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.     

Stability of Moderate Vadasz Number Solutal Convection in a Cylindrical Mushy Layer Subjected to Vertical Vibration

We consider vibration effects on the stability of solutal convection in a mushy layer being cast in a cylindrical geometry. The near eutectic limit is applied and moderate Vadasz numbers are considered to retain the second-order time derivative in the Darcy equation. Since small to moderate radii casting crucibles are the current area of interest, only synchronous modes are analyzed. The results indicate that the presence of vibration in solidifying mushy layers stabilizes the convection, and provides a quantification of the Rayleigh number associated with solutal convection. Of particular interest is the fact that in solidifying systems, the Rayleigh numbers are significantly smaller than that of a passive porous layer.     

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.     

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.     

On Mixed Oscillatory and Steady Modes of Nonlinear Convection in Mushy Layers

We consider the problem of mixed oscillatory and steady modes of nonlinear compositional convection in horizontal mushy layers during the solidification of binary alloys. Under a near-eutectic approximation and the limit of large far-field temperature, we determine a number of two- and three-dimensional weakly nonlinear mixed solutions, and the stability of these solutions with respect to arbitrary three-dimensional disturbances is then investigated. The present investigation is an extension of the problem of mixed oscillatory and steady modes of convection, which was investigated by Riahi (J Fluid Mech 517: 71–101, 2004), where some calculated results were inaccurate due to the presence of a singular point in the equation for the linear frequency. Here we resolve the problem and find some significant new results. In particular, over a wide range of the parameter values, we find that the properties of the preferred and stable solution in the form of particular subcritical mixed standing and steady hexagons appeared to be now in much better agreement with the available experimental results (Tai et al., Nature 359:406–408, 1992) than the one reported in Riahi (J Fluid Mech 517:71–101, 2004). We also determined a number of new types of preferred supercritical solutions, which can be preferred over particular values of the parameters and at relatively higher values of the amplitude of 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.     

On Perturbation and Marginal Stability Analysis of Magneto-Convection in Active Mushy Layer

This present study considers the problem of steady magneto-convection in a horizontal mushy layer with variable permeability and an impermeable mush–liquid interface during directional solidification of binary alloys. We model the flow by introducing a uniform magnetic field in the mushy layer which is considered as a porous medium where Darcy’s law holds and the permeability is a function of the local solid volume fraction. Basic-state solutions are obtained analytically using the no-flow condition. With the help of multiple shooting techniques, we obtain numerical solutions to the linear perturbation system for non-magnetic and magnetic cases. Numerical results are presented showing the effects of the magnetic field and the permeability of the layer. These results demonstrate that the application of an external magnetic field has stabilizing effects on the convection and can reduce the tendency for chimney formation in the mushy layer. In addition, variable permeability, which corresponds to an active mushy layer, indicates more stable and realizable flow system as compared to the case of constant permeability.     

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.     

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.     

Convection in a Horizontal Layer in a Rotating Heat Field

An asymptotic laminar-convection pattern in a plane horizontal liquid layer with a radially nonuniform temperature gradient on its boundaries is investigated. The problem arises in applications connected with modified Czochralski crystal growth technology using the heat field rotation method. An analytical model of the flow is compared with the results of experiments, specially carried out using model fluids and a technological melt. The conditions of adequacy of the model are analyzed and the restrictions on the parameter values and fluid thermophysical properties that ensure the validity of the model are found. The range of variation of the heat field rotation velocity for which the mixing of the melt in the crucible is maximum is determined.     

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.     

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.     

Ferromagnetic Convection in a Rotating Ferrofluid Saturated Porous Layer

The effect of Coriolis force on the onset of ferromagnetic convection in a rotating horizontal ferrofluid saturated porous layer in the presence of a uniform vertical magnetic field is studied. The boundaries are considered to be either stress free or rigid. The modified Brinkman–Forchheimer-extended Darcy equation with fluid viscosity different from effective viscosity is used to characterize the fluid motion. The condition for the occurrence of direct and Hopf bifurcations is obtained analytically in the case of free boundaries, while for rigid boundaries the eigenvalue problem has been solved numerically using the Galerkin method. Contrary to their stabilizing effect in the absence of rotation, increasing the ratio of viscosities, Λ, and decreasing the Darcy number Da show a partial destabilizing effect on the onset of stationary ferromagnetic convection in the presence of rotation, and some important observations are made on the stability characteristics of the system. Moreover, the similarities and differences between free–free and rigid–rigid boundaries in the presence of buoyancy and magnetic forces together or in isolation are emphasized in triggering the onset of ferromagnetic convection in a rotating ferrofluid saturated porous layer. For smaller Taylor number domain, the stress-free boundaries are found to be always more unstable than in the case of rigid boundaries. However, this trend is reversed at higher Taylor number domain because the stability of the stress-free case is increased more quickly than the rigid case.     

