TY - JOUR
T1 - Analytical solutions of one-way coupled magnetohydrodynamic free surface flow
AU - Righolt, B. W.
AU - Kenjeres, S.
AU - Kalter, R.
AU - Tummers, M. J.
AU - Kleijn, C. R.
PY - 2016
Y1 - 2016
N2 - 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(A4)), Bond (Bo ≤ O(A2)) 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 Ha2 ≤ 1, Ca ≤ A4, Bo ≤ A2, 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 Ha2 is increased to 400, Ca to 200A4 or Bo to 100A2.
AB - 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(A4)), Bond (Bo ≤ O(A2)) 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 Ha2 ≤ 1, Ca ≤ A4, Bo ≤ A2, 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 Ha2 is increased to 400, Ca to 200A4 or Bo to 100A2.
KW - Analytical solutions
KW - Free surface
KW - Lubrication theory
KW - Magnetohydrodynamics
KW - Matched asymptotic expansions
KW - Numerical solutions
UR - http://www.scopus.com/inward/record.url?scp=84956796575&partnerID=8YFLogxK
UR - http://resolver.tudelft.nl/uuid:1c1214a5-ab9f-46ab-9de9-80b05e166214
U2 - 10.1016/j.apm.2015.09.101
DO - 10.1016/j.apm.2015.09.101
M3 - Article
AN - SCOPUS:84956796575
SN - 0307-904X
VL - 40
SP - 2577
EP - 2592
JO - Applied Mathematical Modelling: simulation and computation for engineering and environmental systems
JF - Applied Mathematical Modelling: simulation and computation for engineering and environmental systems
IS - 4
ER -