Reverse draining of a magnetic soap film – analysis and simulation of a thin film equation with non-uniform forcing. (English) Zbl 1195.76130
The authors analyse and classify the equilibrium solutions of the one-dimensional thin film equation with no-flux boundary conditions and in the presence of a spatially-dependent external forcing. The paper contains theorems that shed light on the nature of these equilibrium solutions, guarantee their validity, and describe how they depend on the properties of the external forcing. The obtained results are applied to the reverse draining of a one-dimensional magnetic soap film subject to an external non-uniform magnetic field. Numerical simulations which illustrate the convergence of solutions towards the equilibrium configuration, and bifurcation diagrams for steady-state solutions are presented for this case. A remarkable finding is that multiple stable equilibrium solutions exist for fixed parameters, and a rich bifurcation structure relates to these solutions, demonstrating the complexity hidden in a relatively simple evolution equation. Finally, the authors provide a simulation describing how the numerical solutions traverse the bifurcation diagram when the forcing amplitude is slowly increased and then decreased.
Reviewer: Felix Kaplanski (Tallinn)
MSC:
| 76A20 | Thin fluid films |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 76E25 | Stability and instability of magnetohydrodynamic and electrohydrodynamic flows |
| 76D08 | Lubrication theory |
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