1,721,017 research outputs found
Non-linear higher-order boundary value problems describing thin viscous flows near edges
No full textTwo boundary value problems for non-linear higher-order ordinary differential equations are analyzed, which have been recently proposed in the modeling of steady and quasi-steady thin viscous flows over a bounded solid substrate. The first problem concerns steady states and consists of a third-order ODE for the height of the liquid; the ODE contains an unknown parameter, the flux, and the boundary conditions relate, near the edges of the substrate, the height and its second derivative to the flux itself. For this problem, (non-)existence and non-uniqueness results are proved depending on the behavior, as the flux approaches zero, of the "height-function" (the function which relates the height to the flux near the edge out of which the liquid flows). The second problem concerns quasi-steady states and consists of a fourth-order ODE for the (suitably scaled) height of the liquid; non-linear boundary conditions relate the height to the flux near the edges of the substrate. For this problem, the existence of a solution is proved for a suitable class of height-functions. (c) 2008 Elsevier Inc. All rights reserved
Finite speed of propagation and waiting-time phenomena for degenerate parabolic equations with linear growth Lagrangian
Full text linkWe consider a class of degenerate parabolic equations with linear growth Lagrangian. Two prototypes within this class, sharing common features with nonlinear transport equations, are the relativistic porous medium equation and the speed-limited (or flux-limited) porous medium equation. In arbitrary space dimension, we prove that entropy solutions to the Cauchy problem satisfy the finite speed of propagation property. For the two aforementioned prototypes, we provide a condition on the growth of the initial datum which guarantees the occurrence of a waiting-time phenomenon; we also present a heuristic argument in favor of the optimality of such condition
Lower bounds on waiting time for degenerate parabolic equations and systems
No full textWe extend the method in [Dal Passo Giacomelli Gruen, Annali SNS Pisa, 2001] to obtain quantitative estimates of waiting times for free boundary problems associated with degenerate parabolic equations and systems. Our approach is multidimensional, it applies to a large class of equations, including thin-film equations, (doubly) degenerate equations of second and of higher order and also systems of semiconductor equations. For these equations, we obtain lower bounds on waiting times which we expect to be optimal in terms of scaling. This assertion is true for the porous-medium equation which seems to be the only PDE for which two-sided quantitative estimates of the waiting time have been established so far
Rigorous lubrication approximation
No full textWe rigorously carry out a lubrication approximation for a liquid thin film which spreads on a solid, driven by surface tension. We consider a two-dimensional Darcy liquid as simple model case. Of particular interest to us is the codimension-two free boundary, i.e. the triple junctions where solid, liquid and vapor meet. In the considered regime of complete wetting, the contact angle vanishes throughout the evolution. We show in particular that this contact-angle condition is preserved in the lubrication approximation
Scaling laws for droplets spreading under contact-line friction
No full textThis manuscript is concerned with the spreading of a liquid droplet on a plane solid surface. The focus is on effective conditions which relate the speed of the contact line (where liquid, solid, and vapor meet) to the microscopic contact angle. One such condition has been recently proposed by Weiqing Ren and Weinan E [Phys. Fluids 19, 022101, 2007]: it includes into the model the effect of frictional forces at the contact line, which arise from unbalanced components of the Young's stress. In lubrication approximation, the spreading of thin droplets may be modeled by a class of free boundary problems for fourth order nonlinear degenerate parabolic equations. For speed-dependent contact angle conditions of rather general form, a matched asymptotic study of these problems is worked out, relating the macroscopic contact angle to the speed of the contact line. For the specific model of Hen and E, ODE arguments are then applied to infer the intermediate scaling laws and their timescales of validity: in complete wetting, they depend crucially on the relative strength of surface friction (at the liquid-solid interface) versus contact-line friction; in partial wetting, they also depend on the magnitude of the static contact-angle
A local estimate for vectorial total variation minimization in one dimension
Full text linkLet u be the minimizer of vectorial total variation with L2 data-fidelity term on an interval I. We show that the total variation of u over any subinterval of I is bounded by that of the datum over the same subinterval. We deduce analogous statement for the vectorial total variation flow on I
Droplets spreading with contact-line friction: lubrication approximation and traveling wave solutions
No full textWe consider the spreading, driven by surface tension, of a thin liquid droplet on a plane solid surface. In lubrication approximation, this phenomenon may be modeled by a class of free boundary problems for fourth order nonlinear degenerate parabolic equations, the free boundary being defined as the contact line where liquid, solid and vapor meet. Our interest is on an effective free boundary condition which has been recently proposed by Ren and E: it includes into the model the effect of frictional forces at the contact line, which arises from the deviation of the contact angle from its equilibrium value. In this note we outline the lubrication approximation of this condition, we describe the dissipative structure and the traveling wave profiles of the resulting free boundary problem, and we prove existence and uniqueness of a class of traveling wave solutions which naturally emerges from the formal asymptotic analysis
Shear-thinning liquid films: Macroscopic and asymptotic behaviour by quasi self-similar solutions
No full textWe consider the spreading of a thin droplet of viscous liquid on a plane surface driven by capillarity in the complete wetting regime. In the case of constant viscosity, the no-slip condition leads to a force singularity at advancing contact lines. It is well known nowadays that the introduction of appropriate slip conditions removes this paradox and alters only logarithmically the macroscopic behaviour of solutions at intermediate timescales. Here, we investigate a different approach, which consists in keeping the no-slip condition and assuming instead a shear-thinning rheology. This relaxation leads, in lubrication approximation, to fourth order degenerate parabolic equations of quasilinear type. By analysing a class of quasi-self-similar solutions to these equations in the limit of Newtonian rheology, we obtain a scaling law in time for macroscopic quantities (such as macroscopic profile, effective contact-angle) which is only logarithmically affected by the shear-thinning parameters. As opposed to positive slippage models, the scaling law is uniform for large times as far as the macroscopic support is well defined, and thus could also describe the asymptotic behaviour of a large class of solutions for fixed shear-thinning rheology
Propagation of support in one-dimensional convected thin-film flow
No full textIn one space dimension, we study the finite speed of propagation property for zero contact--angle solutions of the thin-film equation in presence of a convective term. In the case of strong slippage, we obtain bounds in terms of the initial mass for both the ``fast" and the ``slow" interfaces, and for both short and (whenever the solution is global) large times, which we expect to be sharp. In the case of weak slippage, we obtain partial results for short times, which include a quantitative bound for moderate growths of the convective term. Our approach is based on energy/entropy methods shaped upon suitable extensions of Stampacchia's Lemma
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