Tài liệu Analytical and numerical analysis of lubrication coating flow models.

The following logos are available for general use. Link directly to these images or right click the image and "Save as..." Analytical and Numerical Analysis of http://www.cgu.edu/images/cgulogo/smallcgu.gif Lubrication Coating Flow Models http://www.cgu.edu/images/cgulogo/cgu124.gif http://www.cgu.edu/images/cgulogo/cgu178.gif Matthew Michal Institute of Mathematical Sciences Claremont Graduate University http://www.cgu.edu/images/cgulogo/cgulg296.gif © Copyright Matthew Michal, 2015 All rights reserved. http://www.cgu.edu/images/cgulogo/cgublk310.gif ProQuest Number: 3740967 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. ProQuest 3740967 Published by ProQuest LLC (2015). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 APPROVAL OF THE REVIEW COMMITTEE This dissertation has been duly read, reviewed, and critiqued by the Committee listed below, which hereby approves the manuscript of Matthew Michal as fulfilling the scope and quality requirements for meriting the degree of Doctor of Philosophy. Marina Chungunova, Chair Institute of Mathematical Sciences Claremont Graduate University Associate Professor Ali Nadim Institute of Mathematical Sciences Claremont Graduate University Professor Chiu-Yen Kao Department of Mathematical Sciences Claremont McKenna College Associate Professor Abstract Analytical and Numerical Analysis of Lubricating Coating Flow Models by Matthew Michal Claremont Graduate University: 2015 For flows on a flat surface, the form of the thin film equation for which Giamcomelli and Otto [48] showed the connection to the Hele-Shaw flow has solutions that match experimental results. However, for thin films on an incline, simulated results lack the instabilities evident in experimental flows. To induce this behavior, we replace the linearized curvature term uxxx with the exact definition of curvature in one dimension. First, we prove the existence of a solution for the thin film equation with a nonlinear curvature term and prove that the energy functional defined by Giacomelli and Otto is nonincreasing for this equation as well. We also prove the arclength dissipation property for both the thin film equation with the linearized curvature term as well as with the nonlinear curvature term. We compare the rates of dissipation for these equations as well as use a finite element method to compare these two rates with that of a Hele-Shaw flow. Due to the difficulties of utilizing fully implicit or explicit numerical schemes for the evolution of the thin film equation because of its nonlinearity, we show the convergence of an iterative scheme with frozen coefficients so that we can apply a “semi-implicit” simulation. We finish with a comparison of previous simulations of the thin film equation on an incline and those with a finite time rupture to new simulations with the nonlinear curvature term. The simulation for thin films on an incline does show some instabilities in the L∞ -norm, and the nonlinear curvature term modifies the evolution of the flow so that the positivity of solutions is preserved for larger time intervals. Acknowledgements My deepest thanks to my professors of mathematics: Dr. Marina Chugunova, my thesis advisor, for her invaluable guidance and infinite patience; Dr. Ali Nadim and Dr. Chiu-Yen Kao, for encouraging my study of computational methods for eigenvalue and spectral problems and for serving on my thesis committee; Dr. Chris Brownell, for mentoring me as an undergraduate mathematician and supporting my first forays into research; Dr. Steven Pauls and Dr. Terence Yi, for supporting my study of upper-level undergraduate mathematics by teaching classes not normally offered in the curricula; And Mrs. Melanie Castro, my high school calculus teacher, for encouraging my study of math and recognizing my gift long before I saw it. The completion of this degree and dissertation would not have been possible without the unending love and support of my family, especially that of my beautiful wife, Sarah. v Table of Contents Table of Contents vi List of Figures ix 1 Introduction 1 1.1 Dynamics of Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Hele-Shaw flows and the Lubrication Approximation . . . . . . . . . . . . . . . . . . 6 1.3 Higher order degenerate non-linear parabolic partial differential equations . . . . . . 8 1.4 Pseudospectral numerical methods and dealiasing techniques in fluid dynamics. . . . 11 2 Existence of nonnegative weak solutions in the lubrication thin film model with a nonlinear curvature term 2.1 13 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.3 Important Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Existence of a Nonnegative Weak Solution . