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.
A long-wave evolution equation is derived using an asymptotic analysis, and the linear stability of a viscoelastic film flowing along the direction of parallel grooves over a uniformly heated topography is studied. A numerical approach adopting spectral collocation technique is used to demarcate the stable and unstable flow regimes. The combined influence of thermocapillarity and viscoelasticity on the films stability is analyzed. By accounting the bottom topography comprising longitudinal gutters, the possibilities of regulating the film dynamics under iso- and nonisothermal conditions and/or optimizing design structure of an apparatus for desirable flow outcomes have been focussed.
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