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.     

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.     

On the Linear Stability of Large Stefan Number Convection in Rotating Mushy Layers for 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 large 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 a large Stefan number scaling allows for the presence of both the stationary and oscillatory modes of convection. In contrast to the problem of a stationary mushy layer, rotating the mushy layer has a stabilising effect on convection. It was observed that increasing the Taylor number or the Stefan number encouraged the oscillatory mode of convection.     

Linear stability of solutal convection in solidifying mushy layers: permeable mush–melt interface

The linear stability theory is used to investigate analytically the effect of a permeable mush–melt boundary condition on the stability of solutal convection in a mushy layer of homogenous permeability at the near eutectic (solid) limit. The results clearly show that, in contrast to the impermeable mush–melt interface boundary condition, the application of the permeable mush–melt interface boundary condition destabilizes the convection in a mushy layer.     

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.     

On Three-Dimensional Non-linear Buoyant Convection in Ternary Solidification

We consider the problem of three-dimensional non-linear buoyant convection in ternary solidification. Under the limit of large far-field temperature, the convective flow is modeled to be in a rectangular cube composing of a horizontal liquid layer above a primary mushy layer, which itself is over a secondary mushy layer. We first apply linear stability analysis to calculate the conditions at the onset of motion. Next, we carry out weakly non-linear analyses to determine solutions in the form of hexagons and their possible stability and to obtain information about tendency for chimney formation. We find that if the flow is driven either from both mushy layers with equal critical conditions at the onset of motion or only by the primary mushy layer, then the flow can be in the form of a double-cell structure vertically with down-hexagons below or above up-hexagons. There is tendency for vertically oriented chimney formation at different horizontal locations in each mushy layer. For the cases where only the critical conditions at the onset of motion are equal in both mushy layers and depending on the values of the mush Rayleigh numbers, the flow can be subcritical (or supercritical) in both mushy layers or mixed subcritical in one layer and supercritical in another layer.     

Stability of Solutal Convection in a Rotating Mushy Layer Solidifying from a Vertical Surface

We consider the effects of rotation in a mushy layer being cast from a vertical surface where the effects of Coriolis acceleration, gravity and centrifugal effects are included. It is demonstrated that the Coriolis acceleration and gravity play a passive role in convection and are excluded from the stability criteria. The stability criteria is presented as the critical centrifugal Rayleigh numbers referenced for locations far away (start of solidification) and close to (nearing end of solidification) the axis or rotation.     

The onset of penetrative double-diffusive convection

The onset of double-diffusive convection in a horizontal fluid layer is studied. The density is assumed to depend quadratically on the temperature and linearly on the solute concentration. Under the Boussinesq approximation, the linear stability of the conduction state is investigated with respect to the oscillatory and steady convection modes. For steady onset, the critical thermal Rayleigh number is found to be a double-valued function of the solutal Rayleigh number as long as the relative maximum of the density profile exists within the fluid layer. Driving mechanisms of the steady convections are discussed.     

The onset of double diffusive convection in a viscoelastic fluid layer

The onset of double diffusive convection in a viscoelastic fluid layer is studied using a linear and a weak nonlinear stability analyses. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. There is a competition between the processes of thermal diffusion, solute diffusion and viscoelasticity that causes the convection to set in through oscillatory mode rather than stationary. The effect of Deborah number, retardation parameter, solutal Rayleigh number, Prandtl number, Lewis number on the stability of the system is investigated. It is shown that the critical frequency increases with Deborah number and solutal Rayleigh number while it decreases with retardation parameter and Lewis number. The nonlinear theory based on the truncated representation of Fourier series method is used to find the heat and mass transfers. The transient behaviour of the Nusselt number and Sherwood number is investigated by solving the finite amplitude equations using Runge-Kutta method. The effect of viscoelastic parameters on heat and mass transfer is brought out.     

