Results 1  10
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24
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 23 (4 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 9 (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 5 (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 4 (3 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.
RayleighBénardMarangoni convection due to evaporation : a linear nonnormal stability analysis
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Numerical Analysis of Marangoni Convection with Freeslip Bottom under Magnetic Field
"... Abstract: In this paper, we use a numerical technique to analyze the onset of Marangoni convection in a horizontal layer of electricallyconducting fluid heated from below and cooled from above in the presence of a uniform vertical magnetic field. The top surface of a fluid is deformable free and t ..."
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Cited by 2 (0 self)
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Abstract: In this paper, we use a numerical technique to analyze the onset of Marangoni convection in a horizontal layer of electricallyconducting fluid heated from below and cooled from above in the presence of a uniform vertical magnetic field. The top surface of a fluid is deformable free and the bottom boundary is rigid and freeslip. The critical values of the Marangoni numbers for the onset of Marangoni convection are calculated and later it is found to be critically dependent on the Hartmann, Crispation and Bond numbers. We found that the presence of Magnetic field always has a stabilizing effect of increasing the critical Marangoni number when the free surface is nondeformable. If the free surface is deformable, then there is a range where the critical Marangoni number will have unstable modes no matter how large magnetic field becomes.
Longwave Marangoni instability with vibration
 J. Fluid Mech
"... The effect of vertical vibration on the longwave instability of a Marangoni system is studied. The vibration augments the stabilizing effect of surface tension in bounded systems. In laterally unbounded systems nonlinear terms can stabilize nonflat states and prevent the appearance of dry spots. T ..."
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Cited by 2 (1 self)
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The effect of vertical vibration on the longwave instability of a Marangoni system is studied. The vibration augments the stabilizing effect of surface tension in bounded systems. In laterally unbounded systems nonlinear terms can stabilize nonflat states and prevent the appearance of dry spots. The effect of a slight inclination of the system is also considered. 1.
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