Well-posedness for the Navier-slip thin-film equation in the case of complete wetting

Giacomelli, Lorenzo; Gnann, Manuel V.; Knüpfer, Hans; Otto, Félix

doi:10.1016/j.jde.2014.03.010

Cited by 45 publications

(108 citation statements)

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“…Compared to the porous-medium case, where solutions are smooth up to the boundary (cf. Angenent [2]), in the thin-film case -unless n = 1 -this is in general not true as proven for source-type solutions in [24], and for general solutions in a neighborhood of n = 2 in the complete wetting regime by Giacomelli, Knüpfer, Otto, and one of the authors of this paper in [24,30]. The partial wetting case was discussed by Knüpfer, and Knüpfer and Masmoudi, respectively, in [40][41][42][43][44] covering the intervals n ∈ (0, 14/5) \ {5/2, 8/3, 11/4}.…”

Section: Introductionmentioning

confidence: 99%

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A dynamical systems approach for the contact-line singularity in thin-film flows

2016

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Abstract. We are interested in a complete characterization of the contact-line singularity of thin-film flows for zero and nonzero contact angles. By treating the model problem of source-type self-similar solutions, we demonstrate that this singularity can be understood by the study of invariant manifolds of a suitable dynamical system. In particular, we prove regularity results for singular expansions near the contact line for a wide class of mobility exponents and for zero and nonzero dynamic contact angles. Key points are the reduction to center manifolds and identifying resonance conditions at equilibrium points. The results are extended to radially-symmetric source-type solutions in higher dimensions. Furthermore, we give dynamical systems proofs for the existence and uniqueness of self-similar droplet solutions in the nonzero dynamic contact-angle case.

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Section: Introductionmentioning

confidence: 99%

“…The study of the regularity of solutions to (1.1a) at the free boundary has attracted increasing interest in the last years [24,27,28,30,36,40,41,43,44]. Physically, the interest lies in a detailed understanding of the regularizing effect of various (nonlinear) slip conditions at the liquid-solid interface for the underlying fluid models.…”

Section: Introductionmentioning

confidence: 99%

A dynamical systems approach for the contact-line singularity in thin-film flows

2016

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“…We expect that the rather explicit characterization of the boundary regularity of sourcetype self-similar solutions in [15] and a corresponding well-posedness result in case of the half-line [14] can lead to new insights here. Yet, the linear operator in this case does seem to have an apparent symmetric structure as the operator L given in (1.19) (cf.…”

Section: Discussionmentioning

confidence: 90%

“…[14]) where h(t, Z(t, x)) := h s (t, Z s (t, x)). Furthermore, the transformation guarantees that the mass M is conserved in time t. This condition would not necessarily be fulfilled if we chose a related coordinate transformation, the hodograph transform (cf.…”

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confidence: 99%

Well-Posedness and Self-Similar Asymptotics for a Thin-Film Equation

Gnann¹

2015

SIAM J. Math. Anal.

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We investigate compactly supported solutions for a thin-film equation with linear mobility in the regime of perfect wetting. This problem has already been addressed by Carrillo and Toscani, proving that the source-type self-similar profile is a global attractor of entropy solutions with compactly supported initial data. Here we study small perturbations of source-type self-similar solutions for the corresponding classical free boundary problem and set up a global existence and uniqueness theory within weighted L 2 -spaces under minimal assumptions. Furthermore, we derive asymptotics for the evolution of the solution, the free boundary, and the center of mass. As spatial translations are scaled out in our reference frame, the rate of convergence is higher than the one obtained by Carrillo and Toscani.Since the boundary is a function of time t, we need a condition determining its evolution which is (1.1c). Viewing (1.1a) as a nonlinear continuity equation, we read off the horizontal fluid velocity as v = ∂ 3 z h. By compatibility, this velocity has to match the speed of the contact lines at the free boundaries. A simple calculation shows that (1.1c) implies conservation of mass (volume) M :=We mention that problem (1.1) can be derived by a lubrication approximation from Darcy's flow in the Hele-Shaw cell [18,21,22]. Equation (1.1a) is a specific case of a larger class of thin-film equations,with a nonlinear mobility h n and where n ∈ (0, 3). Again, (1.2) can be derived from the Navier-Stokes system with an in general nonlinear slip condition at the liquid-solid interface by an asymptotic expansion [8,13,29]. Source-type self-similar solutions.It is instructive to study the case where the initial condition comes from a Dirac mass M δ 0 at time t = −1, where M is the mass of the droplet. Additionally we assume that the solution to (1.1) is self-similar. We note that (1.1a) remains invariant under the two-parameter scaling transformation if we additionally assume conservation of mass, i.e., H * Z * = 1. Hence our source-type self-similar solutions attain the structureThis ansatz automatically yields films for which the mass is constant in time t. We insert ansatz (1.3) in (1.1a) and obtain after some basic manipulations (cf. Appendix A)which is known as the Smyth-Hill profile [34]. Convergence to source-type self-similar solutions.It is known that the long-time dynamics of (1.1) for rather general initial data is governed by the sourcetype self-similar solution (1.4). A first observation supporting this insight was obtained by Bernis [3,4] through the study of the Cauchy problem for the thin-film equation in the 1 + 1-dimensional case. Bernis was able to show that the speed of propagation of the interface is finite and the spreading rate asymptotically matches the one of the source-type solution. For the case of mobility exponent n = 1 (the Darcy flow in the Hele-Shaw cell), a stronger result by Carrillo and Toscani is available [12]: Using analogies to the second-order counterpart of (1.1a), the porous medium equatio...

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“…The analysis of thin film equations in the present of contact lines is well developed research area (cf. [4], [5], [6], [7], [16], [17], [18], [19], [20], [27]). The dynamics of a coupled system for a thin film approximation of the two phase Stokes flow has been considered in [15] as well as [8].…”

Section: Introductionmentioning

confidence: 99%