### Abstract

We study the flow in a layer of conductive liquid under the influence of surface tension, gravity, and Lorentz forces due to imposed potential differences and transverse magnetic fields, as a function of the Hartmann number, the Bond number, the Reynolds number, the capillary number and the height-to-width ratio A. For aspect ratios A 1 and Reynolds numbers Re≤A, lubrication theory is applied to determine the steady state shape of the liquid surface to lowest order. Assuming low Hartmann (Ha ≤ O(1)), capillary (Ca ≤ O(A^{4})), Bond (Bo ≤ O(A^{2})) numbers and contact angles close to 90°, the flow details below the surface and the free surface elevation for the complete domain are determined analytically using the method of matched asymptotic expansions. The amplitude of the free surface deformation scales linearly with the capillary number and decreases with increasing Bond number, while the shape of the free surface depends on the Bond number and the contact angle condition. The strength of the flow scales linearly with the magnetic field gradient and applied potential difference and vanishes for high aspect ratio layers (A → 0). The analytical model results are confirmed by numerical simulations using a finite volume moving mesh interface tracking method, where the Lorentz force is calculated from the equation for the electric potential. It is shown that the analytical result for the free surface elevation is accurate within 0.4% from the numerical results for Ha^{2} ≤ 1, Ca ≤ A^{4}, Bo ≤ A^{2}, Re ≤ A and A ≤ 0.1 and within 2% for A=0.5. For A=0.1, the solution remains accurate within 1% of the numerical solution when either Ha^{2} is increased to 400, Ca to 200A^{4} or Bo to 100A^{2}.

Original language | English |
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Pages (from-to) | 2577-2592 |

Journal | Applied Mathematical Modelling: simulation and computation for engineering and environmental systems |

Volume | 40 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2016 |

### Keywords

- Analytical solutions
- Free surface
- Lubrication theory
- Magnetohydrodynamics
- Matched asymptotic expansions
- Numerical solutions

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## Cite this

*Applied Mathematical Modelling: simulation and computation for engineering and environmental systems*,

*40*(4), 2577-2592. https://doi.org/10.1016/j.apm.2015.09.101