Moderate time scale finite amplitude analysis of large stefan number convection in a gravity modulated mushy layer during the solidification of binary alloys

We investigate the convection amplitude in a binary alloy mushy layer subjected to a vibration body force that is collinear with the gravitational acceleration. The analysis shows that the convection amplitude decreases over time for all vibration frequencies tested. The analysis further reveals that as the vibration frequency increases, the convection amplitude subsequently decreases until a critical vibration frequency; at which the amplitude reaches the lowest value. Further increases in the vibration frequency increase the convection amplitude but gradually.     

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.     

Penetrative convection in sublayer of water including density inversion

Numerical solutions of stability and convective flow in an infinite horizontal water layer, including density inversion, have been obtained using a finite element code. The evolution of the temperature field and flow pattern near the onset of convection are studied in detail. It is known that natural convection develops primarily in the lower unstably stratified layer. Of interest is the penetration of the convection rolls into the upper stably stratified layer and concurrent liquid entrainment as a function of the increasing Rayleigh number at different aspect ratios. Individual convection rolls may grow and expand before splitting up into two roll cells. It is shown that changing the aspect ratio influences critical Rayleigh number, flow symmetry, flow pattern, and transitions between flow patterns. Numerical results on heating from above or from below, agree well with available results in the literature. A correlation to predict critical Rayleigh numbers is given for the case of heating from above.     

The Onset of Double-Diffusive Convection in a Superposed Fluid and Porous Layer Under High-Frequency and Small-Amplitude Vibrations

We numerically simulate the initiation of an average convective flow in a system composed of a horizontal binary fluid layer overlying a homogeneous porous layer saturated with the same fluid under gravitational field and vibration. In the layers, fixed equilibrium temperature and concentration gradients are set. The layers execute high-frequency oscillations in the vertical direction. The vibration period is small compared with characteristic timescales of the problem. The averaging method is applied to obtain vibrational convection equations. Using for computation the shooting method, a numerical investigation is carried out for an aqueous ammonium chloride solution and packed glass spheres saturated with the solution. The instability threshold is determined under two heating conditions—on heating from below and from above. When the solution is heated from below, the instability character changes abruptly with increasing solutal Rayleigh number, i.e., there is a jump-wise transition from the most dangerous shortwave perturbations localized in the fluid layer to the long-wave perturbations covering both layers. The perturbation wavelength increases by almost 10 times. Vibrations significantly stabilize the fluid equilibrium state and lead to an increase in the wavelength of its perturbations. When the fluid with the stabilizing concentration gradient is heated from below, convection can occur not only in a monotonous manner but also in an oscillatory manner. The frequency of critical oscillatory perturbations decreases by 10 times, when the long-wave instability replaces the shortwave instability. When the fluid is heated from above, only stationary convection is excited over the entire range of the examined parameters. A lower monotonic instability level is associated with the development of perturbations with longer wavelength even at a relatively large fluid layer thickness. Vibrations speed up the stationary convection onset and lead to a decrease in the wavelength of most dangerous perturbations of the motionless equilibrium state. In this case, high enough amplitudes of vibration are needed for a remarkable change in the stability threshold. The results of numerical simulation show good agreement with the data of earlier works in the limiting case of zero fluid layer thickness.     

Nonlinear convection induced by inclined thermal and solutal gradients with mass flow

The energy method is developed for the convection problem induced by inclined thermal and solutal gradients, with horizontal mass flow, in a horizontal layer of a saturated porous medium. A non-linear stability analysis is performed and compound matrix method is employed for numerical calculations. For representative parameter values the critical vertical thermal Rayleigh number and wave number are calculated. It is noted that the effect of horizontal thermal or solutal gradient is to switch from stabilizing to destabilizing as their magnitude increases, for zero or small values of mass flow rate. For higher values of mass flow rate the effect is always destabilizing. It is also noted that the horizontal concentration gradient has a stronger destabilizing effect as compared to the horizontal temperature gradient. Received March 18, 2000     