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Dissipation of Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 The arclength dissipation property of positive solutions in the lubrication thin film model with a linearized curvature term 3.1 26 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.1 Bernis and Friedman’s Approximating Problems . . . . . . . . . . . . . . . . 26 3.1.2 Existence of a weak solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 vi 3.1.3 Bernis & Friedman’s Nonnegative Solutions . . . . . . . . . . . . . . . . . . . 31 3.2 Proof of the arclength dissipation property for positive solutions . . . . . . . . . . . 33 3.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4 Convergence of iterations for the frozen coefficents method 42 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Proof of Convergence of Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2.1 4.3 A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5 Numerical analysis of finite time blow up and finite time rupture in two thin film models 48 5.1 Numerical simulations of finite time blow up . . . . . . . . . . . . . . . . . . . . . . . 51 5.2 Numerical simulations of finite time rupture . . . . . . . . . . . . . . . . . . . . . . . 54 Conclusion 57 A Core Code in MATLAB 62 A.1 Programming code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 A.1.1 Pseudospectral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 A.1.1.1 Bernis and Friedman’s Regularization . . . . . . . . . . . . . . . . . 65 A.1.2 Bertozzi Method for Singularity in the Third Derivative . . . . . . . . . . . . 67 A.1.3 The Multistep Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 A.1.4 Linearizing the Thin Film Equation . . . . . . . . . . . . . . . . . . . . . . . 76 A.1.4.1 Order of accuracy of multistep methods using spectral methods . . 78 B Examples and Proofs to Theorems 81 B.1 Proofs to Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 B.1.1 Bernis and Friedman’s Approximating Problems . . . . . . . . . . . . . . . . 81 B.1.2 Existence of Weak Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 B.1.3 Bernis & Friedman’s Proof for Nonnegative Solutions . . . . . . . . . . . . . 88 B.1.4 Bernis & Friedman’s Proof for the Expansion of Support vii . . . . . . . . . . . 96 B.1.5 Bernis & Friedman’s Proof for the Approximation by Positive u . . . . . . . 98 B.2 Important Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 B.3 Existence via Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 B.3.1 Kreiss’ work with A Priori Estimates and Local Existence . . . . . . . . . . . 106 B.3.1.1 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 B.3.1.2 A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 B.3.1.3 Existence via Iteration . . . . . . . . . . . . . . . . . . . . . . . . . 110 B.3.2 Bernis & Friedman’s Approach to the Existence of Solutions via Iteration . . 111 Bibliography 116 viii List of Figures 1 3 2.1 Arclength dissipation for weak solutions when n = 3.1 Upper bound for |ux | < δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Arclength dissipation when n = 1 2 and N = 32 . . . . . . . . . . . . . . . . . . . . . 38 3.3 Arclength dissipation when n = 1 3 and N = 32 . . . . . . . . . . . . . . . . . . . . . 38 3.4 Evolution of the solution for the thin film equation with linearized curvature term with initial data u0 = 7 8 − cos x 1.17 and N = 32 . . . . . . . . . . . 14 and the rate of arclength dissipation when n = 38 . . . 39 3.5 Uniform mesh generated for the initial data u0 = cos(x) + 1.01. . . . . . . . . . . . . 39 3.6 Over the triangularized domain such as on the left, we defined piecewise linear functions that link as seen above. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1 Evolution of iterative solution when n = 1 2 and N = 32 . . . . . . . . . . . . . . . . . 46 4.2 Evolution of iterative solution when n = 1 2 and N = 32 with a monotone decreasing in the H 1 norm after an initial growth. . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1 Visualization of MATLAB’s FFT of u0 = cos(3x − 3π) + sin(2x − 2π) + 0.1. . . . . . 49 5.2 The evolution of the thin film equation in spectral space for u0 = 1.0001 + cos(x − π) with snapshots every 0.1212 seconds where ∆t ∼ ∆x2 5.3 . . . . . . . . . . . . . . . . . 49 Comparison of the solutions for the thin film equations with linearized and nonlinear curvature forms on an incline every 2.25 × 10−7 seconds with initial data u0 = (x−π)2 5.5 + − 1√ e 2(0.15)2 . 0.15 2π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ix 5.4 Comparison of snapshots of solutions of the thin film equation with nonlinear curvature on an incline every 4.9 × 10−8 seconds with the same initial data for grids with 256 and 512 nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.5 Evolution of the Bertozzi scheme when n = 12 . . . . . . . . . . . . . . . . . . . . . . . 