Properties of steady states for thin film equations

We consider nonnegative steady-state solutions of the evolution equation formula here Our class of coefficients f, g allows degeneracies at h = 0, such as f(0) = 0, as well as divergences like g(0) = ±∞. We first construct steady states and study their regularity. For f, g > 0 we construct positive periodic steady states, and non-negative steady states with either zero or nonzero contact angles. For f > 0 and g < 0, we prove there are no non-constant positive periodic steady states or steady states with zero contact angle, but we do construct non-negative steady states with nonzero contact angle. In considering the volume, length (or period) and contact angle of the steady states, we find a rescaling identity that enables us to answer questions such as whether a steady state is uniquely determined by its volume and contact angle. Our tools include an improved monotonicity result for the period function of the nonlinear oscillator. We also relate the steady states and their scaling properties to a recent blow-up conjecture of Bertozzi and Pugh.

[1]  P. Rosenau,et al.  On a nonlinear thermocapillary effect in thin liquid layers , 1994, Journal of Fluid Mechanics.

[2]  P. Rosenau,et al.  Formation of patterns induced by thermocapillarity and gravity , 1992 .

[3]  Carmen Chicone,et al.  The monotonicity of the period function for planar Hamiltonian vector fields , 1987 .

[4]  J. Furter,et al.  Analysis of bifurcations in reaction–diffusion systems with no-flux boundary conditions: the Sel'kov model , 1995, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[5]  Mary C. Pugh,et al.  The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions , 1996 .

[6]  Peter Ehrhard,et al.  Non-isothermal spreading of liquid drops on horizontal plates , 1991, Journal of Fluid Mechanics.

[7]  Francisco Bernis,et al.  Finite speed of propagation for thin viscous flows when 2 ≤ n ≤ 3 , 1996 .

[8]  Michael F. Schatz,et al.  Long-wavelength surface-tension-driven Bénard convection: experiment and theory , 1997, Journal of Fluid Mechanics.

[9]  The steady states of the one-dimensional Cahn-hilliard equation , 1992 .

[10]  Russel E. Caflisch,et al.  Long time existence for a slightly perturbed vortex sheet , 1986 .

[11]  Remarks on periods of planar Hamiltonian systems , 1993 .

[12]  S. Bankoff,et al.  Long-scale evolution of thin liquid films , 1997 .

[13]  S. Bankoff,et al.  Steady thermocapillary flows of thin liquid layers. II. Experiment , 1990 .

[14]  Clarence A. Miller,et al.  Spreading kinetics of a drop on a rough solid surface , 1983 .

[15]  Andrea L. Bertozzi,et al.  THE LUBRICATION APPROXIMATION FOR THIN VISCOUS FILMS: THE MOVING CONTACT LINE WITH A 'POROUS MEDIA' CUT OFF OF VAN DER WAALS INTERACTIONS , 1994 .

[16]  Stephen H. Davis,et al.  Nonlinear theory of film rupture , 1982 .

[17]  Peter Ehrhard The Spreading of Hanging Drops , 1994 .

[18]  R. Schaaf A class of Hamiltonian systems with increasing periods. , 1985 .

[19]  Raymond E. Goldstein,et al.  Instabilities and singularities in Hele–Shaw flow , 1998 .

[20]  The steady states of one dimensional Sivashinsky equations , 1991 .

[21]  M. Bertsch,et al.  Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation , 1995 .

[22]  Mary C. Pugh,et al.  Long-wave instabilities and saturation in thin film equations , 1998 .

[23]  S. G. Bankoff,et al.  Nonlinear stability of evaporating/condensing liquid films , 1988, Journal of Fluid Mechanics.

[24]  Mark A. Lewis,et al.  Spatial Coupling of Plant and Herbivore Dynamics: The Contribution of Herbivore Dispersal to Transient and Persistent "Waves" of Damage , 1994 .

[25]  D. Schaeffer,et al.  Nonlinear behavior of model equations which are linearly ill-posed , 1988 .

[26]  A singular minimization problem for droplet profiles , 1993, European Journal of Applied Mathematics.

[27]  R. Schaaf Global Solution Branches of Two Point Boundary Value Problems , 1991 .

[28]  H. P. Greenspan,et al.  On the motion of a small viscous droplet that wets a surface , 1978, Journal of Fluid Mechanics.

[29]  S. Herminghaus,et al.  Wetting: Statics and dynamics , 1997 .

[30]  Oliver E. Jensen,et al.  The thin liquid lining of a weakly curved cylindrical tube , 1997, Journal of Fluid Mechanics.

[31]  Michael J. Miksis,et al.  The effect of the contact line on droplet spreading , 1991, Journal of Fluid Mechanics.

[32]  Selfsimilar source solutions of a fourth order degenerate parabolic equation , 1997 .

[33]  Francisco Bernis,et al.  Finite speed of propagation and continuity of the interface for thin viscous flows , 1996, Advances in Differential Equations.

[34]  P. Hammond Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe , 1983, Journal of Fluid Mechanics.

[35]  A. Friedman,et al.  Higher order nonlinear degenerate parabolic equations , 1990 .

[36]  E. Tuck,et al.  Thin static drops with a free attachment boundary , 1991, Journal of Fluid Mechanics.

[37]  Goldstein,et al.  Topology transitions and singularities in viscous flows. , 1993, Physical review letters.

[38]  D. v.,et al.  The moving contact line: the slip boundary condition , 1976, Journal of Fluid Mechanics.

[39]  Kadanoff,et al.  Traveling-wave solutions to thin-film equations. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[40]  Andrea L. Bertozzi,et al.  Singularities and similarities in interface flows , 1994 .

[41]  G. I. Barenblatt Scaling: Self-similarity and intermediate asymptotics , 1996 .

[42]  Oron,et al.  Stable localized patterns in thin liquid films. , 1992, Physical review letters.