Asymptotic behaviour of the thin film equation in bounded domains

2001 ◽  
Vol 12 (2) ◽  
pp. 135-157 ◽  
Author(s):  
M. BOWEN ◽  
J. R. KING
Keyword(s):  
Thin Film ◽  
Contact Angle ◽  
Early Time ◽  
Similarity Solutions ◽  
Bounded Domains ◽  
Extinction Time ◽  
Degenerate Diffusion ◽  
Thin Film Equation ◽  
Order Degenerate ◽  
Mass Increase

We investigate the extinction behaviour of a fourth order degenerate diffusion equation in a bounded domain, the model representing the flow of a viscous fluid over edges at which zero contact angle conditions hold. The extinction time may be finite or infinite and we distinguish between the two cases by identification of appropriate similarity solutions. In certain cases, an unphysical mass increase may occur for early time and the solution may become negative; an appropriate remedy for this is noted. Numerical simulations supporting the analysis are included.

scholarly journals Blowup and dissipation in a critical-case unstable thin film equation

2004 ◽  
Vol 15 (2) ◽  
pp. 223-256 ◽  
Author(s):  
T. P. WITELSKI ◽  
A. J. BERNOFF ◽  
A. L. BERTOZZI
Keyword(s):  
Thin Film ◽  
Fourth Order ◽  
Critical Case ◽  
Blow Up ◽  
Critical Mass ◽  
Similarity Solutions ◽  
Fluid Layers ◽  
Parabolic Pde ◽  
Thin Film Equation ◽  
Order Degenerate

We study the dynamics of dissipation and blow-up in a critical-case unstable thin film equation. The governing equation is a nonlinear fourth-order degenerate parabolic PDE derived from a generalized model for lubrication flows of thin viscous fluid layers on solid surfaces. %For a special balance between %destabilizing second-order terms and regularizing fourth-order terms, There is a critical mass for blow-up and a rich set of dynamics including families of similarity solutions for finite-time blow-up and infinite-time spreading. The structure and stability of the steady-states and the compactly-supported similarity solutions is studied.

scholarly journals Dipoles and similarity solutions of the thin film equation in the half-line

Nonlinearity ◽  
2000 ◽  
Vol 13 (2) ◽  
pp. 413-439 ◽  
Author(s):  
Francisco Bernis ◽  
Josephus Hulshof ◽  
John R King
Keyword(s):  
Thin Film ◽  
Similarity Solutions ◽  
Half Line ◽  
Thin Film Equation

Unstable sixth-order thin film equation: I. Blow-up similarity solutions

Nonlinearity ◽  
2007 ◽  
Vol 20 (8) ◽  
pp. 1799-1841 ◽  
Author(s):  
J D Evans ◽  
V A Galaktionov ◽  
J R King
Keyword(s):  
Thin Film ◽  
Blow Up ◽  
Similarity Solutions ◽  
Sixth Order ◽  
Thin Film Equation

Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3

2013 ◽  
Vol 24 (5) ◽  
pp. 735-760 ◽  
Author(s):  
LORENZO GIACOMELLI ◽  
MANUEL V. GNANN ◽  
FELIX OTTO
Keyword(s):  
Thin Film ◽  
Contact Angle ◽  
Wave Solution ◽  
Travelling Wave ◽  
Small Data ◽  
Travelling Wave Solution ◽  
Moving Contact Line ◽  
Source Type ◽  
Moving Contact ◽  
Thin Film Equation

