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ISRN Mathematical Physics
Volume 2013 (2013), Article ID 748613, 24 pages
Research Article

Air-Aided Shear on a Thin Film Subjected to a Transverse Magnetic Field of Constant Strength: Stability and Dynamics

Department of Mathematics and Statistics, University of Constance, 78457 Constance, Germany

Received 28 June 2013; Accepted 22 August 2013

Academic Editors: L. E. Oxman and W.-H. Steeb

Copyright © 2013 Mohammed Rizwan Sadiq Iqbal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The effect of air shear on the hydromagnetic instability is studied through (i) linear stability, (ii) weakly nonlinear theory, (iii) sideband stability of the filtered wave, and (iv) numerical integration of the nonlinear equation. Additionally, a discussion on the equilibria of a truncated bimodal dynamical system is performed. While the linear and weakly nonlinear analyses demonstrate the stabilizing (destabilizing) tendency of the uphill (downhill) shear, the numerics confirm the stability predictions. They show that (a) the downhill shear destabilizes the flow, (b) the time taken for the amplitudes corresponding to the uphill shear to be dominated by the one corresponding to the zero shear increases with magnetic fields strength, and (c) among the uphill shear-induced flows, it takes a long time for the wave amplitude corresponding to small shear values to become smaller than the one corresponding to large shear values when the magnetic field intensity increases. Simulations show that the streamwise and transverse velocities increase when the downhill shear acts in favor of inertial force to destabilize the flow mechanism. However, the uphill shear acts oppositely. It supports the hydrostatic pressure and magnetic field in enhancing films stability. Consequently, reduced constant flow rates and uniform velocities are observed.