The present paper deals with the theoretical study of a weakly non-linear Rayleigh–Benard convection in temperature-dependent viscosity Newtonian liquids with temperature/gravity modulation. The effect of temperature-dependent variable viscosity and temperature/gravity modulation renders the problem analytically intractable. The imposed thermal boundary condition and gravity field consist of a steady part and a time-periodic oscillatory part as described by Venezian. The instability of convective rolls due to temperature/gravity modulation is expanded in terms of the power series of the amplitude of convection, which is quite small. The Ginzburg–Landau equation derived in this problem is a non-autonomous differential equation and is solved using the numerical technique. The authors have demonstrated that one can suppress or enhance heat transfer by tuning the parameters involved. The effect of different parameters on the heat transfer and streamlines is also studied.

The effect of temperature/gravity modulation on finite amplitude cellular convection with variable viscosity effect

The linear stability characteristics of pressure-driven miscible two-fluid flow with same density and varying viscosities in a channel with velocity slip at the wall are examined. A prominent feature of the instability is that only a band of wave numbers is unstable whatever the Reynolds number is, whereas shorter wavelengths and smaller wave numbers are observed to be stable. The stability characteristics are different from both the limiting cases of interface dominated flows and continuously stratified flows in a channel with velocity slip at the wall. The flow system is destabilizing when a more viscous fluid occupies the region closer to the wall with slip. For this configuration a new mode of instability, namely the overlap mode, appears for high mass diffusivity of the two fluids. This mode arises due to the overlap of critical layer of dominant instability with the mixed layer of varying viscosity. The critical layer contains a location in the flow domain at which the base flow velocity equals the phase speed of the most unstable disturbance. Such a mode also occurs in the corresponding flow in a rigid channel, but absent in either of the above limiting cases of flow in a channel with slip. The flow is unstable at low Reynolds numbers for a wide range of wave numbers for low mass diffusivity, mimicking the interfacial instability of the immiscible flows. A configuration with less viscous fluid adjacent to the wall is more stable at moderate miscibility and this is also in contrast with the result for the limiting case of interface dominated flows in a channel with slip, where the above configuration is more unstable. It is possible to achieve stabilization or destabilization of miscible two-fluid flow in a channel with wall slip by appropriately choosing the viscosity of the fluid layer adjacent to the wall. In addition, the velocity slip at the wall has a dual role in the stability of flow system and the trend is influenced by the location of the mixed layer, the location of more viscous fluid and the mass diffusivity of the two fluids. It is well known that creating a viscosity contrast in a particular way in a rigid channel delays the occurrence of turbulence in a rigid channel. The results of the present study show that the flow system can be either stabilized or destabilized by designing the walls of the channel as hydrophobic surfaces, modeled by velocity slip at the walls. The study provides another effective strategy to control the flow system.

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Linear stability analysis of miscible two-fluid flow in a channel with velocity slip at the walls

The combined influence of the porosity of a porous medium and the longitudinal magnetic field on the stability against small disturbances of the pressure-driven laminar fluid flow between two parallel planes is studied. It is assumed that the channel walls have zero conductivity and the region outside the channel is electrically non-conducting. A sixth-order system of disturbance equations is derived and it is solved by the Galerkin method using the Legendre polynomials as its basis. The critical Reynolds number $$Re_c$$
 , the critical wave number $$\alpha _c$$
 and the critical wave speed $$c_c$$
 are computed for a wide range of porosity parameter $$\sigma _p$$
 , magnetic Reynolds number Rm and Alfven number A. Curves of wave number against Reynolds number for neutral stability are presented for a wide range of values of certain parameters. The onset of instability is discussed in detail using the growth rate curves for various parameters present in the problem. The combined influence of the parameters $$\sigma _p$$
 and A to delay the onset of instability in comparison with their influence in isolation is observed and the corresponding results are presented in detail.

The combined effect of porosity of porous media and longitudinal magnetic field on stability of the modified plane Poiseuille flow

Many shear flows exhibit laminar to turbulent transitions at subcritical Reynolds numbers. In this context, the computation of the perturbations that exhibit the largest possible transient growth is of central interest as it often sheds light on the actual bypass transition route. In this work, we consider the effect on the transient growth of a spanwise magnetic field in a boundary layer flow of a liquid metal over an electrically insulating flat plate. We compute the optimal perturbations using non-modal theory and perform a parametric study to measure the influence of the magnetic field on their amplification and orientation with respect to the flow direction. We also perform direct numerical simulations to examine how the optimal perturbations evolve in the nonlinear regime. To assess the influence of the boundary layer development, we consider both a constant Blasius base flow profile and a growing base profile. We show that the properties of the optimal perturbations are not significantly affected by the choice of the base profile, whereas it has a more important impact on their evolution in the nonlinear regime. 

Optimal perturbations and transition in the magnetohydrodynamic boundary layer under the influence of a spanwise magnetic field

The onset of instability against small disturbances of the pressure driven laminar fluid flow through a saturated porous medium in the presence of horizontal magnetic field between the two parallel impermeable plates is studied numerically. Assuming the channel walls have zero conductivity. The Chebyshev collocation method is used to solve the 6th order generalized Orr–Sommerfeld equation. The critical values of the Reynolds number R e c , the wave number α c and the wave speed c c are computed for various sets of the values of the porous parameter σ p , magnetic Reynolds number R m and Alfven number A . The marginal stability curves are drawn for various values of the physical parameters present in the problem. The onset of instability is also discussed in detail using the curves of growth rate of the disturbances and the critical values for various parameters present in the problem. It is observed that the role of σ p , R m and A are to delay the onset of instability of the system. The degree of stabilization is moderate up to certain value of A but it is so rapid when the Alfven number exceeds that value is observed. It is also seen that, the role of the combined influence of porosity of the porous medium and the strength of the magnetic field together suppress the growth of the disturbances very significantly. In this study the results computed are also compared with the existing literature as a particular case for ordinary Newtonian fluids.

Instability of the plane parallel flow through a saturated porous medium in the presence of a longitudinal magnetic field using the Chebyshev collocation method

T. Shobha

Papers

MHD instability of the pressure-driven plane laminar flow in the presence of the uniform coplanar magnetic field: Linear stability analysis

The combined effect of porosity of porous media and longitudinal magnetic field on stability of the modified plane Poiseuille flow