (t) d R 3 dR 4⌫L dR 2 1 = R + + + ⇢ dt2 2 dt R dt ⇢ R L ✓ ◆ L

When considering the collapsing bubble, one must consider the properties of both the contents of the bubble and the properties of the liquid. It is reasonable to assume that the bubble contains some quantity of non-condensible gas, with partial pressure

pG0 [12, 14]. As a violent collapse that we wish to examine happens very quickly, one should assume a zero-mass transport at the bubble’s surface. We also assume that the system is thermally isolated, or adiabatic, and the gas in the bubble obeys

R 3 p = p 0 (5.7) G G0 R ✓ ◆

where pG is the pressure of the gas. Under these assumptions, the Rayleigh-Plesset equation becomes

3 2 2 pB(t) p (t) pG0 R0 d R 3 dR 4⌫L dR 2 1 + = R + + + (5.8) ⇢ ⇢ R dt2 2 dt R dt ⇢ R L L ✓ ◆ ✓ ◆ L 53 Atypicalsolutiontothisequationunderasinusoidalacousticfieldfoundby

Brennan [13] clearly illustrates the implosive nature of this collapse.

Figure 5.6: Figure 4.2 from Brennan’s ”Fundamentals of Multiphase Flows”. ”The nucleus of radius, R0, enters a low-pressure region at a dimensionless time of 0 and is convected back to the original pressure at a dimensionless time of 500. The low-pressure region is sinusoidal and symmetric about 250.” [12]

However, as many bubbles are stable under a oscillating pressure field, further conditions must be placed on the bubble to see this type of collapse. Recall from the

Blake critical radius that any bubble with R0

This allows us to limit bubbles capable of this type of collapse to bubbles with radius

54 3R3 2 R = 0 p p + (5.9) c 2 L V R s ✓ 0 ◆ as bubbles with R0

55 Chapter 6

Non-Spherical Bubbles

This chapter will analyze non- spherical bubble stability under di↵erent axisymmetric shape modes formed by Legendre polynomials.

Stable bubbles naturally tend toward a spherical shape mode due to the surface tension. Bubble that are initially non-spherical will either find a spherical equilib- rium or break apart. This project continues Dr Anthony Harkin’s doctoral research modeling non-spherical bubbles using an equator and pole evaluation to capture the non-spherical dynamics [73]. This model is generated from the velocity potential of a bubble consisting of a breathing mode and an n th Legendre polynomial shape mode. Then the kinematic equations for motion are applied at the equator and north-pole, with all quantities evaluated exactly at those two points. The model is completed by imposing the dynamic boundary condition on the north-pole and equator. This yielded a system of di↵erential equations for radius, shape perturbations, as well as

56 two other variables from the velocity potential [73]. To analyze this model, I worked with Dr. Harkin to develop a MatLab script that solves these numerically and plots the radius and shape perturbations.

6.1 The Equator-Pole Model

Following Harkin’s derivation, the velocity potential for an axisymmetric bubble with abreathingmodeandann th shape mode in spherical coordinates is

A(t) B(t) (r, ✓)= + P (cos(✓)) (6.1) r rn+1 n

and the bubble surface is given by

F (r, ✓, t)=R(t)[1 + ✏(t)Pn(cos(✓))] (6.2)

where P (cos(✓)) is the n th Legendre polynomial, which gives n

0, for odd n 8 k > ( 1) n Pn(cos(✓)) = ↵n = > for even n > 2n k <> 1for✓ =0 > > n > where k = . : 2 Note that for the radius at the north pole and equator, this yields

57 R(t)(1 + ↵n✏)attheequator r = 8 > <> R(t)(1 + ✏)atthenorthpole > Using this, we can rewrite:> the bubble surface equation to get

F (r, ✓, t)=R(t)[1 + ✏(t)P (cos(✓))] r =0 (6.3) n

We apply the kinematic boundary condition, with u(r, ✓)astheusualfluidvelocity, to see