55 5.6 Comparison of solutions of the linearized and nonlinear curvature terms using the Bertozzi scheme and n = 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.7 The linearized curvature equation evolves more quickly to its steady state solution than the nonlinear curvature equation. . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.8 Calculation of the change in mass for the thin film equation with a nonlinear curvature term as well as L2 -norm of the difference of two simulations for the initial data u0 = 0.8 − cos(πx) + 0.25 sin(2πx). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.9 Evolution of the height function under the thin film equations with linearized and nonlinear curvature term respectively with initial data u0 = 0.8−cos(πx)+0.25 sin(2πx). 58 5.10 Snapshot of the evolution of the thin film equation with linearized and nonlinear curvature when t = 7.4849 × 10−5 seconds as both solutions diffuse for initial data (x−π)2 u0 = 5.5 + − 1√ e 2(0.15)2 . 0.15 2π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.11 Evolution of thin film equation with replacement for un with initial data u0 = 0.8 − cos(πx) + 0.25 sin(2πx). Note that when zoomed in, the final time step has a local maximum where there was a global minimum previously. . . . . . . . . . . . . . . . . 59 A.1 Evolution of u0 = sin(x) + 1.1 for t ∈ [0, π] showing 40 snapshots in time when n = 3. 64 A.2 Arc length of u0 = sin(x) + 1.1 for t ∈ [0, π] when n = 3. . . . . . . . . . . . . . . . . 65 A.3 Evolution of solution for initial data u0 = sin(x) + 1.01 for N = 64 nodes, n = 2, and  = 10−4 using Crank-Nicholson time steps with ∆t ∼ 12 ∆x with its rate of arc length dissipation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 A.4 Evolution of solution for initial data u0 = sin(x) + 1.01 for N = 64 nodes, n = 2, and  = 10−5 using Crank-Nicholson time steps with ∆t ∼ 21 ∆x with its rate of arc length dissipation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 x A.5 Evolution of solution for initial data u0 = sin(x) + 1.01 for N = 64 nodes, n = 2, and  = 10−4 using Implicit Euler time steps with ∆t ∼ 12 ∆x with its rate of arc length dissipation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 A.6 Evolution of solution for initial data u0 = sin(x) + 1.01 for N = 64 nodes, n = 2, and  = 10−5 using Implicit Euler time steps with ∆t ∼ 12 ∆x with its rate of arc length dissipation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 A.7 Evolution of the Bertozzi scheme when n = 21 . . . . . . . . . . . . . . . . . . . . . . . 72 A.8 Evolution of the solution for initial data u0 = sin(x) + 1.01 with N = 64 nodes and n = 2 with time steps ∆t ∼ ∆x/2 with snapshots every 2π × 10−1 seconds. . . . . . 76 A.9 Evolution of linearized thin film equation for initial data u0 = sin(x) + 1.01 with N = 64 and n = 2 with time steps ∆t ∼ ∆x/2 with snapshots every 2π × 10−1 seconds. 78 e  (s) % G e 0 (s) B.1 Showing the convergence of G . . . . . . . . . . . . . . . . . . . . . . . 88 B.2 Showing the convergence of ge (s) & ge0 (s) . . . . . . . . . . . . . . . . . . . . . . . . 89 B.3 If M = 1, this gives an example where the family of functions is uniformly bounded and equicontinuous but has two different possible convergent subsequences . . . . . . 100 Chapter 1 Introduction From its founding, mathematics has often been shaped by its environment. Many mathematicians then have developed mathematical techniques and constructed axiomatically the theories that form the bedrock that guide new research in pure mathematics, which were originially meant to mathematically model different aspects of our world. With D’Alembert’s construction of the general wave equation, Fourier followed shortly thereafter with his heat equation. This led to the formation of the study of parabolic systems and, in particular, degenerate parabolic systems. Several mathematical models can be described by the generalized Reynolds equation: ut = (un px )x, where p describes pressure [47, 63]. For mathematical models when pressure is defined by the height of the fluid and n = 1, it models a gravity driven flow such as in the porous media problem. When pressure is defined by the negative curvature of the height function, the equation models the thin film equation studied in this thesis. For pressure defined by the fourth derivative of the height function, the model describes the thickness of the fluid under an elastic plate such as for a semiconductor. In the porous medium problem, mathematicians seeks to model the incompressible flow as it travels through “pores.” These pores contain little volume as compared to the overall volume, which leads to resistances to fluid flow. Therefore, the superficial velocity for the flow is lower as compared to the interstitial velocity with relation Ui = Us  , 1 where  is the void fraction [60]. However, due to the varied media through which the flow could travel,  could be as low as 10% leading to