Nonlinear Double-diffusive Convection Research Articles

Cross Diffusion Effect on Linear and Nonlinear Double Diffusive Convection in a Viscoelastic Fluid Saturated Porous Layer with Internal Heat Source

Double diffusive convection in a binary viscoelastic fluid saturated porous layer in the presence of a cross diffusion effect and an internal heat source is studied analytically using linear and nonlinear stability analysis. The linear stability theory is based on the normal mode technique, while the nonlinear theory is based on a minimal representation of truncated double Fourier series. The modified Darcy law for the viscoelastic fluid of the Oldroyd type is considered to model the momentum equation. The onset criterion for stationary and oscillatory convection and steady heat and mass transfer have been obtained analytically using linear and nonlinear theory, respectively. The combined effect of an internal heat source and cross diffusion is investigated. The effects of Dufour, Soret, internal heat, relaxation and retardation time, Lewis number and concentration Rayleigh number on stationary, oscillatory, and heat and mass transport are depicted graphically. Heat and mass transfer are presented graphically in terms of Nusselt and Sherwood numbers, respectively. It is reported that the stationary and oscillatory convection are significantly influenced with variation of Soret and Defour parameters. An increment of the internal heat parameter has a destabilizing effect as well as enhancing the heat transfer process. On the other hand, an increment of internal heat parameter has a variable effect on mass transfer. It is found that there is a critical value for the thermal Rayleigh number, below which increasing internal heat decreases the Sherwood number, while above it increasing the internal heat increases the Sherwood number.

Weakly Nonlinear Double-Diffusive Oscillatory Magneto-Convection Under Gravity Modulation

An imposed time-periodic gravity field effect on double-diffusive magneto-convection for oscillatory mode has been investigated. The gravity field consisting of steady and periodic modes. A layer is confined with an electrically conducting fluid with Boussines q approximation and heated from below cooled from above. While using the perturbation technique we study nonlinear double-diffusive convection just above the critical state of the onset convection. The growth rate of the disturbances is confined with a critical Rayleigh number to investigate oscillatory convection. Analysis of finite- amplitude convection has been derived through the complex Ginzburg-Landau equation (CGLE). The convective heat and mass transfer obtained through CGLE at third-order under solvability conditions. This convective amplitude is required to estimate heat and mass transfer in terms of the Nusselt and Sherwood numbers. It is found that increasing the frequency of modulation causes diminishing heat and mass transfer. The effect of Prandtl number Pr, magnetic Prandtl number Pm, and amplitude δ enhances heat/mass transfer. It is found that an oscillatory mode of convection enhances the heat and mass transfer than the stationary mode. Further, streamlines, isotherms, and isohalines have their usual nature on double-diffusive magnetoconvection.

Linear and Weak Nonlinear Double Diffusive Convection in a Viscoelastic Fluid Saturated Anisotropic Porous Medium with Internal Heat Source

This paper deals linear and weak nonlinear stability analysis of double-diffusive convection in an anisotropic porous layer with internal heat source saturated by viscoelastic fluid. For linear stability analysis we use normal mode technique and obtained the expression for oscillatory thermal Rayleigh number which is used to plot neutral stability curve for oscillatory case. For nonlinear analysis truncated representation of Fourier series upto two terms is used. The system of time dependent nonlinear equation is solved numerically and plot the curve for heat transfer and mass transfer with respect to time for different parameters. Effect of thermal anisotropy parameter, mechanical anisotropy parameter, relaxation parameter, retardation parameter, internal heat source parameter, solute Rayleigh number, diffusivity ratio, Darcy-Prandtl number on the onset of convection, heat and mass transfers have been discussed. We also draw the stream lines, isotherms, isohalines at different times.

Linear and nonlinear double-diffusive convection in a saturated anisotropic porous layer with Soret effect and internal heat source

Double-diffusive convection in an anisotropic porous layer heated and salted from below with an internal heat source and Soret effect is studied analytically using both linear and nonlinear stability analyses. The generalized Darcy model including the time derivative term is employed for the momentum equation. The critical Rayleigh number and wavenumber for stationary and oscillatory modes and frequency of oscillations are obtained analytically using linear theory. The effects of the anisotropy parameters, concentration Rayleigh number, internal heat source, Soret parameter, Vadasz number and Lewis number on the stationary, oscillatory, and heat and mass transfer are shown graphically. Heat and mass transfer have been obtained in terms of the Nusselt number and Sherwood number.

