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

Berg, Jan Bouwe van den; Bowen, Mark; King, John R.; El-Sheikh, M. M. A.

doi:10.1017/s0956792504005510

Cited by 3 publications

(7 citation statements)

References 15 publications

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“…Similarity solutions of this general form solve many different classes of initial and boundary value problems for (1.1): source-type solutions of the Cauchy problem, draining solutions of the Dirichlet problem, and Boltzmann solutions of the dam-break problem [13,20,22,32,38]. Moreover, these special solutions often act as large-time attractors for solutions starting from much wider classes of initial conditions; we also expect this to be true for the dipole problem.…”

Section: Similarity Solutionsmentioning

confidence: 78%

The Linear Limit of the Dipole Problem for the Thin Film Equation

Bowen

Witelski

2006

SIAM J. Appl. Math.

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We investigate self-similar solutions of the dipole problem for the one-dimensional thin film equation on the half-line {x ≥ 0}. We study compactly supported solutions of the linear moving boundary problem and show how they relate to solutions of the nonlinear problem. The similarity solutions are generally of the second kind, given by the solution of a nonlinear eigenvalue problem, although there are some notable cases where first-kind solutions also arise. We examine the conserved quantities connected to these first-kind solutions. Difficulties associated with the lack of a maximum principle and the non-self-adjointness of the fundamental linear problem are also considered. Seeking similarity solutions that include sign changes yields a surprisingly rich set of (coexisting) stable solutions for the intermediate asymptotics of this problem. Our results include analysis of limiting cases and comparisons with numerical computations.

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Section: Similarity Solutionsmentioning

confidence: 78%

The Linear Limit of the Dipole Problem for the Thin Film Equation

Bowen

Witelski

2006

SIAM J. Appl. Math.

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“…Their n ↑ 2 study via asymptotics and computations yields results that are somewhat similar in spirit to Corollaries 14 and 22. In [24], van den Berg et al consider dipole solutions on a fixed interval [0,1]. In this case, separable solutions are the exact solutions dictated by scaling arguments.…”

Section: 2mentioning

confidence: 99%

“…Assume U (0) = H, U (0) = 0, and U (0) = γ = −H 3 /4 + H. Let U be the resulting solution of equation (12). Let S + (H) and S − (H) be as defined in (24) and α(H, γ) be as defined in (23)…”

Section: 2mentioning

confidence: 99%

“…Let α(H, γ) be as defined in (23) and S + (H) be as defined in (24). If γ ∈ S + (H) and X cut < α(H, γ) then for all x ∈ (X cut , α(H, γ)), U (x) < 0 and…”

Section: 2mentioning

confidence: 99%

“…Finally, since U (α) ≥ 0, the negativity of U implies that U (x) > 0 for x ∈ (X cut , α). (24). U is the solution of equation (12) with U (0) = H, U (0) = 0, and U (0) = γ.…”

Section: 2mentioning

confidence: 99%

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Selfsimilar blowup of unstable thin-film equations

Slepčev¹,

Pugh²

2005

Indiana Univ. Math. J.

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ABSTRACT. We study selfsimilar blowup of long-wave unstable thin-film equations with critical powers of nonlinearities: ut = −(u n uxxx + u n+2 ux)x. We show that the equation cannot have selfsimilar solutions (with zero contact angles) that blow up in finite time if n ≥ 3/2. We show that for 0 < n < 3/2 there are compactly supported, symmetric, selfsimilar solutions (with zero contact angles) that blow up in finite time. Moreover there exist families of these solutions with any number of local maxima. We also study the asymptotic behaviour of these selfsimilar solutions as n approaches 3/2 and obtain a sharp lower bound on the height of solutions one time unit before the blowup. We also prove qualitative properties of solutions; for example, the profile of a selfsimilar solution that blows up in finite time can always be bounded from below by a compactly supported steady state.

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Steady and quasi-steady thin viscous flows near the edge of a solid surface

Баренблатт

Bertsch²,

Giacomelli³

2010

Eur. J. Appl. Math

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Abstract.A new approach is proposed to the description of thin viscous flows near the edges of a solid surface. For a steady flow, the lubrication approximation and the no-slip condition are assumed to be valid on most of the surface, except for relatively small neighborhoods of the edges, where an universality principle is postulated: the behavior of the liquid in these regions is universally determined by flux, external conditions and material properties. The resulting mathematical model is formulated as an ordinary differential equation involving the height of the liquid film and the flux as unknowns, and analytical results are outlined. The form of the universal functions which describe the behavior in the edge regions is also discussed, obtaining conditions of compatibility with lubrication theory for small fluxes. Finally, an ordinary differential equation is introduced for the description of moderate time asymptotic profiles of a liquid film which flows off a bounded solid surface.

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

Cited by 3 publications

References 15 publications

The Linear Limit of the Dipole Problem for the Thin Film Equation

The Linear Limit of the Dipole Problem for the Thin Film Equation

Selfsimilar blowup of unstable thin-film equations

Steady and quasi-steady thin viscous flows near the edge of a solid surface

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