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Examining phototactic bioconvection in non-scattering suspensions with upper heating and collimated irradiation, this study delves into the intricate dynamics influenced by light and microorganisms. The study focuses on the linear stability of the basic state, examining neutral curves. The numerical analysis involves solving a system of equations using the MATLAB bvp4c solver. The investigation considers the impact of parameters, such as the thermal Rayleigh number, critical total intensity, and Lewis number, on the critical bioconvection Rayleigh number. As the critical total intensity varies, a transition from a stationary to an oscillatory solution (and vice versa) is observed. Phototactic microorganisms are incorporated into the model, and the results show how varying parameters affect convection patterns and stability. The findings reveal interesting phenomena, including Hopf bifurcations and limit cycles. [ABSTRACT FROM AUTHOR]

On the basis of Darcy–Brinkman model, linear stability analysis is used to study bio-thermal convection in a suspension of gyrotactic microorganisms in a highly porous medium heated from below. A Galerkin method is performed to solve the governing equations generating a correlation between the traditional thermal Rayleigh number and the critical value of the bioconvection Rayleigh number. The effects of three variables including the bioconvection Péclet number, the gyrotaxis number and the modified Darcy number on both the wave number and the critical bioconvection Rayleigh number are analyzed and shown graphically. Results indicate that the critical bioconvection Rayleigh number becomes larger with increasing Darcy number. [ABSTRACT FROM AUTHOR]

Abstract: The linear stability of triply diffusive convection in a binary Maxwell fluid saturated porous layer is investigated. Applying the normal mode method theory, the criterion for the onset of stationary and oscillatory convection is obtained. The modified Darcy–Maxwell model is used as the analysis model, this allows us to make a thorough investigation of the processes of viscoelasticity and diffusions that causes the convection to set in through oscillatory rather than stationary. The effects of Vadasz number, normalized porosity parameter, relaxation parameter, Lewis number and solute Rayleigh number on the system are presented numerically and graphically. [Copyright &y& Elsevier]

The onset of double diffusive convection in a viscoelastic fluid-saturated porous layer is studied when the fluid and solid phase are not in local thermal equilibrium. The modified Darcy model is used for the momentum equation and a two-field model is used for energy equation each representing the fluid and solid phases separately. The effect of thermal non-equilibrium on the onset of double diffusive convection is discussed. The critical Rayleigh number and the corresponding wave number for the exchange of stability and over-stability are obtained, and the onset criterion for stationary and oscillatory convection is derived analytically and discussed numerically. [ABSTRACT FROM AUTHOR]

Abstract: Stability analysis of double-diffusive convection for viscoelastic fluid with Soret effect in a porous medium is investigated using a modified-Maxwell–Darcy model. We use the linear stability analysis to investigate how the Soret parameter and the relaxation time of viscoelastic fluid effect the onset of convection and the selection of an unstable wavenumber. It is found that the Soret effect is to destabilize the system for oscillatory convection. The relaxation time also enhances the instability of the system. The effects of Soret coefficient and relaxation time on the heat transfer rate in a porous medium are studied using the nonlinear stability analysis, the variation of the Nusselt number with respect to the Rayleigh number is derived for stationary and oscillatory convection modes. Some previous results can be reduced as the special cases of the present paper. [ABSTRACT FROM AUTHOR]

Abstract: On the basis of Brinkman model, the onset of double-diffusive (thermosolutal) convection with a reaction term in a horizontal sparsely packed porous media is studied using the normal mode analysis. The effects of Brinkman term, the reaction term, the normalized porosity ε and Lewis number Le on the Rayleigh number for stationary and oscillatory case are presented graphically. The Darcy number destabilize the system in case of both stationary and oscillatory mode. The effects of Lewis number and reaction term depend on the value of the values of solutal Rayleigh number. Furthermore, some results of Darcy model can be recovered in limit cases. [Copyright &y& Elsevier]

• Instabilities with the surface tension and the buoyancy for a non-Newtonian fluid in a fluid-porous system are studied with a deformable surface. • The influences of different key parameters on Rayleigh and Marangoni instabilities in short- and long-wave branches are investigated. • It is found that there is an interval of the Marangoni number in short-wave branches for which the critical Rayleigh number is absent. • Opposite effects between long- and short-wave branches in the coupling mode are observed. This paper describes the linear stability analysis of Rayleigh–Marangoni–Bénard convection with a deformable surface in a fluid overlying a highly porous layer. Using the Chebyshev tau-QZ method, we investigate the oscillatory mode of both the Rayleigh and Marangoni instabilities for non-Newtonian fluids. The numerical results indicate that a deformable upper surface destabilizes the system. Moreover, the Marangoni instabilities in long-wave branches first decrease and then increase with increasing depth ratio, while the opposite effects are observed in short-wave branches. The Rayleigh instabilities decrease monotonically as the depth ratio increases. For certain reference parameters, the system becomes more stable as the Biot number and Galileo number increase. A greater strain retardation time and a smaller stress relaxation time lead to more stable Marangoni convection in long-wave branches, whereas these viscoelastic times have the opposite effects in short-wave branches. The influence of viscoelasticity on the Rayleigh instability is also investigated. Finally, the coupling mode of the two instabilities is studied in detail. Variations in the Marangoni number influence long- and short-wave branches differently. Interestingly, there is an interval of the Marangoni number in short-wave branches for which no critical Rayleigh number exists. [ABSTRACT FROM AUTHOR]

By utilizing a randomly swimming model, a linear stability analysis is applied to investigate the stability of bioconvection in a horizontal suspension layer of motile gyrotactic micro-organisms with heated from below. The micro-organisms under consideration are orientated by a balance between a gravitational torque, due to them being bottom heavy, and viscous torque arising from local fluid velocity gradients. The obtained eigenvalue problem containing thermal Rayleigh number and bioconvection Rayleigh number is solved numerically using one-term Galerkin method. The case of non-oscillatory instability is analyzed, the relationship among thermal Rayleigh number, bioconvection Rayleigh number, Lewis number, critical wavenumber and the shape of microorganisms are discussed. We point out that the heating from below makes the layer more unstable. When increasing the value of thermal Rayleigh number to 1750, the suspension becomes unstable itself, which imply that bioconvection Rayleigh number has nothing to do with the stability of this system. We also find that Lewis number has no effect on critical value of thermal Rayleigh number, but has a great influence on critical bioconvection Rayleigh number. The increasing cell eccentricity enlarges the critical value of bioconvection Rayleigh number, which means that the suspension is more stable. [ABSTRACT FROM AUTHOR]

The qualitative effect of variable internal heat source and variable gravity on the onset of convection in a horizontal fluid saturated sparsely packed porous layer is investigated using linear stability analysis. The single term Galerkin technique is used to compute the value of the critical Rayleigh number and the corresponding wave number. Three different sets of variable gravity and heat source functions are chosen and their influence on the onset of convection is discussed. The eigenvalues are obtained for free-free velocity boundary conditions with isothermal temperature conditions on the boundaries. The influence of porous parameter on the onset of convection is brought out. It is found that the variable heat source and variable gravity does not affect the shape and size of the convective cell at the onset of convection. However the porous parameter do so. [ABSTRACT FROM AUTHOR]