an interstitial velocity ten times that of the superficial velocity. These lower porosity values lead to larger effects of viscous forces as compared to inertial forces. Frequently, this implies lower Reynolds numbers and a Darcian flow model. Developed by Henry Darcy and his colleagues in the mid-19th century, Darcy flows are described by the flux term q, dependent on the permeability of the media (κ), pressure drop (p), and viscosity (µ), or in the general form as simply κ q = − ∇p. µ However, this model describes flow in a aquifer where a steady state is reached. The Reynolds equation form of the porous media problem, also described as the groundwater flow equation, mirrors Fourier’s heat equation/flux, but instead is dependent on parameter for specific storage Ss , which characterizes the release of flow in an aquifer and source term G, Ss ∂h = −∇ · q + G. ∂t The thin film equation is a special form of the porous media problem using a Hele-Shaw flow. In lubrication theory, flow is considered where the film thickness H is much less than the length of the flow L (i.e.  = H/L  1). The pressure is independent of spatial direction perpendicular to the plane, and the gradient of the pressure is dependent on the curvature of the height function and the viscosity µ. When put together in the Reynolds equation, we see n ut = − u ∂2u µ 2 ∂x ! ! = −µ(un uxxx )x. x x In the derivation of this model from Hele-Shaw flow [48], we see that Hele-Shaw conditions hold for the dimensionless thin film equation when n = 1. In construction of semiconductors, the placement of impurities called dopants play an important part in the electric characteristics of the semiconductor which arises from the wrinkling upon the annealing of SiGe films bonded to Si substrates [70, 47]. The behavior of the dopants in silicon is described by oxidation-enhanced and oxidation-retarded diffusion. These two processes conflict and are modelled by King in 1989 [70] with a description of solutions in the different boundary 2 layers. From the generalized Reynolds equation, we can see the equation ut = (un uxxxxx )x, with pressure given by p = uxxxx [47, 63]. When King first introduced the model in 1986, he assumed the case with n = 3 to model the behavior of the Newtonian viscous fluids not slipping along the substrate. With the cases of n = 2 where there are Navier-slip conditions and n = 1 modeling the thin film flows in Hele-Shaw cells, different authors model the behavior of the dopants with the addition of extra terms. To include shearing forces Σ, we can add the term uxxx , and to include the bending moment of an overlying plate, we can include the negative curvature m = −uxx. 1.1 Dynamics of Thin Films For small values of h in the thin film equation, the equation must be corrected to include the van der Waals molecular forces [26]. For solutions to this modified thin film equation with initial data in C 4,α , the flow has a rupture in finite time, and the energy of the mean of the initial data is more than the energy of the initial data itself. Craster and Matar experimentally and numerically studied the effects of changing the mass of the added surfactant and the thickness of the film [35]. By increasing mass while holding the thickness constant, the fluid would not develop as many fingers, and most of the fluid would remain together as thick fingers, which reduced the effect of surface tension on the flow. Lower mass amounts of surfactant led to much more fingering that was wider and of the same thickness as the main mound. Increasing the thickness of the flow led to a more “stable” spreading pattern. Edmonstone, Craster, and Matar studied the effects of the mass of the surfactant and a parameter representing the preference of the surfactant to form micelles (both dimensionless parameters) and their relationship to the spreading of a droplet [43]. Holding the second parameter constant, they noticed that increasing the surfactant mass will cause a second front to develop just beyond the end of the drop off of the main mound, and that instabilities due to transverse perturbations can cause the formation of fingers in the fronts. In 2009, Craster and Matar reviewed and published the results of previous authors on thin films [36]. They highlighted both experimental applications 3 to the theory as well as mentioning where future research is necessary. Grotberg used the thin film equation to model the liquid lining of the acinar region of the respiratory system [54]. To model the low and high surface tension regions that describe the area, they built a system where the thin film approximation covers the high surface tension portion with only a boundary layer including the low surface tension region. The length of the region is O(Σ2 ), where Σ is the ratio of the thickness of the low surface tension region flow to the difference between