Moderate Stefan Number Convection in Rotating Mushy Layers: A New Darcy Equation Formulation

The coriolis effect on a solidifying mushy layer is considered. A near-eutectic approximation and large far-field temperature is employed in the current study for moderate Stefan numbers. The linear stability theory is used to investigate analytically the Coriolis effect on convection in a rotating mushy layer for a new formulation of the Darcy equation. It was found that only stationary convection is possible for moderate Stefan numbers. In contrast to the problem of a stationary mushy layer, rotating the mushy layer has a stabilizing effect on convection. It was also discovered that fot Taylor numbers larger than three (i.e., Ta > 3),increasing the retardability coefficient (hence increasing the solid fraction) destablished the convection.     

Stefan Number Effect on the Transition from Stationary to Oscillatory Convection in a Solidifying Mushy Layer Subjected to Rotation

The current study investigates the Stefan number effect on the transition from stationary to oscillatory convection in a rotating mushy layer where the near eutectic approximation is applied. It is found that for rotating solidifying systems exhibiting a Stefan number of unit order (i.e., St=1), stationary convection is only possible up to Ta=3. Beyond Ta=3, for St=1, it is found that the oscillatory mode is the most dangerous mode of convection. A map showing the region of occurrence of the oscillatory mode is also presented for a range of Stefan numbers. The map reveals that the oscillatory mode is the most dangerous mode for intermediate values of Stefan number whilst the stationary mode is the most dangerous mode for very small and very large values of Stefan number. It is also demonstrated that increasing the rotation rate serves to render the oscillatory mode as the becoming the most dangerous mode of convection.     

Weak Nonlinear Analysis of Moderate Stefan Number Oscillatory Convection in Rotating Mushy Layers

We consider the solidification of a binary alloy in a mushy layer subject to Coriolis effects. A near-eutectic approximation and large far-field temperature is employed in order to study the dynamics of the mushy layer with a Stefan number of unit order of magnitude. The weak nonlinear theory is used to investigate analytically the Coriolis effect in a rotating mushy layer for a new moderate time scale proposed by the author. It is found that increasing the Taylor number favoured the forward bifurcation.     

Coriolis effect on convection for a low Prandtl number fluid

The problem of finite-amplitude thermal convection in a horizontal layer of a low Prandtl number heated from below and rotating about a vertical axis is studied. Linear stability and weak non-linear theories are used to investigate analytically the Coriolis effect on gravity-driven convection. The non-linear steady problem is solved by perturbation techniques, and the preferred mode of convection is determined by a stability analysis. Finite-amplitude results, obtained by using a weak amplitude, correspond to both stationary and oscillatory convections. These amplitude equations permit to identify from the post-transient conditions that the fluid is subject to Pitchfork bifurcation in the stationary convection and Hopf bifurcation in the oscillatory convection. Thereafter, in the small perturbations hypothesis, an amplitude solution is evaluated and drawn in time and space scales.     

Moderate Time Scale Finite Amplitude Analysis of Large Stefan Number Convection in Rotating Mushy Layers

We investigate the steady state convection amplitude for solutal convection occurring during the solidification of a rotating mushy layer in a binary alloy system for a new Darcy equation formulation. We adopt a large far field temperature and assume that the initial composition is very close to the eutectic composition. The linear stability analysis showed that rotation stabilised solutal convection. The results of the weak non-linear analysis of stationary convection indicates the presence of Hopf bifurcation, associated with the oscillatory mode, developing at Ta = 3.     

Effect of Coriolis force on the thermosolutal convection threshold in a rotating annular Hele-Shaw cell

In this study, the linear stability analysis is used to determine the onset of thermosolutal convection in fluids confined in rotating annular Hele-Shaw cell. The fluid layer is submitted to radial gradients of temperature and concentration. The effects of both Coriolis force and curvature parameter on the stationary and oscillatory convection are investigated when the Prandtl number is of the order of unity or larger than unity.     

The Effect of Rotation on the Onset of Double Diffusive Convection in a Horizontal Anisotropic Porous Layer

The effect of rotation and anisotropy on the onset of double diffusive convection in a horizontal porous layer is investigated using a linear theory and a weak nonlinear theory. The linear theory is based on the usual normal mode technique and the nonlinear theory on the truncated Fourier series analysis. Darcy model extended to include time derivative and Coriolis terms with anisotropic permeability is used to describe the flow through porous media. The effect of rotation, mechanical and thermal anisotropy parameters, and the Prandtl number on the stationary and overstable convection is discussed. It is found that the effect of mechanical anisotropy is to allow the onset of oscillatory convection instead of stationary. It is also found that the existence of overstable motions in case of rotating porous medium is not restricted to a particular range of Prandtl number as compared to the pure viscous fluid case. The finite amplitude analysis is performed to find the thermal and solute Nusselt numbers. The effect of various parameters on heat and mass transfer is also investigated.     

