Results 1  10
of
11
Linear Stability of Steady States for Thin Film and CahnHilliard Type Equations
, 2000
"... . We study the linear stability of smooth steady states of the evolution equation h t = (f(h)h xxx ) x (g(h)h x ) x ah under both periodic and Neumann boundary conditions. If a 6= 0 we assume f 1. In particular we consider positive periodic steady states of thin lm equations, where a = 0 and f; ..."
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Cited by 12 (3 self)
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. We study the linear stability of smooth steady states of the evolution equation h t = (f(h)h xxx ) x (g(h)h x ) x ah under both periodic and Neumann boundary conditions. If a 6= 0 we assume f 1. In particular we consider positive periodic steady states of thin lm equations, where a = 0 and f; g might have degeneracies such as f(0) = 0 as well as singularities like g(0) = +1. If a 0, we prove each periodic steady state is linearly unstable with respect to volume (area) preserving perturbations whose period is an integer multiple of the steady state's period. For area preserving perturbations having the same period as the steady state, we prove linear instability for all a if the ratio g=f is a convex function. Analogous results hold for Neumann boundary conditions. The rest of the paper concerns the special case of a = 0 and power law coecients f(y) = y n and g(y) = By m . We characterize the linear stability of each positive periodic steady state under perturbations of t...
Pattern formation with a conservation law
 Nonlinearity
"... Abstract. Pattern formation in systems with a conserved quantity is considered ..."
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Cited by 3 (0 self)
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Abstract. Pattern formation in systems with a conserved quantity is considered
Stochastic geometry of polygonal networks  an alternative approach to the hexagonsquaretransition in Bénard convection
, 1997
"... The tools of stochastic geometry are applied to the transition from hexagonal to square cells recently observed in surfacetensiondriven Benard convection. By means of this method we study the metrical and topological evolution of Benard cells as a function of the temperature di#erence across the l ..."
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Cited by 2 (1 self)
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The tools of stochastic geometry are applied to the transition from hexagonal to square cells recently observed in surfacetensiondriven Benard convection. By means of this method we study the metrical and topological evolution of Benard cells as a function of the temperature di#erence across the layer. We find distinct di#erences in the metric of the three cell types. While sidelength, area and perimeter of hexagons and squares grow monotoneously, the particular quantities of the pentagons change with onset of the transition in a steplike manner. Below the transition the pentagons behave similiar to hexagons, above close to squares which underlines their mediating character within the transition. The relation between perimeter and area plays obviously a decisive role for the stability of one cell class. We show that the perimeterarea ratio of a square Benard cell exceeds that of the hexagonal one by an unexpected high value. We find that the Benard pattern obeys to the AboavWeair...
TIME INTEGRATION AND STEADYSTATE CONTINUATION METHOD FOR LUBRICATION EQUATIONS
, 903
"... Abstract. The partial di erential equations describing the evolution of thin liquid lms on solid substrates in lubrication approximation are highly nonlinear. Thus, the classical semiimplicit time integrator with a constant linear part fails due to a strong variation of the Jacobian between two co ..."
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Cited by 1 (1 self)
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Abstract. The partial di erential equations describing the evolution of thin liquid lms on solid substrates in lubrication approximation are highly nonlinear. Thus, the classical semiimplicit time integrator with a constant linear part fails due to a strong variation of the Jacobian between two consecutive timesteps. We propose to integrate the equations using an exponential propagation. This method requires to evaluate the rightmost eigenvalues of the Jacobian Matrix. Because the discretized system is large and sparse, we adapt di erent classical iterative methods (Chebyschev acceleration, shiftinvert Cayley transform). We show that the Cayley transform is the most powerful method. Furthermore, the rightmost spectrum determined in such a way can also be employed in a continuation technique that allows to follow steady state solutions in parameter space. In this way time stepping and bifurcation analysis can be coupled. The resulting common numerical framework is exempli ed using (i) dewetting on a horizontal homogeneous substrate, and (ii) the depinning of pinned drops. Both examples are treated in two and threedimensional settings.
Scalable Parallel Finite Element Computations of RayleighBenardMarangoni Problems in a Microgravity Environment
"... 1. RayleighBenardMarangoni Flows 1. Microgravity Flow Problems 2. Mathematical Model and Finite element Method 3. Solution Algorithm 4. Parallelization 5. Performance 2. Evaluation Criteria and Performance 3. MGFLO Software Package 1. Code Modules 2. List of Routines 3. Sample Input File 4. Sample ..."