the two thicknesses. Stone et al. [96] and Squires and Quake [95] organized and reviewed results of microfluidics including their relation to thin films. Weinstein and Ruschak [100] and Craster and Matar [36] reviewed the work on thin films and the current research. Gomba focused on developing a method of conserving the volume rather than holding the height of the fluid at a constant height behind the contact line (or a constant flux flow) [50]. The stability analysis becomes time dependent, and the perturbations match the velocity of the leading contact line with the maximum amplitude of the perturbations decreasing to an asymptotic value that matches the experimental values. Bertozzi, Munch, and Shearer noticed that when adding a nonconvex flux to the thin film equation, with n = 3 and parameter  to describe a thin film with conflicting thermal surface tension gradient and gravity terms, a jump in initial data develops into a double wave, which includes an undercompressive wave [11]. This undercompressive wave approaches a shock as  → 0, which serves as an accumulation point for a countable family of compressive waves, with each bifurcation creating a stable and unstable travelling wave. In response to their previous work, the authors built a spectral framework to analyze these compressive and undercompressive waves [16]. Huppert noted that an instability in a viscous current down a slope developed independently of the coefficient of viscosity [64]. The experimental values suggested instead that the instability partially depended on the cross-sectional area and only occurred after the flow had traveled a critical length proportional to the square root of the cross-sectional area. Ward continued this work by adding particles into the current down the slope [99]. For low concentrations of the particleladen “slurry,” Huppert’s predicted values scaled with the experimental data. However, for higher concentrations, the flow developed more perturbations due to the particles, but the average front position also scaled accordingly. The deviations from the expected positions of the front for high concentrations of particles are dependent on the volume of the mixture, the angle of inclination, 4 and the sizes of the particles. Perazzo noticed for non-Newtonian fluids on an incline, the thin film equation gives three separate traveling wave solutions rather than the single downslope wave solution seen for Newtonian fluids [87]. The first of the extra two waves is a downslope wave in the form of a hypergeometric function with a complex solution unless the free surface tends to the horizontal. The second is an upslope wave. When tan(α)  |hx | instead, there are once again only downslope waves in the form of a generalized, non-Newtonian version of Huppert’s prediction as a unique self-similar solution. Chugunova, Pugh, and Taranets showed the necessary parameters for solutions to the thin film equations on the outer surface of a rotating cylinder [28, 27]. Hinch and Kelmanson published results describing the evolution of perturbations in a viscous thin film flow outside of a rotating cylinder. They showed a drift rate and exponential decay with fundamental modes proportional to h̄4 and h̄7 respectively [58] and the development of shock-like perturbations for low surface tension flow [59]. For vertical cylinders, Siddiqui, Mahmood, and Ghori showed the advantages that the homotopy perturbation method had over traditional perturbation methods to approximate solutions [93], while Hayad and Sajid showed that the homotopy analysis method was superior when there are strong nonlinearities, as the homotopy perturbation method will be divergent there [56]. In partially-filled horizontally-rotating cylinders, Hosoi and Mahadevan [61] and Thoroddsen and Mahadevan [97] studied the axial instabilities that form from inertial forces. Joseph et al. broke down these flows that also included particles into three regimes that describe the suspension of the particles in the flow [68]. Knoll et al. studied copolymers that form in the thin films in cylinders [72]. Seeing the experimental data shown in the previous papers, we used the nonlinear curvature term as compared to the linearized form that the lubrication approximation equation describes. In Chapter 2, we look for weak solutions to this equation and show that the energy functional dissipates for these weak solutions. Then, in Chapter 5, we study the numerical simulations of viscous thin film flows on an incline. 