Natural convection due to heat and mass transfer in a composite system

A numerical study is performed to analyse heat and mass transfer phenomena due to natural convection in a composite cavity containing a fluid layer overlying a porous layer saturated with the same fluid. The flow in the porous region is modelled using Brinkman–Forchheimer-extended Darcy model that includes both the effect of macroscopic shear (Brinkman effect) and flow inertia (Forchheimer effect). The vertical walls of the two-dimensional enclosure are isothermal whilst the horizontal walls are adiabatic. The two regions are coupled by equating the velocity and stress components at the interface. The resulting coupled equations in non-dimensional form are solved by an alternating direction implicit method by transforming them into parabolic form by the addition of false transient terms. The numerical results show that the amount of fluid penetration into the porous layer depends strongly upon the Darcy, thermal and solutal Rayleigh numbers. Average Nusselt number decreases while average Sherwood number increases with an increase of the Lewis number. The transfer of heat and mass on the heated wall near the interface depends strongly on the Darcy number. Received on 11 May 1998     

Weak Non-Linear Analysis of Moderate Stefan Number Stationary Convection in Rotating Mushy Layers

The effects of rotation on a mushy layer, during the solidification of binary alloys, is considered. A near-eutectic approximation and large far-field temperature are employed in order to decouple the mushy layer from the overlying liquid melt. The current study employs a new moderate time scale for mushy layers exhibiting Stefan numbers of unit order of magnitude. The weak non-linear theory is used to evaluate the leading order amplitude. The results of the weak non-linear theory are then used to establish the nature of the bifurcation, that is whether the bifurcation is forward or inverse.     

Stability Analysis of a Porous Layer Heated from Below and Subjected to Low Frequency Vibration: Frozen Time Analysis

We investigate the convection amplitude in an infinite porous layer subjected to a vibration body force that is collinear with the gravitational acceleration and heated from below. The analysis focuses on the specific case of low frequency vibration where the frozen time approximation is used. The results reveal that for moderate Vadasz numbers, increasing the magnitude of the acceleration stabilizes the convection. The results of the large Vadasz number analysis reveals that the acceleration plays a passive role in the stability of convection and the classical stability criteria for Rayleigh–Benard convection applies.     

Numerical Investigation on Marginal Stability and Convection with and without Magnetic Field in a Mushy Layer

In this article, an investigation is conducted to analyze the marginal stability with and without magnetic field in a mushy layer. During alloy solidification, such mushy layer, which is adjacent to the solidification front and composed of solid dendrites and liquid, is known to produce vertical chimneys. Here, we carry out numerical investigation for particular range of parameter values, which cover those of available experimental studies, to determine the convective flow at the onset of motion. The governing coupled non-linear partial differential equations are non-dimensionalised and solved to get the steady basic-state solution. The thickness of the mushy layer is determined as a part of the solution. Using multiple shooting technique, we determine the steady-state solutions in a range of critical Rayleigh number. We analyse the effect of several parameters, Chandrasekhar number Q, and Robert’s number τ on the problem. It was found that an increase in Q has a stabilizing effect on solidification and the critical Rayleigh number increases on increasing Q. It was also found that for moderate or small values of Robert’s number τ the critical Rayleigh number is mostly insensitive.     

Double-diffusive convection in an inclined porous layer with a concentration-based internal heat source

The thermosolutal instability of double-diffusive convection in an inclined fluid-saturated porous layer with a concentration-based internal heat source is investigated. The linear instability of small-amplitude perturbations to the system is analyzed with respect to transverse and longitudinal rolls. The resultant eigenvalue problem is solved numerically utilizing the Chebyshev tau method. It is shown that an increasing inclination angle causes a strong stabilization in the transverse rolls irrespective of the internal heat source or vertical solutal Rayleigh number. Furthermore, substantial qualitative changes are demonstrated in the linear instability thresholds with variations in the inclination angle and concentration-based heat source.     

The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium: double diffusive case

The effects of both horizontal and vertical hydrodynamic, thermal and solutal heterogeneity, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, are studied analytically using linear stability theory for the case of weak heterogeneity. The Brinkman model is employed. It is found that the effect of such heterogeneity on the critical value of the Rayleigh number Ra based on mean properties is of second order if the properties vary in a piecewise constant or linear fashion. The effects of horizontal heterogeneity and vertical heterogeneity are then comparable once the aspect ratio is taken into account, and to a first approximation are independent.