@F + u F =0 (6.4) @t ·r

Looking at each term individually, we see

@F = R˙ (t)+R˙ (t)✏(t)P (cos(✓)) + R(t)˙✏P (cos(✓)) and @t n n 1 d F = 1+ R(t)✏(t) P (cos(✓)) r r d✓ n

u(r, ✓)=urrˆ + u✓✓ˆ A(t) (n +1)B(t) B(t) d = + P (cos✓) + P (cos✓) ✓ˆ r2 rn+2 n rn+2 d✓ n  

Inserting these into 6.4 yields

✏u R(t) d R˙ (t)+(˙✏R + ✏R˙ )P (cos(✓)) u + ✓ P (cos(✓)) = 0 (6.5) n r r d✓ n

58 Evaluating this at the equator and north pole, respectively, we see:

2 2 A(t) ↵n(n +1)B(t) n ↵n 1✏B(t) ˙ R(t)(1+↵n✏)+˙✏↵R(t)= 2 2 n+2 n+2 n+2 n+3 R(t) (1 + ↵n✏) R(t) (1 + ↵n✏) R(r) (1 + ↵n✏) (6.6)

A(t) (n +1)B(t) R˙ (1 + ✏)+˙✏R(t)= (6.7) R(t)2(1 + ✏)2 R(t)n+2(1 + ✏)n+2

To apply the dynamic boundary conditions, we must first consider the pressure balance equation given by

@2 p +2u = p +2H + p (6.8) B @n2 L visc

Where is the surface tension, u is the liquid velocity, H is the mean curvature of the bubble surface, and pvisc is a pressure correction term that accounts for the viscous boundary layer around the bubble, and is given by

2n(n +2)µB(t) p = P (cos(✓)) visc rn+3 n

And for even n,themeancurvatureisgivenby

R H = 2+(n2 + n +2)✏ P (cos(✓)) (6.9) 2r2 n ✓ ◆ Examining Bernoulli’s equation, we get the following for the pressure of the liquid

59 directly outside the bubble:

@ ⇢ p = p1(t) ⇢ (6.10) L @t 2r ·r

Evaluating 6.8 with 6.9 6.10 and 6.1 at the equator and north pole, respectively, we get the following coupled di↵erential equations for A˙(t)andB˙ (t)

2 A˙(t) B˙ (t)↵n p1(t) pB 1 B(t)↵n 1 + = + R(t)(1 + ↵ ✏) R(t)n+1(1 + ↵ ✏)n+1 ⇢ 2 R(t)n+2(1 + ↵ ✏)n+2 n n  n 1 A(t) (n +1)B(t)↵ 2 + n 2 R(t)2(1 + ↵ ✏)2 R(t)n+2(1 + ↵ ✏)n+2  n n 2 2 2 2 2(1 + ↵n✏) +3↵n 1n ✏ + ↵nn✏(n +1)(1+↵n✏) + 2 2 2 2 3/2 ⇥ ⇢R(t) (1 + ↵n✏) + ↵n 1n ✏ ⇤ 2µ 2A(t) ↵ (2n +1)(n +2)B(t) ⇥ + n ⇤ ⇢ R(t)3(1 + ↵ ✏)3 R(t)n+3(1 + ↵ ✏)n+3 L  n n (6.11)

2 A˙(t) B˙ (t) p1(t) p (2 + (n + n +2)✏ + = B + R(t)(1 + ✏) R(t)n+1(1 + ✏)n+1 ⇢ ⇢R(t)(1 + ✏)2 1 A(t) (n +1)B(t) 2 + (6.12) 2 R(t)2(1 + ✏) R(t)n+2(1 + ✏)n+2  2µ 2A(t) (2n +1)(n +2)B(t) + ⇢ R(t)3(1 + ✏)3 R(t)n+3(1 + ✏)n+3 L 

Note that the shape modes obtained from the Orgus-Prosperetti surface transport theorem can be recovered by linearizing 6.6, 6.7, 6.11, 6.12 [75].

