We consider the gravity-driven laminar flow of a shallow fluid layer down an uneven incline with the principal objective of investigating the effect of bottom topography and surface tension on the stability of the flow. The equations of motion are approximations to the Navier-Stokes equations which exploit the assumed relative shallowness of the fluid layer. Included in these equations are diffusive terms that are second order relative to the shallowness parameter. These terms, while small in magnitude, represent an important dependence of the flow dynamics on the variation in bottom topography and play a significant role in theoretically capturing important aspects of the flow. Some of the second-order terms include normal shear contributions, while others lead to a nonhydrostatic pressure distribution. The explicit dependence on the cross-stream coordinate is eliminated from the equations of motion by means of a weighted residual approach. The resulting mathematical formulation constitutes an extension of the modified integral-boundary-layer equations proposed by Ruyer-Quil and Manneville ͓Eur. Phys. J. B 15, 357 ͑2000͔͒ for flows over even surfaces to flows over variable topography. A linear stability analysis of the steady flow is carried out by taking advantage of Floquet-Bloch theory. A numerical scheme is devised for solving the nonlinear governing equations and is used to calculate the evolution of the perturbed equilibrium flow. The simulations are used to confirm the analytical predictions and to investigate the interfacial wave structure. The bottom profile considered in this investigation corresponds to periodic undulations characterized by measures of wavelength and amplitude. Conclusions are drawn on the combined effect of bottom topography and surface tension.

The two-dimensional problem of gravity-driven laminar flow of a thin layer of fluid down a heated wavy inclined surface is discussed. The coupled effect of bottom topography, variable surface tension and heating has been investigated both analytically and numerically. A stability analysis is conducted while nonlinear simulations are used to validate the stability predictions and also to study thermocapillary effects. The governing equations are based on the Navier–Stokes equations for a thin fluid layer with the cross-stream dependence eliminated by means of a weighted residual technique. Comparisons with experimental data and direct numerical simulations have been carried out and the agreement is good. New interesting results regarding the combined role of surface tension and sinusoidal topography on the stability of the flow are presented. The influence of heating and the Marangoni effect are also deduced.

The stability of a conducting fluid flow over a rotating disk with a uniform magnetic field applied normal to the disk, is investigated. It is assumed that the magnetic field is unaffected by the motion of the fluid. The mean flow and linear stability equations are solved for a range of magnetic field-strength parameters and the absolute/convective nature of the stability is investigated. It is found that increasing the magnetic field parameter is in general stabilizing.

We have carried out a theoretical analysis of the linear thermoconvective stability of a ferrofluid, which is confined between two horizontal plates maintained at different constant temperatures and which is subject to an external uniform magnetic field in the vertical direction. The effects of the spin viscosity, vortex viscosity and magnetization relaxation are considered and discussed. The eigenvalue problem is solved by employing the Chebyshev pseudospectral method. It is found that the presence of magnetic field complements the buoyancy force in destabilizing the fluid at lower values of the magnetic field only and when the applied field is increased, the effect is reversed and the flow becomes more stable.

The linear stability of a rotating fluid  in  the annulus  between two concentric cylinders is investigated in the presence of a magnetic field  which is  azimuthal as well as in axial direction. Several results of MHD stability have been derived by using the inner product method. It is shown that when the swirl velocity component is large, the hydromagnetic effects become small compared with those due to swirl. The presence of a velocity field and imposed magnetic field will lead to the basic state to more stability. Keywords: Hydromagnetic Stability; Rotating Fluid. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i2.3945                 J. Sci. Res. 2 (2), 250-256 (2010)