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1. RayleighBenardMarangoni Flows 1. Microgravity Flow Problems 2. Mathematical Model and Finite element Method 3. Solution Algorithm 4. Parallelization 5. Performance 2. Evaluation Criteria and Performance 3. MGFLO Software Package 1. Code Modules 2. List of Routines 3. Sample Input File 4. Sample Output File 5. User specified conditions 6. Usage 4. Test Problem and Performance Results 5. Experimental Studies 6. Conclusions and Recommendations Acknowledgments Glossary References Authors Dr G.F. Carey: project director, professor in Aerospace Engineering at the University of Texas at Austin, director of the CFD Lab. Dr R. Mclay: Research Associate at the University of Texas at Austin. Dr. C. Harlé: Post Doctoral associate, CFD Lab. Dr. B. Davis: Post Doctoral associate, CFD Lab. Dr. H Swinney: Professor at the University of Texas at Austin, director of the Center for Nonlinear Dynamics. Dr. S. Van Hook: Post Doctoral associate, Center for Nonlinear Dynamics. Abstract MGFLO is desi...
Instabilities and Spatiotemporal Chaos of Longwave Hexagon Patterns in Rotating Marangoni Convection
, 2002
"... We consider surfacetension driven convection in a rotating uid layer. For nearly insulating boundary conditions we derive a longwave equation for the convection planform. Using a Galerkin method and direct numerical simulations we study the stability of the steady hexagonal patterns with respec ..."
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We consider surfacetension driven convection in a rotating uid layer. For nearly insulating boundary conditions we derive a longwave equation for the convection planform. Using a Galerkin method and direct numerical simulations we study the stability of the steady hexagonal patterns with respect to general sideband instabilities. In the presence
unknown title
, 2006
"... Under consideration for publication in J. Fluid Mech. 1 The reopening of a collapsed, fluidfilled elastic tube ..."
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Under consideration for publication in J. Fluid Mech. 1 The reopening of a collapsed, fluidfilled elastic tube
Instabilities and Spatiotemporal Chaos of Longwave Hexagon Patterns in Rotating
, 2002
"... We consider surfacetension driven convection in a rotating fluid layer. For nearly insulating boundary conditions we derive a longwave equation for the convection planform. Using a Galerkin method and direct numerical simulations we study the stability of the steady hexagonal patterns with respect ..."
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We consider surfacetension driven convection in a rotating fluid layer. For nearly insulating boundary conditions we derive a longwave equation for the convection planform. Using a Galerkin method and direct numerical simulations we study the stability of the steady hexagonal patterns with respect to general sideband instabilities. In the presence of rotation steady and oscillatory instabilities are identified. One of them leads to stable, homogeneously oscillating hexagons. For sufficiently large rotation rates the stability balloon closes, rendering all steady hexagons unstable and leading to spatiotemporal chaos.
Harry SwinneyFinite Element Study of A Heated Thin Fluid Layer Including Surfactant Effect
"... December 2002To my wife Zhen and my parents Xianmin and Aijuan. Acknowledgments First I wish to thank my advisor, Graham F. Carey, for his insight, expertise, and encouragement which greatly helped me throughout this Ph.D. work. Dr. Van Nguyen directed us to related literature and studies on monol ..."
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December 2002To my wife Zhen and my parents Xianmin and Aijuan. Acknowledgments First I wish to thank my advisor, Graham F. Carey, for his insight, expertise, and encouragement which greatly helped me throughout this Ph.D. work. Dr. Van Nguyen directed us to related literature and studies on monolayer surfactants and we also thank her for helpful suggestions on the surfactant extension problem. My committee members, Clint Dawson,David Dolling,Harry Swinney and Robert McLay have provided helpful suggestions and I also benefited from the interaction with Dr. Swinney’s program. I would like to thank everyone in our CFD lab, particularly Ben and Bill for their excellent system support and technical assistance. I also would like to express my appreciation to Bill, Damien, Michael and Jeremy for the lastminute help with LaTex before the deadline to file the final paperwork. The past four years would have been much more difficult without the encouragement and support of my friends. Pat has always been there patiently helping me with all kinds of paperwork. My wife and I had lots of happy hours with our neighbors Fang Yin and Yi Zhou. I really enjoyed the discussions with Shuyu and Saeedi on topics