In one space dimension, we consider source-type (self-similar) solutions to the thin-film equation with vanishing slope at the edge of their support (zero contact-angle condition) in the range of mobility exponents $n\in\left(\frac 3 2,3\right)$. This range contains the physically relevant case n=2 (Navier slip). The existence and (up to a spatial scaling) uniqueness of these solutions has been established in [3] (Bernis, F., Peletier, L. A. & Williams, S. M. (1992) Source type solutions of a fourth-order nonlinear degenerate parabolic equation. Nonlinear Anal. 18, 217–234). It is also shown there that the leading-order expansion near the edge of the support coincides with that of a travelling-wave solution. In this paper we substantially sharpen this result, proving that the higher order correction is analytic with respect to two variables: the first one is just the spatial variable whereas the second one is a (generically irrational, in particular for n=2) power of it, which naturally emerges from a linearisation of the operator around the travelling-wave solution. This result shows that – as opposed to the case of n=1 (Darcy) or to the case of the porous medium equation (the second-order analogue of the thin-film equation) – in this range of mobility exponents, source-type solutions are not smooth at the edge of their support even when the behaviour of the travelling wave is factored off. We expect the same singular behaviour for a generic solution to the thin-film equation near its moving contact line. As a consequence, we expect a (short-time or small-data) well-posedness theory – of which this paper is a natural prerequisite – to be more involved than in the case n=1.

On continuous branches of very singular similarity solutions of the stable thin film equation. II – Free-boundary problems

2011 ◽  
Vol 22 (3) ◽  
pp. 245-265 ◽  
Author(s):  
J. D. EVANS ◽  
V. A. GALAKTIONOV
Keyword(s):  
Thin Film ◽  
Cauchy Problem ◽  
Free Boundary ◽  
Boundary Problem ◽  
Similarity Solutions ◽  
Thin Film Equation ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Image Position ◽  
Similar Solutions

We discuss the fourth-order thin film equation with a stable second-order diffusion term, in the context of a standard free-boundary problem with zero height, zero contact angle and zero-flux conditions imposed at an interface. For the first critical exponent where N ≥ 1 is the space dimension, there are continuous sets (branches) of source-type very singular self-similar solutions of the form For p ≠ p0, the set of very singular self-similar solutions is shown to be finite and consists of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of second kind similarity solutions of the pure thin film equation Such solutions are detected by a combination of linear and non-linear ‘Hermitian spectral theory’, which allows the application of an analytical n-branching approach. In order to connect with the Cauchy problem in Part I, we identify the cauchy problem solutions as suitable ‘limits’ of the free-boundary problem solutions.

On continuous branches of very singular similarity solutions of a stable thin film equation. I – The Cauchy problem

2011 ◽  
Vol 22 (3) ◽  
pp. 217-243 ◽  
Author(s):  
J. D. EVANS ◽  
V. A. GALAKTIONOV
Keyword(s):  
Thin Film ◽  
Cauchy Problem ◽  
Similarity Solutions ◽  
Diffusion Term ◽  
Source Type ◽  
Thin Film Equation ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Image Position ◽  
Harmonic Equation

We consider the fourth-order thin film equation, with a stable second-order diffusion term. For the first critical exponent, where N ≥ 1 is the space dimension, the Cauchy problem is shown to admit countable continuous branches of source-type self-similar very singular solutions of the form These solutions are inherently oscillatory in nature and will be shown in Part II to be the limit of appropriate free-boundary problem solutions. For p ≠ p0, the set of very singular solutions is shown to be finite and to be consisting of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of similarity solutions of the second kind of the pure thin film equation Such solutions are detected by the ‘Hermitian spectral theory’, which allows an analytical n-branching approach. As such, a continuous path as n → 0+ can be constructed from the eigenfunctions of the linear rescaled operator for n = 0, i.e. for the bi-harmonic equation ut = −Δ2u. Numerics are used, wherever appropriate, to support the analysis.

scholarly journals Smooth zero-contact-angle solutions to a thin-film equation around the steady state

2008 ◽  
Vol 245 (6) ◽  
pp. 1454-1506 ◽  
Author(s):  
Lorenzo Giacomelli ◽  
Hans Knüpfer ◽  
Felix Otto
Keyword(s):  
Thin Film ◽  
Contact Angle ◽  
Steady State ◽  
Thin Film Equation
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