60 6.2 Uncoupling the Di↵erential Equations

Note that 6.6, 6.7, 6.11, 6.12 are all linear in R,˙ ✏˙, A˙ and B˙ ,andtheperiodicnature of the system lends itself to Floquet analysis. To make this analysis less cumbersome, we define the right hand sides of the above equations as follows

2 2 A(t) ↵n(n +1)B(t) n ↵n 1✏B(t) feq = 2 2 n+2 n+2 n+2 n+3 (6.13) R(t) (1 + ↵n✏) R(t) (1 + ↵n✏) R(t) (1 + ↵n✏) A(t) (n +1)B(t) f = (6.14) np R(t)2(1 + ✏)2 R(t)n+2(1 + ✏)n+2 2 p1(t) pB 1 B(t)↵n 1 d = + eq ⇢ 2 R(t)n+2(1 + ↵ ✏)n+2  n 1 A(t) (n +1)B(t)↵ 2 + n 2 R(t)2(1 + ↵ ✏)2 R(t)n+2(1 + ↵ ✏)n+2  n n 2 2 2 2 2(1 + ↵n✏) +3↵n 1n ✏ + ↵nn✏(n +1)(1+↵n✏) + 2 2 2 2 3/2 ⇥ ⇢R(t) (1 + ↵n✏) + ↵n 1n ✏ ⇤ 2µ 2A(t) ↵ (2n +1)(n +2)B(t) ⇥ + n ⇤ (6.15) ⇢ R(t)3(1 + ↵ ✏)3 R(t)n+3(1 + ↵ ✏)n+3 L  n n 2 p1(t) p (2 + (n + n +2))✏ d = B + np ⇢ ⇢R(t)(1 + ✏)2 1 A(t) (n +1)B(t) 2 + 2 R(t)2(1 + ✏) R(t)n+2(1 + ✏)n+2  2µ 2A(t) ↵ (2n +1)(n +2)B(t) + n (6.16) ⇢ R(t)3(1 + ↵ ✏)3 R(t)n+3(1 + ↵ ✏)n+3 L  n n

Then we form the following matrix equations

(1 + ↵n✏(t)) ↵nR(t) R˙ (t) feq 2 3 2 3 = 2 3 (6.17) 6 (1 + ✏(t)) R(t) 7 6 ✏˙(t) 7 6fnp7 6 7 6 7 6 7 4 5 4 5 4 5 61 And

1 1 ˙ R(t)(1 + ↵ ✏(t)) Rn+1(t)(1 + ↵ ✏)n+1 A(t) deq 2 n n 3 2 3 = 2 3 (6.18) 1 1 ˙ 6 7 B(t) dnp 6 R(t)(1 + ✏(t)) Rn+1(t)(1 + ✏)n+1 7 6 7 6 7 6 7 6 7 6 7 4 5 4 5 4 5 Assuming that both matrices are non-singular, we write

1 1 (1 + ↵n✏(t)) ↵nR(t) n+1 n+1 R(t)(1 + ↵n✏(t)) R (t)(1 + ↵n✏) MR✏ = 2 3 ,MAB = 2 3 1 1 (1 + ✏(t)) R(t) 6 7 6 7 6 R(t)(1 + ✏(t)) Rn+1(t)(1 + ✏)n+1 7 6 7 6 7 4 5 4 5 Which gives closed form inverses

R(t) ↵nR(t) 1 1 MR✏ = 2 3 Det(MR✏) 6 (1 + ✏(t)) (1 + ↵n✏(t))7 6 7 4 5 and

1 1 1 n+1 n+1 n+1 n+1 1 2R (t)(1 + ✏) R (t)(1 + ↵n✏) 3 MAB = Det(MAB) 1 1 6 7 6 R(t)(1 + ✏(t)) R(t)(1 + ↵ ✏(t)) 7 6 n 7 4 5 Using this to rearrange 6.17 and 6.18, we get the following systems of di↵erential equations