Linear and nonlinear double diffusive convection in a rotating sparsely packed porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated rotating porous layer is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and nonlinear stability analyses. The Brinkman model that includes the Coriolis term is employed as the momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for the energy equation. The onset criterion for stationary, oscillatory, and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal diffusion, solute diffusion, and rotation that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number, and Taylor number on the stability of the system is investigated. The nonlinear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study.     

Linear and non-linear double diffusive convection in a rotating porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated rotating porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and non-linear stability analyses. The Darcy model that includes the time derivative and Coriolis terms is employed as momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusions that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number and Taylor number on the stability of the system is investigated. The non-linear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out.     

Onset of Double-Diffusive Reaction–Convection in an Anisotropic Rotating Porous Layer

The linear and non-linear stability of a rotating double-diffusive reaction–convection in a horizontal anisotropic porous layer subjected to chemical equilibrium on the boundaries is investigated considering a Darcy model that includes the Coriolis term. The effect of Taylor number, mechanical, and thermal anisotropy parameters, reaction rate, solute Rayleigh number, Lewis number, and normalized porosity on the stability of the system is investigated. We find that the Taylor number has a stabilizing effect, chemical reaction may be stabilizing or destabilizing and that the anisotropic parameters have significant influence on the stability criterion. The effect of various parameters on the stationary, oscillatory, and finite-amplitude convection is shown graphically. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer.     

The Effect of Rotation on the Onset of Double Diffusive Convection in a Sparsely Packed Anisotropic Porous Layer

The effect of rotation on the onset of double diffusive convection in a sparsely packed anisotropic porous layer, which is heated and salted from below, is investigated analytically using the linear and nonlinear theories. The Brinkman model that includes the Coriolis term is employed for the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes and a dispersion relation are obtained analytically using linear theory. The effect of anisotropy parameters, Taylor number, Darcy number, solute Rayleigh number, Lewis number, Darcy–Prandtl number, and normalized porosity on the stationary, oscillatory and finite amplitude convection is shown graphically. It is found that contrary to its usual influence on the onset of convection in the absence of rotation, the mechanical anisotropy parameter show contrasting effect on the onset criterion at moderate and high rotation rates. The nonlinear theory based on the truncated representation of Fourier series method is used to find the heat and mass transfers. The effect of various parameters on heat and mass transfer is shown graphically. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study.     

Three-Dimensional Convection in an Inclined Porous Layer Heated from below and Subjected to Gravity and Coriolis Effects

The linear stability theory is used to investigate analytically the effects of Coriolis acceleration on gravity driven convection in a rotating porous layer. The stability of a basic solution is analysed with respect to the onset of stationary convection. It was discovered that increasing the Taylor number caused degeneracy to polyhedric cells for a specific range of inclination angles. The effects of the magnitude of the horizontal wavenumber is discussed in relation to the magnitude of the Taylor number.     

Coriolis effect on thermal convective instability of viscoelastic fluids in a rotating porous cylindrical annulus

Linear stability analysis of thermal convection is studied for a viscoelastic fluid in a rotating porous cylindrical annulus. The modified Darcy–Jeffrey model with the addition of the Coriolis term in a rotating frame of reference is applied to characterize the non-Newtonian rheology in porous media. We investigate how the interaction among the Coriolis force, viscoelasticity, and bounded sidewalls affects the preferred mode at the onset of convection. The results show that for a slowly rotating case, the oscillatory mode is always preferred for any considered cylindrical radii. However, for a moderately rotating case, the oscillatory preferred mode only arises intermittently as the outer cylindrical radius gradually increases. This result is quite different from the case for viscoelastic fluids in a rotating porous layer or in a porous cylinder without rotation. Further, we discover that for a pair of given cylindrical radii when the Taylor number exceeds a critical value depending on the viscoelastic parameters, the oscillatory convection does not occur. We also examine how the variations of the Taylor number and the viscoelastic parameters affect the patterns of temperature disturbance at the onset of convection.     

Natural convection in a rotating anisotropic porous layer with internal heat generation

In this article, linear and nonlinear thermal instability in a rotating anisotropic porous layer with heat source has been investigated. The extended Darcy model, which includes the time derivative and Coriolis term has been employed in the momentum equation. The linear theory has been performed by using normal mode technique, while nonlinear analysis is based on minimal representation of the truncated Fourier series having only two terms. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. Effects of parameters on critical Rayleigh number has also been investigated. A weak nonlinear analysis based on the truncated representation of Fourier series method has been used to find the Nusselt number. The transient behavior of the Nusselt number has also been investigated by solving the finite amplitude equations using a numerical method. Steady and unsteady streamlines, and isotherms have been drawn to determine the nature of flow pattern. The results obtained during the analysis have been presented graphically.     

