A unified linear theory of rotating hydromagnetic flow between two parallel infinite plates subject to imposition of axial velocity

• Published in: Zeitschrift für angewandte Mathematik und Mechanik Vol. 98; no. 8; pp. 1369 - 1389
• Main Authors: Badeti, Satyanarayana, Vempaty, Somaraju, Suripeddi, Srinivas
• Format: Journal Article
• Language: English
• Published: Weinheim Wiley Subscription Services, Inc 01.08.2018
• Subjects: Axisymmetric flow; Boundary layers; Electric currents; Flux; hydromagnetics; injection and suction; Parameters; Plates (structural members); rotating fluids; Rotation; Suction; Thickness; Velocity
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.201700174

Uniformly valid solutions for a linear, steady, axisymmetric, hydromagnetic flow confined between two electrically insulated, infinite, differentially rotating plates subject to constant axial velocity of fluid through injection at bottom plate and an equal suction at the top plate are presented. The applied magnetic field is in the direction of the rotation vector Ωẑ. The dynamics change depending on the parameter ranges defined by the three nondimensional parameters, E(=υL2Ω) the Ekman number, α2(=σB02ρΩ) the Elsasser number and R(=ULΩ) the injection Rossby number. There are broadly four parameter ranges and the transition from one dynamics to the other is explained by a critical study of the flow in the hitherto unexplored transition parameter range E1/2α−1≪R≫E1/2α. It is found that the dynamics of the flow change even for α≪1 in the presence of the imposed axial velocity. The viscous Ekman‐Hartmann layer at the bottom boundary blown up by the injection of the fluid and it transforms into a new inviscid resistive boundary layer of thickness O(Rα2). The suction layer at the top boundary characterized by a force balance between viscous and inertial forces has a thickness O(ER). In the interior, the meridional circulations of mass flux as well as electric current flux have z−dependent structure for α≪1. For α≫1, the electric current circulation loses its z−dependent structure while the mass flux circulation continues to retain its z−dependent structure until α→RE−1/2. The interior spins up to the azimuthal velocity on the bottom plate in a time O(R−1) which happens to be less than the hydrodynamic and hydromagnetic spin‐up time. Uniformly valid solutions for a linear, steady, axisymmetric, hydromagnetic flow confined between two electrically insulated, infinite, differentially rotating plates subject to constant axial velocity of fluid through injection at bottom plate and an equal suction at the top plate are presented. The applied magnetic field is in the direction of the rotation vector Ωẑ. The dynamics change depending on the parameter ranges defined by the three nondimensional parameters, E(=υL2Ω) the Ekman number, α2(=σB02ρΩ) the Elsasser number and R(=ULΩ) the injection Rossby number. There are broadly four parameter ranges and the transition from one dynamics to the other is explained by a critical study of the flow in the hitherto unexplored transition parameter range E1/2α−1≪R≫E1/2α.…