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Aeronautical Engineering applied magnetic field approximation aspect ratio assumed Blake boundary conditions channel height corresponding current equation derivatives differential equation Due to Fringing effective pump length effects caused electrical electromagnetic pumps equation 31 evaluated Figure VII first-order perturbation fluid interaction fluid velocity function of aspect Hartmann number Hartmann-type boundary layer ideal efficiency ideal performance ideal pressure head induced effects induced magnetic field inertia forces inertia terms integrated interaction parameter laboratory coordinates laminar-flow induction pump Laplace's equation linear long pump longitudinal velocity magnetic field intensity magnetic Reynolds number magnetic vector potential magnetic-field intensity magnetohydrodynamic magnitude momentum equations neglected no-slip condition node position obtained Ohm's Law ohmic heating ohmic losses pondermotive region resulting equations shear stress short pumps sinusoidal magnetic field skin friction slip ratio slip velocities steady coordinate system thin boundary layers Turcotte typical pump unit width upstream velocity profile viscous effects viscous loss walls wave length zero zero-order solution