Stability of Solutal Convection in a Gravity Modulated Mushy Layer During the Solidification of Binary Alloys

The linear stability theory is used to investigate analytically the effects of gravity modulation on solutal convection in the mushy layer of solidifying binary alloys. The gravitational field consists of a constant part and a sinusoidally varying part, which is synonymous to a vertically oscillating mushy layer subjected to constant gravity. The linear stability results are presented for both the synchronous and subharmonic solutions. It is demonstrated that up to the transition point between the synchronous and subharmonic regions, increasing the frequency of vibration rapidly stabilizes the solutal convection. Beyond the transition point, further increases in the frequency tend to destabilize the solutal convection, but gradually. It is also demonstrated that the effect of increasing the ratio of the Stefan number and the solid composition (₀) is to destabilize the solutal convection.     

Thermal instability in a rotating anisotropic porous layer saturated by a viscoelastic fluid

Linear and non-linear thermal instability in a rotating anisotropic porous medium, saturated with viscoelastic fluid, has been investigated for free-free surfaces. The linear theory is being related to the normal mode method and non-linear analysis is based on minimal representation of the truncated Fourier series analysis containing only two terms. The extended Darcy model, which includes the time derivative and Coriolis terms has been employed in the momentum equation. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. A weak non-linear theory based on the truncated representation of Fourier series method is used to find the thermal Nusselt number. The transient behaviour of the Nusselt number is also investigated by solving the finite amplitude equations using a numerical method. The results obtained during the analysis have been presented graphically.     

The Onset of Double Diffusive Convection in a Couple Stress Fluid Saturated Anisotropic Porous Layer

The double diffusive convection in a horizontal couple stress fluid saturated anisotropic porous layer, which is heated and salted from below, is studied analytically. The modified Darcy equation that includes the time derivative term is used to model 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 parameter, solute Rayleigh number, Lewis number, couple stress parameter, and Vadasz number on the stationary, oscillatory, and finite amplitude convection is shown graphically. It is found that the thermal anisotropy parameter, couple stress parameter, and solute Rayleigh number have stabilizing effect on the stationary, oscillatory, and finite amplitude convection. The mechanical anisotropy parameter has destabilizing effect on stationary, oscillatory, and finite amplitude convection. The Lewis number has stabilizing effect in the case of stationary and finite amplitude modes, with dual effect in the case of oscillatory convection. Vadasz number advances the onset of oscillatory convection. The heat and mass transfer decrease with an increase in the values of couple stress parameter, while both increase with an increase in the value of solute Rayleigh number and mechanical anisotropy parameter. The thermal anisotropy parameter and Lewis number have contrasting effect on the heat mass transfer.     

Coriolis Effect on Convection in a Rotating Porous Layer Subjected to Variable Gravity

We consider the effects of rotation in a porous layer heated from below and subjected to a variable gravity field. The study is presented for large Vadasz numbers where no oscillatory convection is possible. It is demonstrated that the Coriolis acceleration stabilizes the convection in a variable gravity field, whilst the effect of gravity parameter stabilses the convection when reduced and destabilizes the convection when increased.     

Non-Linear Two Dimensional Double Diffusive Convection in a Rotating Porous Layer Saturated by a Viscoelastic Fluid

Double diffusive convection in a rotating anisotropic porous layer, saturated by a viscoelastic fluid, heated from below and cooled from above has been studied making linear and non-linear stability analyses. The fluid and solid phases are considered to be in equilibrium. In momentum equation, we have employed the Darcy equation which includes both time derivative and Coriolis terms. The linear theory based on normal mode method is considered to find the criteria for the onset of stationary and oscillatory convection. A weak non-linear analysis based on minimal representation of truncated Fourier series analysis containing only two terms has been used to find the Nusselt number and Sherwood number as functions of time. We have solved the finite amplitude equations using a numerical scheme. The results obtained, during the above analyses, have been presented graphically and the effects of various parameters on heat and mass transfer have been discussed. Finally, we have drawn the steady and unsteady streamlines, isotherms, and isohalines for various parameters.     

Inertial and coriolis effects on oscillatory flow in a horizontal dendrite layer

Flow instability due to oscillatory modes of disturbances in a horizontal dendrite layer during alloy solidification is investigated under an external constraint of rotation. The flow in the dendrite layer, which is modeled as flow in a porous layer and with the inertial effects included, is assumed to rotate about the vertical axis at a constant angular velocity. The investigation is an extension of the work in Riahi (On stationary and oscillatory modes of flow instablity in a rotating porous layer during alloy solidification. J. Porous Media, 6, 177–187, 2003), which was for the case in the absence of the inertial effects. Results of the stability analyses indicate, in particular, that the Coriolis effect can enhance the physical domain for the oscillatory flow, while the inertial effect tends to reduce such domain. Sufficiently strong inertial effect can eliminate presence of the oscillatory mode only for the rotation rate beyond some value. The effect of interaction between the local volume fraction of solid and the flow associated with the Coriolis term was found to be stabilizing.