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. 1. Introduction A nonlinear fourth-order degenerate parabolic differential equation of the form: where , , and are arbitrary continuous functions of the interfacial thickness , represents a scalar conservation law associated to the flow of a thin viscous layer on an incline under different conditions. The stability and dynamics of equations of the type of (1) is a subject of major interest [1–18] because of their robustness in regimes where viscosity dominates inertia [19]. Such studies have focused attention primarily on the isothermal and nonisothermal instability analysis, mainly for nonconducting fluids. Since the investigation of Chandrasekar [20] on the stability of a flow between coaxial rotating cylinders in the presence of a magnetic field held in the axial direction, the laminar flow of an electrically conducting fluid under the presence of a magnetic field has been studied extensively. For instance, Stuart [21] has reported on the stability of a pressure flow between parallel plates under the application of a parallel magnetic field. Among other earlier investigations, Lock [22] examined the stability when the magnetic field is applied perpendicular to the flow direction and to the boundary planes. Hsieh [23] found
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