Pressure-dipole solutions of the thin-film equation

2018 ◽  
Vol 30 (2) ◽  
pp. 358-399
Author(s):  
M. BOWEN ◽  
T. P. WITELSKI
Keyword(s):  
Thin Film ◽  
Accumulation Point ◽  
Full Time ◽  
Infinite Domain ◽  
Compactly Supported ◽  
Sign Changes ◽  
Thin Film Equation ◽  
Self Similar ◽  
Type Boundary ◽  
Similar Solutions

We investigate self-similar sign-changing solutions to the thin-film equation, ht = −(|h|nhxxx)x, on the semi-infinite domain x ≥ 0 with zero-pressure-type boundary conditions h = hxx = 0 imposed at the origin. In particular, we identify classes of first- and second-kind compactly supported self-similar solutions (with a free-boundary x = s(t) = Ltβ) and consider how these solutions depend on the mobility exponent n; multiple solutions can exist with the same number of sign changes. For n = 0, we also construct first-kind self-similar solutions on the entire half-line x ≥ 0 and show that they act as limiting cases for sequences of compactly supported solutions in the limit of infinitely many sign changes. In addition, at n = 1, we highlight accumulation point-like behaviour of sign-changes local to the moving interface x = s(t). We conclude with a numerical investigation of solutions to the full time-dependent partial differential equation (based on a non-local regularisation of the mobility coefficient) and discuss the computational results in relation to the self-similar solutions.

scholarly journals On a fractional thin film equation

2020 ◽  
Vol 9 (1) ◽  
pp. 1516-1558
Author(s):  
Antonio Segatti ◽  
Juan Luis Vázquez
Keyword(s):  
Thin Film ◽  
Porous Medium ◽  
Porous Medium Equation ◽  
Order Theory ◽  
Compactly Supported ◽  
Limit Value ◽  
Medium Equation ◽  
Thin Film Equation ◽  
Self Similar ◽  
Similar Solutions

Abstract This paper deals with a nonlinear degenerate parabolic equation of order α between 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit value α = 4 while the Porous Medium Equation is the limit α = 2. We prove existence of a nonnegative weak solution for a general class of initial data, and establish its main properties. We also construct the special solutions in self-similar form which turn out to be explicit and compactly supported. As in the porous medium case, they are supposed to give the long time behaviour or the wide class of solutions. This last result is proved to be true under some assumptions. Lastly, we consider nonlocal equations with the same nonlinear structure but with order from 4 to 6. For these equations we construct self-similar solutions that are positive and compactly supported, thus contributing to the higher order theory.

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.

Effective and microscopic contact angles in thin film dynamics

2000 ◽  
Vol 11 (2) ◽  
pp. 181-201 ◽  
Author(s):  
MICHIEL BERTSCH ◽  
ROBERTA DAL PASSO ◽  
STEPHEN H. DAVIS ◽  
LORENZO GIACOMELLI
Keyword(s):  
Thin Film ◽  
Solid Surface ◽  
Contact Line ◽  
Liquid Films ◽  
Contact Angles ◽  
Intermediate Asymptotics ◽  
Quantitative Expression ◽  
Thin Film Equation ◽  
Self Similar ◽  
Similar Solutions

We introduce and analyse a class of quasi-self-similar solutions of the thin film equation to describe the dynamics of expanding liquid films on a solid surface. Using these solutions as intermediate asymptotics profiles, we obtain a quantitative expression for the shape of the film and a relation between the speed of the contact line and the macroscopic and microscopic contact angles.

Blow-up similarity solutions of the fourth-order unstable thin film equation

2007 ◽  
Vol 18 (2) ◽  
pp. 195-231 ◽  
Author(s):  
J. D. EVANS ◽  
V. A. GALAKTIONOV ◽  
J. R. KING
Keyword(s):  
Thin Film ◽  
Fourth Order ◽  
Blow Up ◽  
Similarity Solutions ◽  
Diffusion Term ◽  
Thin Film Equation ◽  
Self Similar ◽  
Image Position ◽  
General Aspects ◽  
Similar Solutions

We study blow-up behaviour of solutions of the fourth-order thin film equationwhich contains a backward (unstable) diffusion term. Our main goal is a detailed study of the case of the first critical exponentwhereN≥ 1 is the space dimension. We show that the free-boundary problem with zero contact angle and zero-flux conditions admits continuous sets (branches) of blow-up self-similar solutions. For the Cauchy problem inRN×R+, we detect compactly supported blow-up patterns, which have infinitely many oscillations near interfaces and exhibit a “maximal” regularity there. As a key principle, we use the fact that, for small positiven, such solutions are close to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equationwhich are better understood and have been studied earlier [19]. We also discuss some general aspects of formation of self-similar blow-up singularities for other values ofp.