˙ R(t) 1 R(t) ↵nR(t) feq 2 3 = 2 3 2 3 (6.19) Det(MR✏) 6 ✏˙(t) 7 6 (1 + ✏(t)) (1 + ↵n✏(t))7 6fnp7 6 7 6 7 6 7 4 5 4 5 4 5

62 and

1 1 ˙ A(t) 1 Rn+1(t)(1 + ✏)n+1 Rn+1(t)(1 + ↵ ✏)n+1 deq 2 3 = 2 n 3 2 3 (6.20) Det(M ) ˙ AB 1 1 B(t) 6 7 dnp 6 7 6 R(t)(1 + ✏(t)) R(t)(1 + ↵ ✏(t)) 7 6 7 6 7 6 n 7 6 7 4 5 4 5 4 5 6.3 Nondimensionalizing the Di↵erential Equations

We then went to dimensionless variables via the following transformations

˜ R(t) t ˜ A(t)T ˜ B(t)T R(t)= ,⌧= , A(t)= 3 , B(t)= n+3 (6.21) R0 T R0 R0

Where R0 is the equilibrium radius and T is the natural frequency of the bubble.

These transformations lead to the following identities

dR˜ R d✏ 1 dA˜ R3 dB˜ Rn+3 R˙ (t)= 0 , ✏˙ = , A˙(t)= 0 , B˙ (t)= 0 (6.22) d⌧ T d⌧ T d⌧ T 2 d⌧ T 2

First we define

˜ ˜ 2 2 ˜ A(t)R0 ↵n(n +1)B(t)R0 n ↵n 1✏B(t)R0 f˜eq = (6.23) 2 2 n+2 n+2 n+2 n+3 T R˜ (t)(1 + ↵n✏) T R˜(t) (1 + ↵n✏) T R˜(t) (1 + ↵n✏)

A˜(t)R0 (n +1)B˜(t)R0 f˜np = (6.24) T R˜(t)2(1 + ✏)2 T R˜(t)n+2(1 + ✏)n+2

63 To nondimensionalize deq and dnp,wemustadditionallyconsidertheoverallunits

p1(t) p ✏ of B and . ⇢ ⇢R(t)(1 + ✏)2

kg kg kg m [P ]= , [⇢]= , []= ms2 m3 s2

m2 giving both terms units of .Sincekg do not appear in the dimensions of the term, s2 we use transformations to give dimensionless terms rather than true dimensionless variables.

R0 ⇢˜ p˜ p˜B 1 ˜ 2 = , 3 = ⇢, 2 = p , 2 = pB T R0 R0T 1 R0T

This gives us

2 2 T p˜1(t) p˜B 1 B˜(t)R0↵n 1 d˜eq = + 2 ˜ n+2 n+2 R0 ⇢˜ 2 "T R(t) (1 + ↵n✏) # 2 1 A˜(t)R (n +1)B˜(t)↵ R 0 + n 0 ˜ 2 2 ˜ n+2 n+2 2 "T R(t) (1 + ↵n✏) T R(t) (1 + ↵n✏) # 2 2 2 2 2 ˜T 2(1 + ↵n✏) +3↵n 1n ✏ + ↵nn✏(n +1)(1+↵n✏) + ˜ 3 2 2 2 2 3/2 ⇥ ⇢˜R(t)R0 (1 + ↵n✏) + ↵n 1n ✏ ⇤ ˜ ˜ 2µ 2A(t) ⇥ ↵n(2n +1)(n +2)⇤ B(t) + (6.25) ˜ 3 3 ˜ n+3 n+3 ⇢L "T R(t) (1 + ↵n✏) T R(t) (1 + ↵n✏) #

(6.26)

64 2 2 2 T p˜1(t) p˜B ˜T (2 + (n + n +2))✏ d˜np = + 2 ˜ 3 2 R0 ⇢˜ ⇢˜R(t)R0(1 + ✏) 2 1 A(t)R (n +1)B˜(t)R 0 + 0 2 "T R˜(t)2(1 + ✏) T R˜(t)n+2(1 + ✏)n+2 # 2µ 2A˜(t) (2n +1)(n +2)B˜(t) + (6.27) ⇢L "T R˜(t)3(1 + ✏)3 T R˜(t)n+3(1 + ✏)n+3 #

