In the present study, the onset of Double diffusive reaction-convection in a fluid layer with viscous fluid, heated and salted from below subject to chemical equilibrium on the boundaries, has been investigated. Linear and nonlinear stability analysis have been performed. For linear analysis normal mode technique is used and for nonlinear analysis minimal representation of truncated Fourier series is used. The effect of Lewis number, solute Rayleigh number, reaction rate and Prandtl number on the stability of the system is investigated. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the heat and mass transfer.

This paper focuses on analytical and numerical investigation of Double-diffusive bioconvection in a porous media saturated by nanofluid using the modified mass flux condition. Normal mode technique is employed to solve the governing equations of the Brinkman-Darcy model. The Galerkin weighted residual method (single-term and six-term) is used to obtain numerical solution of the mathematical model. It is found that due to the presence of gyrotactic microorganisms, Rayleigh number is decreased substantially which shows that convection sets in earlier as compared to nanofluid without microorganisms and this destabilizing effect is more predominant for faster swimming microorganisms. modified Darcy number number, Soret parameter, and porosity postpone the onset of the bioconvection, whereas nanoparticle Rayleigh number, bioconvection Rayleigh number, nanoparticle Lewis number, Dufour parameter, Pé clet number, and Lewis number pre-pone the onset of bioconvection under certain conditions.

A theoretical analysis of thermo-convective instability in an electrically conducting two component fluid layer is carried out when the gravity field vary with time in a sinusoidal manner. Newtonian liquid is considered between two horizontal surfaces, under a constant vertical magnetic field. The disturbance is expanded in terms of power series of amplitude of convection, which is assumed to be small. We use the linear matrix differential operator method to find the Ginzburg-Landau amplitude equation for the modulated problem. Use the solution of the Ginzburg-Landau equation in quantifying the amount of heat and mass transports in terms of Nusselt and Sherwood numbers. It is found that, the effect of magnetic field is to stabilize the system. Effect of various parameters on the heat and mass transport is also discussed. -Further, it is found that the heat and mass transports can be controlled by suitably adjusting frequency and amplitude of gravity modulation.

The effect of chemical reaction and external vertical magnetic field on the onset of the Double diffusive convection in couple stress fluid between infinite horizontal parallel plates  has been studied.  The effectiveness of vertical magnetic field and chemical reaction were gauged by determining the values of  Chandrashekhar number (Q) and Damk hler number  in terms of other controlling parameters and shown their effect on stability of system through graphs. The entire investigation is performed in two parts: linear and weakly non-linear stability analysis. A comparative study is presented in stationary case of linear stability analysis for four types of bounding surfaces: (a) Realistic bounding surfaces i.e. Rigid-Rigid, Rigid-Free and Free-Rigid (R/R, R/F and F/R) (b) Non Realistic bounding surface i.e. Free-Free (F/F). However, oscillatory case and weakly non-linear stability analysis are restricted for Free-Free (F/F) boundary surfaces. Graphical representations are used to illustrate how different parameters affect stationary, oscillatory, finite-amplitude states and  the amount of heat and mass transfer. By analysing the linear stability analysis, it is observed that the onset of convection is more dominant in oscillatory case than stationary. The stability criteria for Q came out as (in decreasing order) F/F>F/R>R/R>R/F which is different from the criteria came out for rest of the controlling parameters i.e F/R>R/R>F/F>R/F in stationary case. It is also reported that the Q, Couple stress parameter (C) and ratio of heat capacities on heat transfer  is responsible for the delay of the onset of convection while  (impact of chemical reaction) enhances the onset of convection. Non-linear stability analysis using the truncated representation of Fourier series method predicts the occurrence of sub-critical instability in the form of finite amplitude motion. The effect of Q, Lewis number Le, and solute Rayleigh number RaS, increased the amount of heat and mass transfer while C decreased. We also draw streamlines, isotherms, isohalines and magnetic streamlines for different time intervals (unsteady) i.e. for (0.01, 0.03, 0.009, 0.006) and showed the pattern of the onset of convection.

In this study, a meshfree framework based on the reproducing kernel collocation method is proposed for incremental-iterative analysis of Double-diffusive natural convection in a porous enclosure, in which the forward difference method is adopted for temporal discretization, and the two-step version of Newton-Raphson method is used for iteration. As the Double-diffusive convection problem is composed of multi phases and is influenced by both material and geometric parameters, the resulting system is highly nonlinear and complicated. From the numerical investigation, the partially heated boundary with different buoyancy ratios can yield monocellular flow problems with opposite phenomena depending on the contribution of thermal/solute buoyancy force. For the domains with burrowing inside, the key feature is the contour of stream function, which is separated into two vortexes by the hole in the simply connected domain while the two vortexes are not separated completely in the multiply connected domain due to the geometric compression of two holes. It is further shown that the framework is capable of solving various Double-diffusive convection problems with satisfactory accuracy and efficiency by uniform discretization as well as few source points in the approximation.

Entropy generation of Double diffusive natural convection in a three dimensional differentially heated enclosure has been performed numerically. Vertical walls of enclosure are heated differentially and remaining walls are adiabatic. The obtained results were presented via iso-concentration, iso-temperatures, velocity vector projection, particle trajectories, velocity profiles, iso-entropy, local Nusselt and Sherwood numbers, average Nusselt and Sherwood numbers and Bejan numbers at different value of buoyancy ratio (-2£N£2) and Rayleigh numbers (103£Ra£105). The Lewis number is fixed at Le=2. It is found that both Rayleigh number and buoyancy ratio play dominant role on entropy generation and heat and mass transfer as well as fluid flow. A special case occurred between -1.5£N£-1 and complex structure is observed.

The nonlinear stability of stationary and oscillatory Double-diffusive convection in an Oldroyd-B fluid layer is investigated using a perturbation method. The cubic Landau equations are derived and based on which the stability of stationary and oscillatory bifurcating solutions in the neighborhood of their critical values is discussed. The boundary between stationary and oscillatory convection demarcated by identifying a codimension-two points in the viscoelastic parameters plane. The bifurcating solution is found to be subcritical depending on the choices of physical parameters. Heat and mass transport are estimated in terms of Nusselt numbers. The effect of Prandtl number is observed only in the case of oscillatory motions and increase in its value is to decrease the heat and mass transfer. Besides, increasing relaxation and retardation parameters is to decrease and increase the amount of heat and mass transfer, respectively in the stationary case, while these parameters found to exhibit an opposing kind of behavior in the case of oscillatory motions.

This paper deals with the theoretical investigation of the effect of rotation in a magnetized Ferro fluid with internal angular momentum, heated and soluted from below subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained. A linear stability analysis theory and normal mode analysis method have been carried out to study the onset of convection. The influence of various parameters like rotation, solute gradient, magnetization and internal angular momentum parameters (i.e. coupling parameter, spin diffusion parameter and heat conduction parameter) has been analyzed on the onset of stationary convection. The critical magnetic thermal Rayleigh number for the onset of instability is also determined numerically for sufficiently large values of buoyancy magnetization parameter M1 and results are depicted graphically. The principle of exchange of stabilities is found to hold true for the Ferro fluid with internal angular momentum heated from below in the absence of rotation, coupling between vorticity and spin, micro inertia and solute gradient. The oscillatory modes are introduced due to the presence of the rotation, coupling between vorticity and spin, micro inertia and solute gradient, which were non-existent in their absence. In this paper, an attempt is also made to obtain the sufficient conditions for the non-existence of over stability.