scholarly journals The self-similar solution for draining in the thin film equation

2004 ◽  
Vol 15 (3) ◽  
pp. 329-346 ◽  
Author(s):  
JAN BOUWE VAN DEN BERG ◽  
MARK BOWEN ◽  
JOHN R. KING ◽  
M. M. A. EL-SHEIKH
Keyword(s):  
Thin Film ◽  
Contact Angle ◽  
Boundary Conditions ◽  
Finite Domain ◽  
Series Solutions ◽  
Self Similar Solution ◽  
Power Series Solutions ◽  
Thin Film Equation ◽  
Self Similar ◽  
Similar Solutions

We investigate self-similar solutions of the thin film equation in the case of zero contact angle boundary conditions on a finite domain. We prove existence and uniqueness of such a solution and determine the asymptotic behaviour as the exponent in the equation approaches the critical value at which zero contact angle boundary conditions become untenable. Numerical and power-series solutions are also presented.

Self-similar solutions of the second kind for the modified porous medium equation

1994 ◽  
Vol 5 (3) ◽  
pp. 391-403 ◽  
Author(s):  
Josephus Hulshof ◽  
Juan Luis Vazquez
Keyword(s):  
Porous Medium ◽  
Porous Medium Equation ◽  
Self Similarity ◽  
Compactly Supported ◽  
Medium Equation ◽  
Self Similar ◽  
Image Position ◽  
Similar Solutions

We construct compactly supported self-similar solutions of the modified porous medium equation (MPME)They have the formwhere the similarity exponents α and β depend on ε, m and the dimension N. This corresponds to what is known in the literature as anomalous exponents or self-similarity of the second kind, a not completely understood phenomenon. This paper performs a detailed study of the properties of the anomalous exponents of the MPME.

The Influence of Boundaries on Gravity Currents and Thin Films: Drainage, Confinement, Convergence, and Deformation Effects

2022 ◽  
Vol 54 (1) ◽  
pp. 27-56
Author(s):  
Zhong Zheng ◽  
Howard A. Stone
Keyword(s):  
Thin Film ◽  
Oil And Gas ◽  
Gravity Currents ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Self Similar ◽  
Similar Solutions ◽  
Film Flows ◽  
Gravity Surface

Thin film flows, whether driven by gravity, surface tension, or the relaxation of elastic boundaries, occur in many natural and industrial processes. Applications span problems of oil and gas transport in channels to hydraulic fracture, subsurface propagation of pollutants, storage of supercritical CO2 in porous formations, and flow in elastic Hele–Shaw configurations and their relatives. We review the influence of boundaries on the dynamics of thin film flows, with a focus on gravity currents, including the effects of drainage into the substrate, and the role of the boundaries to confine the flow, force its convergence to a focus, or deform, and thus feedback to alter the flow. In particular, we highlight reduced-order models. In many cases, self-similar solutions can be determined and describe the behaviors in canonical problems at different timescales and length scales, including self-similar solutions of both the first and second kind. Additionally, the time transitions between different solutions are summarized. Where possible, remarks about various applications are provided.

Existence of self-similar solutions of the two-dimensional Navier–Stokes equation for non-Newtonian fluids

2019 ◽  
Vol 26 (1/2) ◽  
pp. 167-178 ◽  
Author(s):  
Dongming Wei ◽  
Samer Al-Ashhab
Keyword(s):  
Newtonian Fluids ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Infinite Domain ◽  
Navier Stokes Equation ◽  
Continuity Equations ◽  
Self Similar ◽  
Similar Solutions ◽  
Two Parameters

The reduced problem of the Navier–Stokes and the continuity equations, in two-dimensional Cartesian coordinates with Eulerian description, for incompressible non-Newtonian fluids, is considered. The Ladyzhenskaya model, with a non-linear velocity dependent stress tensor is adopted, and leads to the governing equation of interest. The reduction is based on a self-similar transformation as demonstrated in existing literature, for two spatial variables and one time variable, resulting in an ODE defined on a semi-infinite domain. In our search for classical solutions, existence and uniqueness will be determined depending on the signs of two parameters with physical interpretation in the equation. Illustrations are included to highlight some of the main results.

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