To complete the nondimensionalization, we only need to treat the the left hand side of the di↵erential equations.

dR˜(⌧) R d✏ 1 R˙ (t)(1 + ↵ ✏)+˙✏(t)↵ R(t)= o (1 + ↵ ✏)+ ↵ R R˜(⌧)(6.28) n n d⌧ T n d⌧ T n 0 dR˜(⌧) R d✏ 1 R˙ (t)(1 + ✏)+˙✏(t)R(t)= o (1 + ✏)+ R R˜(⌧)(6.29) d⌧ T d⌧ T 0

65 dA˜ 3 dB˜ n+3 A˙(t) B˙ (t)↵ R ↵nR + n = d⌧ 0 + d⌧ 0 n+1 n+1 2 ˜ 2 n+1 ˜n+1 n+1 R(t)(1 + ↵n✏) R(t) (1 + ↵n✏) T RR0(1 + ↵n✏) T R0 R (1 + ↵n✏) dA˜ R2 dB˜ ↵ R2 = 0 + n 0 2 2 n+1 n+1 d⌧ T R˜(1 + ↵n✏) d⌧ T R˜ (1 + ↵n✏) 2µ 2A˜(t) ↵ (2n +1)(n +2)B˜(t) + n ˜ 3 3 ˜ n+3 n+3 ⇢L "T R(t) (1 + ↵n✏) T R(t) (1 + ↵n✏) #

(6.30)

˜ ˜ A˙(t) B˙ (t) dA R3 dB Rn+3 + = d⌧ 0 + d⌧ 0 n+1 n+1 2 ˜ 2 n+1 ˜n+1 n+1 R(t)(1 + ✏) R(t) (1 + ✏) T RR0(1 + ✏) T R0 R (1 + ✏) dA˜ R2 dB˜ R2 = 0 + 0 d⌧ T 2R˜(1 + ✏) d⌧ T 2R˜n+1(1 + ✏)n+1 2µ 2A˜(t) (2n +1)(n +2)B˜(t) + (6.31) ⇢L "T R˜(t)3(1 + ✏)3 T R˜(t)n+3(1 + ✏)n+3 #

6.4 Numerical experiments

I ran numerical simulations of this system and saw the following

66 Figure 6.1: Evolution of bubble radius and size of the deformation with initial radius 2 equal to R0, the equilibrium radius, and initial deformation size 10 R0.Thisfluid has the same properties as water.

67 Figure 6.2: Evolution of bubble radius and size of the deformation with initial radius 1 equal to R0, the equilibrium radius, and initial deformation size 10 R0.Thisfluid has the same properties as water.

68 Figure 6.3: Evolution of bubble radius and size of the deformation with initial radius 3 1.05R0 where R0 is the equilibrium radius and initial deformation size 10 R0.The theoretical fluid used is 100 times more viscous than water, with the same surface tension.

69 Figure 6.4: Evolution of bubble radius and size of the deformation with initial radius 2 1.05R0 where R0 is the equilibrium radius and initial deformation size 10 R0.The theoretical fluid used is 100 times more viscous than water, with the same surface tension.

70 Figure 6.5: Evolution of bubble radius and size of the deformation with initial radius 3 1.1R0 where R0 is the equilibrium radius and initial deformation size 10 R0.The theoretical fluid used is 5 times more viscous than water, with the same surface tension.

71 Figure 6.6: Evolution of bubble radius and size of the deformation with initial radius 1 1.1R0 where R0 is the equilibrium radius and initial deformation size 10 R0.The theoretical fluid used is 5 times more viscous than water, with the same surface tension.

72 Figure 6.7: Evolution of bubble radius and size of the deformation with initial radius 2 1.05R0 where R0 is the equilibrium radius and initial deformation size 10 R0.The theoretical fluid used is 10 times more viscous than water, with the same surface tension.

