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Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der Waals forces. (English) Zbl 1143.76473

Summary: We consider a fully practical finite-element approximation of the following system of nonlinear degenerate parabolic equations:
\[ \frac{\partial u}{\partial t} + \frac{1}{2} \nabla . (u^2 \nabla [\sigma(v)])- \frac{1}{3} \nabla .(u^3 \nabla w) = 0, \qquad w =- c\Delta u + a u^{-3}- \delta u^{-\nu}, \]
\[ \frac{\partial v}{\partial t} + \nabla . (uv\nabla [\sigma(v)])- \rho\Delta v-\frac{1}{2} \nabla .(u^2v \nabla w) = 0. \]
The above models a surfactant-driven thin-film flow in the presence of both attractive, \(a>0\), and repulsive, \(\delta>0\) with \(\nu>3\), van der Waals forces; where \(u\) is the height of the film, \(v\) is the concentration of the insoluble surfactant monolayer and \(\sigma(v):=1-v\) is the typical surface tension. Here \(\rho\geq 0\) and \(c>0\) are the inverses of the surface Peclet number and the modified capillary number. In addition to showing stability bounds for our approximation, we prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system, (i) in one space dimension when \(\rho>0\); and, moreover, (ii) in two space dimensions if in addition \(\nu\geq 7\). Furthermore, iterative schemes for solving the resulting nonlinear discrete system are discussed. Finally, some numerical experiments are presented.
See also the authors and H. Garcke [SIAM J. Numer. Anal. 41, No. 4, 1427–1464 (2003; Zbl 1130.76361)].

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76A20 Thin fluid films
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76D08 Lubrication theory
76D45 Capillarity (surface tension) for incompressible viscous fluids

Citations:

Zbl 1130.76361

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