5 1.2 Hele-Shaw flows and the Lubrication Approximation If a fluid has a low velocity field, the viscosities are very large, or the length scales of the flow are very small, we can approximate the flow as a Stokes flow. A Stokes flow is described by the differential equations:    ∇p = µ∇2 u + f,   ∇ · u = 0. Therefore, Stokes flows are dependent on time only as their boundary conditions are, and the solutions are independent of solutions at other points in time. If the flow is moving through a porous medium, we can further simplify the model as a Hele-Shaw flow. In a Hele-Shaw flow, the x and y directions are parallel to flat plates, and the z direction is perpendicular to the plates. However, there is only a gap of 2H between the plates, which are at z = ±H. We say that the gap is asymptotically small and the flow is guided by the equations and boundary condition   2 2   , u = ∇p z −H   2µ    2 ∇ p = 0,        ∇p · n̂ = 0, where p = p(x, y, t) and n̂ is normal to the boundaries. The Hele-Shaw cell describes an experiment where a fluid is inserted between the plates, and the fluid is bounded by another already-present fluid. The Hele-Shaw cell experiments are very useful in describing real-world phenomena such as “fingering" and are useful for both oil companies to better find oil and for environmental agencies attempting to find the flow of water through the soil, since Hele-Shaw flows predict how flows will interact. These flows showcase different behaviors of both Newtonian and non-Newtonian fluids. More specifically, the dynamics of non-Newtonian fluids are useful for tertiary oil recovery, filtration, gel permeation chromatography, and injection molding as described by Azaiez and Singh [2]. Therefore, in their paper, they described how the stability is improved by replacing the characteristics of Newtonian fluids with that of a non-Newtonian fluid due to its shear-thinning behavior. The 6 difference in maximum growth rates and the spectrum of unstable wave numbers between Newtonian and non-Newtonian fluids also leads to more frequent finger instabilities in non-Newtonian fluids. Constantin et al. showed that thin necks of fluid between two larger droplets will not break in a finite time using the lubrication approximation and that the minimum width of the neck decreases similar to 1/t4 for large t [33]. Also, they found similarity solutions that match theoreticallyexpected shapes. Dupont et al. showed the dynamics of a singularity, or separation of flow, in finite time in a lubrication approximated flow under the correct initial and boundary conditions [42]. Without inertial forces, the height of the region from the “pinch” grows quadratically. With inertial forces, the region grows linearly. In their paper in 2002, Lee, Lowengrub, and Goodman described the pinchoff in mixtures of fluids and the linear stability and the propagation of the interfaces of these mixtures [77]. Böckmann and Müller showed that for an autocatalytic reaction front in a quasi-2D cell, a buoyancy-driven instability develops as a fingering deformation [19]. The higher wave numbers are stable, but the lower wave numbers for perturbations are unstable. Martin et al. furthered their work with an autocatalytic reaction front by adding a chemical reaction term [79]. They noted that the regime would benefit from a 2D model to describe the development of the unstable fingering, since they observed “there is little chance for these fortuitous predictions to hold for other chemical reaction parameters.” Almgren showed that despite Constantin and Pugh’s work on circles, singularities can form from the circles [1]. The lubrication approximation can model the flow well near the singularity, but it does not stop the singularity from forming, due to the global forces acting upon it, despite the resistance of the local forces. Goldstein et al. studied the possible pinching behavior of a gravity-driven Hele-Shaw flow [49]. They showed that the pinching must occur in at least infinite time with simulations suggesting earlier (in finite time) in unstable systems above a bifurcation point. Ruyer-Quil studied how to correct Hele-Shaw flow for inertial forces [90], and Crowdy studied exact solutions on a fluid annulus in a rotating Hele-Shaw cell [37]. Seis studied the complex spectra of the porous media problem and its symmetries [91]. Katz and Worster developed a finite volume scheme that simulates the directional solidification of a binary alloy in a Hele-Shaw cell without needing internal boundary conditions between regions of the volume using Darcy’s law and the Enthalpy Method [69]. They found that for the three regions of the volume – solid, “mush”, and liquid – the mush-liquid interface becomes unstable or develops boundary layer convection, depending on the method. Hernandez et al. studied an 7 air-filled porous medium problem with a constant forced fluid flow rate and determined a method of simulating the forced evolution of the liquid-air interface [57]. The non-local dynamics of the interface experience “roughening” and experience two scaling regimes separated by a saturation time after which the roughness and growth exponents become equal. Otto proved long-time existence of weak solutions