An analytical study of nonlinear double-diffusive convection in a porous medium under temperature/gravity modulation

The article deals with nonlinear thermal instability problem of double-diffusive convection in a porous medium subjected to temperature/gravity modulation. Three types of imposed time-periodic boundary temperature (ITBT) are considered. The effect of imposed time-periodic gravity modulation (ITGM) is also studied in this problem. In the case of ITBT, the temperature gradient between the walls of the fluid layer consists of a steady part and a time-dependent periodic part. The temperature of both walls is modulated in this case. In the problem involving ITGM, the gravity field has two parts: a constant part and an externally imposed time-periodic part. Using power series expansion in terms of the amplitude of modulation, which is assumed to be small, the problem has been studied using the Ginzburg–Landau amplitude equation. The individual effects of temperature and gravity modulation on heat and mass transports have been investigated in terms of Nusselt number and Sherwood number, respectively. Further the effects of various parameters on heat and mass transports have been analyzed and depicted graphically.

Linear and Nonlinear Double-Diffusive Convection in a Fluid-Saturated Porous Layer with Cross-Diffusion Effects

The double-diffusive convection in a horizontal fluid-saturated porous layer, which is heated and salted from below in the presence of Soret and Dufour effects, is studied analytically using both linear and nonlinear stability analyses. The linear analysis is based on the usual normal mode technique, while the nonlinear analysis is based on truncated representation of Fourier series. The generalized Darcy model that includes the time derivative is employed for the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes, and frequency of oscillations are obtained analytically using linear theory. The effects of solute Rayleigh number, Lewis number, normalized porosity parameter, Vadasz number, Soret and Dufour parameters on the stationary, oscillatory convection, and heat and mass transfers are shown graphically. The Vadasz number has dual effect on the threshold of the oscillatory convection. Some known results are recovered as special cases of the present problem.

Linear and Non-linear Double Diffusive Convection in a Fluid-Saturated Anisotropic Porous Layer with Cross-Diffusion Effects

The double diffusive convection in a horizontal anisotropic porous layer saturated with a Boussinesq binary fluid, which is heated and salted from below in the presence of Soret and DuFour effects is studied analytically using both linear and non-linear stability analyses. The linear analysis is based on the usual normal mode technique, while the non-linear analysis is based on a minimal representation of double Fourier series. The generalized Darcy model including the time derivative term is employed for the momentum equation. The critical Rayleigh number, wavenumbers for stationary and oscillatory modes, and frequency of oscillations are obtained analytically using linear theory. The effects of anisotropy parameter, solute Rayleigh number, and Soret and DuFour parameters on the stationary, oscillatory convection, and heat and mass transfer are shown graphically. Some known results are recovered as special cases of the present problem.

An analytical study of linear and nonlinear double diffusive convection in a fluid saturated anisotropic porous layer with Soret effect

The double diffusive convection in a horizontal anisotropic porous layer saturated with a Boussinesq fluid, which is heated and salted from below in the presence of Soret coefficient is studied analytically using both linear and nonlinear stability analyses. The normal mode technique is used in the linear stability analysis while a weak nonlinear analysis based on a minimal representation of double Fourier series method is used in the nonlinear analysis. The generalized Darcy model including the time derivative term is employed for the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes and frequency of oscillations are obtained analytically using linear theory. The effect of anisotropy parameters, solute Rayleigh number, Soret parameter and Lewis number on the stationary, oscillatory, finite amplitude convection and heat and mass transfer are shown graphically.

An analytical study of linear and non-linear double diffusive convection with Soret and Dufour effects in couple stress fluid

The onset of double diffusive convection in a two component couple stress fluid layer with Soret and Dufour effects has been studied using both linear and non-linear stability analysis. The linear theory depends on normal mode technique and non-linear analysis depends on a minimal representation of double Fourier series. The effect of couple stress parameter, the Soret and Dufour parameters, and the Prandtl number on the stationary and oscillatory convection are presented graphically. The Dufour parameter enhances the stability of the couple stress fluid system in case of both stationary and oscillatory mode. The effect of positive Soret parameter is to destabilize the system in case of stationary mode while it stabilizes the system in case of oscillatory mode. The negative Soret parameter enhances the stability in both stationary and oscillatory mode. The couple stress parameter enhances the stability of the system in both stationary and oscillatory modes. The Dufour parameter increases the heat transfer while the couple stress parameter has reverse effect. The Soret parameter has negligible influence on heat transfer. Both Dufour and Soret parameters increases the mass transfer while the couple stress parameter has dual effect depending on the value of the Rayleigh number.

An analytical study of linear and non-linear double diffusive convection with Soret effect in couple stress liquids

The double diffusive convection in a two-component couple stress liquid layer with Soret effect is studied using both linear and non-linear stability analyses. The linear theory is based on normal mode technique and the non-linear analysis is based on a minimal representation of double Fourier series. The effect of couple stress parameter, the Soret parameter, the solute Rayleigh number, the Prandtl number and the diffusivity ratio on the stationary, oscillatory and finite amplitude convection are shown graphically. It is found that the effects of couple stress are quite large and the positive Soret number enhances the stability while the negative Soret number enhances the instability. The non-linear theory predicts that, finite amplitude motions are possible only for negative Soret parameter. The transient behaviour of thermal and solute Nusselt numbers has been investigated by solving numerically a fifth order Lorenz model using Runge–Kutta method.