73 Figure 6.8: Evolution of bubble radius and size of the deformation with initial radius 2 1.2R0 where R0 is the equilibrium radius and initial deformation size 10 R0.The theoretical fluid used is 10 times more viscous than water, with the same surface tension.

74 Figure 6.9: Evolution of bubble radius and size of the deformation with initial radius 3 1.1R0 where R0 is the equilibrium radius and initial deformation size 10 R0.The theoretical fluid used is 5 times more viscous than water, with the same surface tension.

75 Figure 6.10: Evolution of bubble radius and size of the deformation with initial 1 radius 1.1R0 where R0 is the equilibrium radius and initial deformation size 10 R0. The theoretical fluid used is 5 times more viscous than water, with the same surface tension.

76 Chapter 7

Finite Di↵erence Scheme

Here I will derive and discuss the Finite Di↵erence scheme used to model this bubble formation and cavitation in non-Newtonian fluids. I will discuss its limitations and applications. I will present the results of my numerical simulations here.

7.1 Finite Di↵erence Schemes for Burger’s Equa-

tion

IamcombiningthetwoequationstoformatimefractionalBurgersequationthat can describe bubbling, non-Newtonian fluids. First, I will find a scheme to solve this new equation numerically, both with smooth and bubble-containing initial conditions, then I will add outside force and find specific conditions to create bubbles in an ini- tially smooth fluid.

77 Generalized time fractional Burger’s type equation

@2u(x, t) @u(x, t) D↵u(x, y)=d u(x, t)p t @x2 @t

Power-law non-Newtonian fluid Burger’s type equation

⌘ 1 @u(x, y) µ @ @u(x, t) @u(x, t) @u(x, t) = u(x, t) @t ⇢ @x @x @x @t " # My equation

⌘ 1 µ @ @u(x, t) @u(x, t) @u(x, t) D↵u(x, y)= u(x, t)p t ⇢ @x @x @x @t " # 7.2 Deriving the Finite Di↵erence Scheme

In terms of putting this fractional derivative into a finite di↵erence scheme, weights with the ”rounded down” derivative are used since the traditional derivative ap- proximations are not easily modified for the fractional case. This means that the

0 <↵<1willhaveadi↵erentschemethanthe1<↵<2case.Thismakessense since 0 <↵<1isequivalenttosubdi↵usionandsubdispersionwhere1<↵<2 should model superdi↵usion and superdispersion. The scheme I am currently working with is for 0 <↵<1, and uses

78 n 1 ↵ n k 0 Dt u(x, t) uj (!n 1 k !n k)uj !n 1uj ⇡ Xk=1 1 ↵ 1 ↵ Where ! =(1+k) k k

The rest of the scheme I am working on right now is standard finite di↵erence, with one exception. Li, Zhang and Ran showed that by rewriting the advection term as @u 1 @u @ up = up + up+1 @x p +2 @x @x  you can linearize by taking up from the previous step and decrease computational time while preserving stability.

Now to create the rest of the scheme, we must look at the di↵usion term. I will be using a centered approximation for the partial derivative to match the scheme by

Li et al[1].

79 ⌘ 1 n n ⌘ 1 n n @ @u(x, t) @u(x, t) @ uj+1 uj 1 uj+1 uj 1 = @x @x @x @x 2h 2h " # " # n n ⌘ 1 n n @ u u u u j+1 j 1 n n j+1 j 1 = sign(uj+1 uj 1) @x 2h 2h " # n n ⌘ @ n n uj+1 uj 1 = sign(uj+1 uj 1) @x 2h  n n ⌘ n n @ uj+1 uj 1 = sign(uj+1 uj 1) @x 2h n n ⌘ 1 n n uj+1 uj 1 = sign(uj+1 uj 1)⌘ 2h

This goes to

⌘ 1 2 n n ⌘ 1 n n n @u(x, t) @ u(x, t) uj+1 uj 1 uj+1 2uj + uj 1 ⌘ ⌘ @x @x2 ⇡ 2h h2 Combining this with the scheme from [1], we get:

↵ ⌘ 1 n n 1 k 0 ⌘⌫⌧ (2 ↵) n n uj k=1 a(n 1) k an k uj an 1uj = uj uj ⇢ x xx P ↵ ⌧ (2 ↵) n 1 p n n 1 p n 1 u u + u u p +2 j j x j j x h i

In order to turn this into a programmable finite di↵erence scheme, we need to isolate each un and make a linear equation. Note that all terms with n 1are j

80 essentially constants. We will go term by term.

n 1 n n n 1 n n n n n uj+1 uj 1 uj+1 2uj + uj 1 u u = j x j xx 2h h2 n 1 n 1 n 1 uj+1 uj 1 Let = Cn 1 2h Cn 1 n n n = uj+1 2uj + uj 1 h2

Which gives us

n n n 1 p n n 1 p n n 1 p n n 1 p n 1 n 1 p uj+1 uj 1 (uj+1 ) uj+1 (uj 1 ) uj 1 u u + u u = u + j j x j j x j 2h 2h

So

n n n n n n n 1 k 0 uj+1 2uj + uj 1 n 1 p uj+1 uj 1 uj k=1 a(n 1) k an k uj an 1uj = Cn 1 + uj h2 2h n 1 p n n 1 p n P (uj+1 ) uj+1 (uj 1 ) uj 1 + 2h

7.3 Numerical Experiments

Implementing this in MatLab, I applied it to a number of initial conditions for a variety of ↵ and ⌘ values for appropriate viscosities.

81 Figure 7.1: This is a shear-thinning fluid. For smooth initial conditions, the fluid velocity propagates smoothly toward an equilibrium solution. Note the bowing of the steady state for choices of ↵ further from 1.

82 Figure 7.2: For smooth initial conditions, the shear-thickening fluid velocity propa- gates smoothly toward an equilibrium solution. Note that the bowing of the steady state for choices of ↵ further from 1 does not seem to depend on fluid type, only ↵.

83 Figure 7.3: With step function initial conditions, we see bubbles occur under cer- tain conditions. For larger values of ↵,theshear-thickeningfluidresistsbreakingto form bubbles, but as ↵ decreases, the sharp changes in velocity indicative of bubble formation begin to occur.

84 Figure 7.4: For larger values of ↵,thereisagainresistancetofluidrupture,andthe graph only begins to feature the sharp changes in u indicative of bubble formation at ↵ =0.5.

85 Figure 7.5: For smaller ⌘,thepowerlawindex,weseethefluidbreakingforallvalues of ↵ rather than only for small alpha.

86 Figure 7.6: For this shear-thinning fluid, we see both increased resistance to fluid rupture and a slower decay to a steady state.

87 Figure 7.7: Here we see clear fluid rupture for all values of ↵ and for ↵>1, we see the break continue indicating the bubble’s continued existance.

88 Figure 7.8: Finally, we see as with ⌘ = .5, bubble formation for all values of ↵.We also see more bubbles staying formed.

89 Chapter 8

Future Work

Future work along this trajectory could develop several lines of research.

First, the finite di↵erence scheme given in Chapter 7 could be expanded several ways: an acoustic field could be added to examine the full cavitation process in non-

Newtonian fluids, the model could be expanded into 2- and 3- dimensions, and the bubble nuclei seeding from Chapter 5 could be incorporated into the initial condi- tions. Looking forward, this is a very rich field and should support a considerable amount of future reseach.

Second, looking further into cavitation, preliminary work suggests that adding noise into the pressure field holding a a single bubble may change the collapse behav- ior. Given the damaging nature of collapsing cavitation bubbles, this is an interesting, and perhaps important, area for future work.

90 Third, and lastly, future research on non-spherical bubbles might apply the work to non-Newtonian fluids or use the existing model to examine bubble behavior beyond average radius during the cavitation collapse process. Ultrasonic cavitation is a rela- tively new medical technique and developing a theoretical background for cavitation that applies to biological fluids could support this in the future.

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