Thermal Convection in a Plane Layer Rotating About a Horizontal Axis

The thermal convection of a fluid in a plane vertical layer with a cylindrical lateral boundary, which rotates uniformly about a horizontal symmetry axis, is investigated experimentally. The structure and excitation limit of the convective flows are studied as functions of the rotation frequency, the temperature difference between the layer boundaries, and the layer thickness. The determining dimensionless parameters are found. It is shown that the period-average gravity action produces convection in the form of hexagonally distributed cells stationary in the reference system tied to the cavity.     

Stability of Double-Diffusive Natural Convection in a Vertical Porous Layer

Transport in Porous Media - The stability of double-diffusive buoyant flow in a vertical layer of Darcy porous medium whose boundaries are held at different constant temperatures and solute...     

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.     

Convection in a Horizontal Fluid Layer with a Uniform Internal Heat Source

On the basis of a numerical simulation of convection in a horizontal fluid layer with a uniform heat source it is concluded that the convective heat flux is constant over the entire convection layer not only in the case of steady-state external conditions but also in the case of heating (cooling) of the fluid layer at a constant rate. The convective heat flux is mainly determined by the Rayleigh number and depends only slightly on the layer heating (cooling) rate.     

Mixed Convection in a Ventilated Cavity Filled with a Triangular Porous Layer

Accurate prediction of the macroscopic flow parameters needed to describe flow in porous media relies on a good knowledge of flow field distribution at a much smaller scale—in the pore spaces. The extent of the inertial effect in the pore spaces cannot be underestimated yet is often ignored in large-scale simulations of fluid flow. We present a multiscale method for solving Oseen’s approximation of incompressible flow in the pore spaces amid non-periodic grain patterns. The method is based on the multiscale finite element method [MsFEM Hou and Wu in J Comput Phys 134:169–189, 1997)] and is built in the vein of Crouzeix and Raviart elements (Crouzeix and Raviart in Math Model Numer Anal 7:33–75, 1973). Simulations of inertial flow in highly non-periodic settings are conducted and presented. Convergence studies in terms of numerical errors relative to the reference solution are given to demonstrate the accuracy of our method. The weakly enforced continuity across coarse element edges is shown to maintain accurate solutions in the vicinity of the grains without the need for any oversampling methods. The penalisation method is employed to allow a complicated grain pattern to be modelled using a simple Cartesian mesh. This work is a stepping stone towards solving the more complicated Navier–Stokes equations with a nonlinear inertial term.     

Theoretical Approach to the Onset of Convection in a Porous Layer

We examine the effect of viscous forces on the displacement of one fluid by a second, immiscible fluid along parallel layers of contrasting porosity, absolute permeability and relative permeability. Flow is characterized using five dimensionless numbers and the dimensionless storage efficiency, so results are directly applicable, regardless of scale, to geologic carbon storage. The storage efficiency is numerically equivalent to the recovery efficiency, applicable to hydrocarbon production. We quantify the shock-front velocities at the leading edge of the displacing phase using asymptotic flow solutions obtained in the limits of no crossflow and equilibrium crossflow. The shock-front velocities can be used to identify a fast layer and a slow layer, although in some cases the shock-front velocities are identical even though the layers have contrasting properties. Three crossflow regimes are identified and defined with respect to the fast and slow shock-front mobility ratios, using both theoretical predictions and confirmation from numerical flow simulations. Previous studies have identified only two crossflow regimes. Contrasts in porosity and relative permeability exert a significant influence on contrasts in the shock-front velocities and on storage efficiency, in addition to previously examined contrasts in absolute permeability. Previous studies concluded that the maximum storage efficiency is obtained for unit permeability ratio; this is true only if there are no contrasts in porosity and relative permeability. The impact of crossflow on storage efficiency depends on the mobility ratio evaluated across the fast shock-front and on the time at which the efficiency is measured.     

Marangoni Mixed Convection Boundary Layer Flow

The paper deals with a steady coupled dissipative layer, called Marangoni mixed convection boundary layer, which can be formed along the interface of two immiscible fluids, in surface driven flows. The mixed convection boundary layer is generated when besides the Marangoni effects there are also buoyancy effects due to gravity and external pressure gradient effects. We shall use a model proposed by Golia and Viviani (L’ Aerotecnica missili e Spazio 64 (1985) 29–35, Meccanica 21 (1986) 200–204) wherein the Marangoni coupling condition has been included into the boundary conditions at the interface. The similarity equations are first determined, and the pertinent equations are solved numerically for some values of the governing parameters and the features of the flow and temperature fields as well as the interface velocity and heat transfer at the interface are analysed and discussed.