to the lubrication approximation with contact angle π/4 and hx = 1 [86]. Giacomelli and Otto rigorously showed the connection between Darcy flow/Hele-Shaw cells and lubrication flow, in addition to the appropriate scaling [48]. They also showed that the lubrication approximation preserves the quality of vanishing contact angles in regimes of complete wetting seen in Darcy flow/Hele-Shaw models. Bertozzi and Pugh studied the modified lubrication equation to show the formulation of singularities in similarity solutions [14] and showed the conditions for the positive solutions for the original lubrication equation [12]. Brenner and Bertozzi together showed the spreading of a droplet using similarity solutions [20]. In Chapter 3, we compare this connection between Hele-Shaw flows and the lubrication approximation. We show numerically, using finite element methods, that the arclength dissipation property shown earlier in Chapter 3 is also seen in the Hele-Shaw cell at similar rates. 1.3 Higher order degenerate non-linear parabolic partial differential equations While Bernis and Friedman focused on larger n, in the thin film equation, and approximations to achieve positive results with n < 3, this paper sought to describe solutions for small n (0 < n < 1/2) √ [4]. These solutions developed finite time singularities on domains with measures smaller than 2 2 but have classical solutions before such time. Also, for n > 7/2, solutions retain their property of positivity. Bernis and McLeod built a similar solution for a fourth order degenerate partial differential equation from a fourth order ordinary differential equation [8]. They showed that these solutions are unique with both integral and asymptotic methods. Using a source type form and a shooting technique, Bernis, Peletier, and Williams built a symmetric solution to the thin film equation for 0 < n < 3 [9]. They also proved uniqueness of the source solution, as well as gave an explicit solution. Ferreira and Bernis found that source-type solutions in higher dimensions have a unique 8 C 1 radial self-similar nonnegative solution for 0 < n < 3, but the thin film equation has no such solution for higher n [46]. For this solution, the height function propagates at finite speed but loses the non-zero contact angle property for the desired boundary conditions given by the source-type equation’s boundary condition. Building on their previous year’s work, the authors noted that as n → 3, the limit of the source-type solution is a scaled delta function [6]. Rescaling the solution first leads to a limit having the form of a parabolic arch with a nonzero contact angle. Bernoff and Witelski noted that linear stability analysis shows stable source-type solutions in the thin film equation [10]. For the special case of n = 1 in the thin film equation, the solutions to the equation come in the form of the Gegenbauer polynomials. Grün showed that the thin film equation can have the waiting time phenomenon for dimensions N < 4 when under Navier’s slip conditions or some other weaker conditions specified in the paper [55]. Using a Hardy-type inequality on infinite cones, the author found that solutions with convex initial data had a finite speed of propagation when n ∈ [2, 3). Chou et al. showed that borderline solutions are global H 1 solutions starting at a boundary point of the set of stable maximal solutions to the non-homogeneous Laplace equation [25]. These borderline solutions in a convex set, Ω, blow up in finite time, but, after some time, it becomes uniformly bounded and decays to zero as time, t, goes to infinity. Laugesen first showed a new energy functional R hp h2x where p < 0 and .5 < n < 3 [74]. He saw that the functional dissipates for all positive smooth solutions and noted that the Hele-Shaw model follows n = 1 and p = −.5. Also, they are continuous in Ω × (0, ∞)\S, where the Hausdorff dimension H n−4/(p−1) is zero. Chugunova and Taranets applied Laugesen’s work to thin films on an incline to show dissipation for Laugesen’s functional as well as a new α-energy functional [31]. Du and Fan showed when the same energy functional in the form of R h−p h2x (p > 0 in this case) dissipates, there is a suitable weak, nonnegative solution [41]. Using this functional, we can bound from above the Hausdorff dimension of the rupture set of h by 1 + 6/n and the Hausdorff dimension of the set of rupture times by 1 − 2(n − 2)2 /(8 − n). Using the thin film equation, Chugunova et al. studied the dynamics of the flow on a rotating horizontal cylinder including the effects of gravity [28]. They showed existence of nonnegative weak solutions as well as how these solutions cannot outgrow linear growth both anaytically and numerically. Laugensen and Pugh characterized the linear stability of a viscous thin film liquid with variants that can include gravity or van der 9