The joint effect of pulsating throughflow and external electric field on the outset of convective instability in a Horizontal porous medium layer saturated by a dielectric nanofluid is investigated. Pulsating throughflow alters the basic profiles for temperature and the volumetric fraction of nanoparticle from linear to nonlinear with layer height, which marks the stability expressively. To treat this problem, the Buongiorno’ s two-phase mathematical model is used taking the flux of volumetric fraction of nanoparticle is vanish on the Horizontal boundaries. Using the framework of linear stability theory and frozen profile approach, the stability equations are derived and solved analytically applying the Galerkin weighted residuals method with thermal Rayleigh-Darcy number D R as the eigenvalue. The effect of increasing the external AC electric Rayleigh-Darcy number Re, the modified diffusivity ratio A N and the nanoparticle Rayleigh number N R is to favorable for the convective motion, while the Lewis number e L and porosity parameter have dual influence on the stability scheme in the existence of pulsating throughflow. The frozen profile method shows that the result of pulsating throughflow in both directions is stabilizing. An enlarged amplitude of throughflow fluctuations offers to increased stability by an amount that vary on frequency. It is also found that the oscillatory mode of convection is not favorable for nanofluids if the vertical nanoparticle flux is vanish on the boundaries.

An analysis is made for the effect of throughflow on the onset of convection in a rectangular box under the assumption that total flux (sum of diffusive, thermophoretic, and convective) is zero on the boundaries. A linear stability analysis and Galerkin weighted residual method are used to obtain the Rayleigh number and stability curves for the onset of convection. Three dominating combination of parameters are extracted from the nondimensional analysis. All rescaled parameters promote the convection. Aspect ratios, throughflow, and nanoparticles play an important role in the formulation of cell distribution and development of convection. Oscillatory convection is possible for permissible range of nanofluid parameters. It is also found that the size of a cellular mode is altered by throughflow and nanoparticles.

The effect of vertical through flow and time-periodic gravity field has been investigated on Darcy convection.The amplitude of gravity modulation is considered to be very small and the disturbances are expanded in terms of power series of amplitude of convection. A weak nonlinear stability analysis has been performed for the stationary mode of convection. As a consequence heat transport evaluated in terms of the Nusselt number, which is governed by the non-autonomous Ginzburg-Landau equation. throughflow can stabilize or destabilize the system for stress free and isothermal boundary conditions. The amplitude and frequency of modulation, Prandtl Darcy number on heat transport have been analyzed and depicted graphically. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system. Finally flow patterns are presented in terms of streamlines and isotherms.

The paper deals with the combined effect of non-Newtonian saturating fluid and Horizontal flow rate on the thermal convection in a highly permeable, porous plane layer saturated with a power-law model. Asymmetric boundary conditions are assumed, with a cooled free surface at the top and a heated, impermeable, rigid wall at the bottom. The generalised Forchheimer equation is employed to model the power-law fluid movement. Convection cells emerge in the power-law fluid because of vertical temperature gradient imposed by the thermal boundaries. The onset of this scenario can be studied using linear stability theory, which leads to an eigenvalue problem. The latter is solved either numerically, employing shooting schemes, or analytically, using one-order Galerkin approach. The present study is considered an extension of the classical Prats problem. When the Peclet number, which defines the flow rate, is negligible, the configuration switches to the special case of Darcy-Rayielgh instability. The results show that the form drag exhibits a stronger stabilizing influence in shear-thinning fluids compared to shear-thickening and Newtonian ones since the saturating fluid is described by the power-law model. This scenario appears in the specific range of the Peclet number. In general, this investigation can be used to understand the heat transfer process in subsurface hydrocarbon reservoirs where the fluid may exhibit non-Newtonian behaviour.

The effect of vertical throughflow on the onset of bio-thermal convection in a water-based nanofluid containing gyrotactic microorganisms is investigated using more realistic boundary conditions. The Galerkin weighted residual method is used to obtain numerical solutions of the mathematical model. The effects of bioconvection Rayleigh number, gyrotaxis number, bioconvection Pé clet number, Lewis number, Pé clet number, particle density increment number, modified diffusitivity ratio, and nanoparticle Rayleigh number on thermal Rayleigh number are examined. The combined effect of Brownian motion and thermophoresis of nanoparticles, vertical throughflow, and gyrotactic microorganisms on the thermal Rayleigh number is found to be destabilizing and its value is decreased by first to third orders of magnitude as compared to regular fluids. Critical wave number is dependent on bioconvection parameters, nanofluid parameters as well as throughflow parameter. The results obtained using passive boundary conditions are compared with those of active boundary conditions. The present study may find applications in seawater convection at the ocean crust.

The instability of non-Newtonian power law fluid in double diffusive convection in a porous medium with vertical throughflow is investigated. The lower and upper boundaries are taken to be permeable, isothermal and isosolutal. For vertical throughflow the linear stability of flow is determined by the power law index (n), non-Newtonian Rayleigh number (Ra), Buoyancy ratio (N), Pé clet number (Pe) and Lewis number (Le). The eigenvalue problem is solved by two-term Galerkin approximation to obtain the critical value of Rayleigh number and neutral stability curves. It is observed that the neutral stability curves, as well as the critical wave number and Rayleigh number, are affected by the parameters such as Pé clet number, buoyancy ratio and Lewis number. The neutral stability curves indicate that power law index n has destabilizing nature when it takes values for dilatant fluid at low Pé clet numbers while for the pseudoplastic fluids it shows stabilizing effect. In the absence of buoyancy ratio and vertical throughflow, the present numerical results coincide with the solution of standard Horton-Rogers-Lapwood Problem. The numerical analysis of linear stability for the limiting case of absolute pseudoplasticity is also done by using Galerkin method.

Providing an appropriate definition of a Horizontal subbundle of a Lie algebroid will lead to construction of a better framework on Lie algebriods. In this paper, we give a new and natural definition of a Horizontal subbundle using the prolongation of a Lie algebroid and then we show that any linear connection on a Lie algebroid generates a Horizontal subbundle and vice versa. The same correspondence will be proved for any covariant derivative on a Lie algebroid.