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, λ = .125, β = .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.