Numerical and experimental modeling of viscous fingering during liquid–liquid miscible displacement

Viscous fingering takes place when the viscous forces of a displacing phase has greater momentum than that of the displaced phase. Viscous fingering is an extremely important phenomenon in many applications of enhanced oil recovery, underground liquid waste disposal, and geothermal energy production. While the onset and propagation of viscous fingers during liquid–liquid displacement is considered to be of severe engineering consequences, little has been done to mathematically model the onset and propagation of a viscous finger. Viscous finger under double diffusive conditions is even less understood. In this paper, two-dimensional non-linear double diffusive convection in a multi-porous cavity is considered, both numerically and experimentally. The Darcy equation, including Brinkman term to account for the viscous effects, is used as the momentum balance equation. The model consists of two rectangular cavities filled with a porous medium. The smaller cavity is located at the top of the larger one. The larger cavity is filled initially with glycerin while the smaller one contains fresh water. At the initial time, the fresh water is injected with either constant flow rate (numerical) or constant hydrostatic head (experiments) and the viscous fingering formation is studied in details. The momentum, solutal, energy and continuity equations are solved numerically using the finite element technique. This transient problem is solved to study the thermal displacement, the isothermal displacement and the microgravity displacement of glycerin by water to understand the onset and the propagation of viscous fingering. For each case, the variation of the distance between the tip of the finger with time is studied in details. The effects of aspect ratio and displacement velocity are studied, both in the context of onset and propagation of viscous fingers. Experimentally, an ingenious method is developed for visualizing 2-D flow in a porous medium. A carbonate formation is used as the porous medium. Chemical dyeing is used to delineate the propagating front of a viscous finger. Initial series of experiments are conducted under isothermal conditions.

Heat and mass transfer in a multi-porous cavity

In this paper, two-dimensional non-linear double diffusive convection in a multi-porous cavity is considered. The Darcy equation, including Brinkman term to account for the viscous effects, is used as the momentum equation. The model consists of two rectangular cavities filled with glass beads having a diameter d 1 of either 5.25 mm (Case 1) or 3.25 mm (Case 2). The smaller cavity is located at the top left corner of the larger one. The larger cavity is filled initially with hot salty fluid while the smaller one contains initially cold fresh fluid. At the initial time, the obstacle between the two cavities was released and the double diffusive phenomena were studied in details. The momentum, solutal, energy and continuity equations are solved numerically using the finite element technique. This transient problem is solved for two different Darcy numbers. For each Darcy number, the influence of the solutal Rayleigh number on double diffusive convection was studied in details. The permeability in the horizontal and vertical direction was assumed identical. A comparison of the intruding force between this case and the open flow case studied by Saghir et al [Int. J. Heat. Mass Transfer, in press] showed that it is inversely proportional to the Darcy number. Finite element modelling results indicate that salinity induces stronger convection than the thermal ones.

Steady Nonlinear Double-Diffusive Convection in a Porous Medium Based upon the Brinkman–Forchheimer Model

A theoretical study of the problem of steady nonlinear double-diffusive convection through a porous medium is presented. The Brinkman–Forchheimer model is used to represent the porous medium. A variational formulation is given to deal with the weak solution and the existence, regularity, and uniqueness results are discussed.

The effect of rotation on linear and non-linear double-diffusive convection in a sparsely packed, porous medium

The linear and non-linear stability of rotating double-diffusive convection in a sparsely packed porous medium is investigated considering a non-Darcy equation. In the case of linear theory both marginal and overstable motions are discussed. In the former case it is shown that the effect of Taylor number and porous parameter is to make the system more stable. In the latter case, however, it is shown that the bottom-heavy solute gradient and rotation destabilize the system under certain conditions. By drawing the stability boundaries in the Rayleigh number plane it is shown that the effect of rotation and porous parameter is to decrease the region of instabilities. Using the theory of self-adjoint operator the variation of critical eigenvalue with physical and boundary parameters is studied. In the case of non-linear theory, both steady and unsteady cases have been considered. In the unsteady case the transient behaviour concerning the variation of Nusselt number with time has been investigated, by solving numerically a seventh-order Lorenz model using the Runge-Kutta-Gill method. Interesting results are obtained by comparing these results with those of the steady case. Finally, the effect of porous parameter on streamfunction, isotherms, isohalines and zonal velocity is studied.

Nonlinear double-diffusive convection with local heat and solute sources

Effects of local heat and solute sources on nonlinear double diffusive convection has been investigated. These sources introduce three new parameters. Dependence of the horizontal cells' size, critical solute Rayleigh number, and heat and solute fluxes on these new parameters is determined. In particular, it is found that local sources can significantly reduce the horizontal cells' size. For the case where the strength of the local solute source is insignificant, the preferred horizontal flow cross-section pattern is found to be squares. On the other hand, if the strength of the local solute source is significant, then hexagons become the preferred pattern when flow amplitude ϵ exceeds a critical value. However, both squares and hexagons can be stable for an even higher value of ϵ. The convective motion is upward at the center of the hexagonal cells if a solute sink acts on the flow. When a solute source acts on the flow, downward motion at the cells' center will occur. Furthermore, the local solute source is also found to be able to provide finite amplitude instability.

Double-diffusive convection in an inclined fluid layer

The nonlinear double-diffusive convection in a Boussinesq fluid with stable constant vertical solute gradient, and bound by two differentially heated rigid inclined parallel plates is considered. The analysis was carried out by a Galerkin method for the cases when the angle of inclination was 0°, −45° and +45° (positive angle denotes heating from below, and negative angle denotes heating from above). The counter-rotating cells predicted by the linear theory merge into single cells with the same sense of rotation within a very short period of time even under slightly supercritical conditions. This is consistent with the experimental observations. Furthermore, as observed in the experiments, the evolution of instability is more rapid when heating is from above than when heating is from below. Our results for a salt-heat system are in excellent agreement with those based on the limiting case of Lewis number → 0 and Schmidt number → ∞.

Nonlinear periodic convection in double-diffusive systems

We study two examples of two-dimensional nonlinear double-diffusive convection (thermohaline convection, and convection in an imposed vertical magnetic field) in the limit where the onset of marginal overstability just precedes the exchange of stabilities. In this limit nonlinear solutions can be found analytically. The branch of oscillatory solutions always terminates on the steady solution branch. If the steady solution branch is subcritical this occurs when the period of the oscillation becomes infinite, while if it is supercritical, it occurs via a Hopf bifurcation. A detailed discussion of the stability of the oscillations is given. The results are in broad agreement with the largeramplitude results obtained previously by numerical techniques.

Nonlinear Double-diffusive Convection in a Moderate Prandtl Number Fluid

Recent work on nonlinear double-diffusive convection in a low Prandtl number fluid is extended to the case of a moderate Prandtl number G'. The boundary layer method is used by assuming a large Rayleigh number R for different ranges of the diffusivity ratio -c and the solute Rayleigh number R; It is found that the heat and solute fluxes F and F; increase with G', Rand n;', and that F; F- 1 is independent of G'. The flow is shown to have a solute layer of thickness proportional to F-1-2, The horizontal wavenumber which maximizes F is found to be independent of G', but it increases with Rand R;1

Nonlinear Double-diffusive Convection in a Low Prandtl Number Fluid

Nonlinear double-diffusive convection is studied using the modal equations of cellular convection. The boundary layer method is used by assuming a large Rayleigh number R for a fluid of low Prandtl number (J, and different ranges of the diffusivity ratio 7: and. the solute Rayleigh number Rs. The heat and solute fluxes are found to increase with R(J and decrease with Rs. The effect of the solute is stabilizing, although the convection in a fluid with large (J is less affected by the solute concentration. The flow is shown to have a solute layer which thickens as (J, R, .-1 or R;l decreases. It is proved that it is only for this layer that the solute affects the boundary layer structure.

Nonlinear double-diffusive convection

The two-dimensional motion of a fluid confined between two long horizontal planes, heated and salted from below, is examined. By a combination of perturbation analysis and direct numerical solution of the governing equations, the possible forms of large-amplitude motion are traced out as a function of the four non-dimensional parameters which specify the problem: the thermal Rayleigh number RT, the saline Rayleigh number ES, the Prandtl number σ and the ratio of the diffusivities τ. A branch of time-dependent asymptotic solutions is found which bifurcates from the linear oscillatory instability point. In general, for fixed σ, τ and RS, as RT increases three further abrupt transitions in the form of motion are found to take place independent of the initial conditions. At the first transition, a rather simple oscillatory motion changes into a more complicated one with different structure, at the second, the motion becomes aperiodic and, at the third, the only asymptotic solutions are time independent. Disordered motions are thus suppressed by increasing RT. The time-independent solutions exist on a branch which, it is conjectured, bifurcates from the time-independent linear instability point. They can occur for values of RT less than that at which the third transition point occurs. Hence for some parameter ranges two different solutions exist and a hysteresis effect occurs if solutions obtained by increasing RT and then decreasing RT are followed. The minimum value of RT for which time-independent motion can occur is calculated for fourteen different values of σ, τ and RS. This minimum value is generally much less than the critical value of time-independent linear theory and for the larger values of σ and RS and the smaller values of τ, is less than the critical value of time-dependent linear theory.