Nonlinear evolution analysis and stability for convection in a mushy layer with permeable interface

We consider the problem of two- and three-dimensional nonlinear buoyant flows in horizontal mushy layers during the solidification of binary alloys. We study the nonlinear evolution of such flow based on a recently developed realistic model for the mushy layer with permeable interface. The evolution approach is based on a Landau type equation for the amplitude of the secondary nonlinear solution, which can be in the form of rolls, squares, rectangles or hexagons. Using both analytical and computational methods, we calculate the solutions to the evolution equation near the onset of motion for both subcritical and supercritical regimes and determine the stable solutions. We find, in particular, that for several investigated cases with different parameter regimes, secondary solution in the form of subcritical down-hexagons or supercritical up-hexagons can be stable. However, the preferred solution for smallest values of the Rayleigh number and the amplitude of motion is in the form of subcritical down-hexagons. This result appears to agree with the experimental observation on the form of the convective flow near the onset of motion.     

Effect of Low Rotation Rate on Steady Convection During the Solidification of a Ternary Alloy

We consider the problem of steady convective flow during the directional solidification of a horizontal ternary alloy system rotating at a constant and low rate about a vertical axis. Under the limit of large far-field temperature, the flow region is modeled to be composed of two horizontal mushy layers, which are referred to here as a primary layer over a secondary layer. We first determine the basic state solution and then carry out linear stability analysis to calculate the neutral stability boundary and the critical conditions at the onset of motion. We find, in particular, that there are two flow solutions and each solution exhibits two neutral stability boundaries, and the flow can be multi-modal in the low rotating rate case with local minima on each neutral boundary. The critical Rayleigh number and the wave number as well as the vertical volume flux increase with the rotation rate, but the flow is found to be less stabilizing as compared to the binary alloy counterpart flow. The effects of low rotation rate increase the solid fraction and the liquid fraction at certain vertically oriented fluid lines, and the highest value of such increase is at a horizontal level close to the interface between the two mushy layers.     

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 an Initially, Stably Stratified Fluid Subjected to a Step Change in Temperature

The onset of convective instability in an initially quiescent, stably stratified fluid layer between two horizontal plates is analyzed with linear theory. The bottom boundary is heated suddenly from below, subjected to a step change in surface temperature. The critical time t _c to mark the onset of Rayleigh-Bénard convection is predicted by propagation theory. This theory uses the length scaled by , where α denotes thermal diffusivity. Under the normal mode analysis the dimensionless disturbance equations are obtained as a function of τ(=αt/d ²) and ζ(=Z/), where d is the fluid layer depth and Z is the vertical distance. The resulting equations are transformed to self-similar ones by using scaling and finally fixing τ as τ_c under the frame of coordinates τ and ζ. For a given γ, Pr and τ_c, the minimum value of Ra is obtained from the marginal stability curve. Here γ denotes the temperature ratio to represent the degree of stabilizing effect, Pr is the Prandtl number and Ra is the Rayleigh number. With γ=0, the minimum Ra value approaches the well-known value of 1708 as τ_c increases. However, it is inversely proportional to τ_c ^3/2 as τ_c decreases. With increasing γ, the system becomes more stable. It is interesting that in the present system, propagation theory produces the stability criteria to bound the available experimental data over the whole domain of time. Received 5 November 2001 and accepted 29 March 2002 Published online: 2 October 2002 RID="*" ID="*" This work has been supported by both SK Chemicals Co. Ltd. and LG Chemical Ltd., Seoul under the Brain Korea 21 Project of the Ministry of Education. Communicated by H.J.S. Fernando     

Double-diffusive parallel flow induced in a horizontal Brinkman porous layer subjected to constant heat and mass fluxes: analytical and numerical studies

Fluid flow and heat and mass transfer induced by double-diffusive natural convection in a horizontal porous layer subjected to vertical gradients of temperature and concentration are studied analytically and numerically using the Brinkman-extended Darcy model. Both cases of rigid and free horizontal boundaries are examined in the present study. The parameters governing the problem are the Rayleigh number R_T, the Lewis number Le, the buoyancy ratio N, the Darcy number Da and the aspect ratio Ar. The analytical solution is based on the parallel flow approximation. The critical Rayleigh number corresponding to the onset of the parallel flow in this system is determined analytically as a function of Le, N and Da. For sufficiently small Da, both free and rigid boundaries yield results which are identical to those predicted by the Darcy model. The present investigation shows that there exists a region in the plane (N, Le) where the convective flow is not possible in the layer regardless of the Rayleigh and Darcy numbers considered. Received on 21 December 1998     

An experimental evaluation of the behaviour of mono and polydisperse polystyrenes in Cross-Slot flow

The behaviour of a number of mono and polydisperse polystyrenes are probed experimentally in complex extensional flow within a Cross-Slot geometry using flow-induced birefringence. Polystyrenes with similar molecular weight (M _w) and increasing polydispersity (PD) illustrated the effect of PD on the principal stress difference (PSD) pattern in extensional flow. Monodisperse materials exhibited only slight asymmetry at moderate flowrates, although increased asymmetry and cusping was observed at high flowrates. The response of monodisperse materials of different M _w at various flowrates is presented and characterised by Weissenberg numbers for both chain stretch and orientation using a theory for linear entangled polymers. The comparison of stress profiles against Weissenberg number for each process is used to determine whether the PSD pattern observed is independent of M _w and elucidate which relaxation mechanism is dominant in the flow regimes probed. For monodisperse materials, at equivalent chain orientation Weissenberg number (We _τd), different molecular weight materials were seen to exhibit similar steady state PSD patterns independent of We _τR (chain stretch We). Whilst no obvious critical Weissenberg number (We) was found for the onset of increased asymmetry and cusping, it was found to occur in the “orientating flow without chain stretch” regime.     

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.     

Onset of Buoyancy-Driven Convection in Porous Media Saturated with Cold Water Cooled from Above

When porous media saturated with initially stagnant cold water around the density maximum temperature are cooled from above, convection may be induced in an unstable lower layer. In this study, the onset of buoyancy-driven convection during time-dependent cooling is investigated using the propagation theory, which transforms disturbance equations similarly, and also considering the density inversion effect. The critical Darcy–Rayleigh number Ra _D,c is found as a function of the dimensionless density maximum temperature θ _max. For Ra _D > Ra _D,c the dimensionless critical time τ _c to mark the onset of instability is presented as a function of Ra _D and θ _max. These critical conditions are compared with previous theoretical results.     

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.     

Hydromagnetic free convection flow with induced magnetic field effects

An exact solution is presented for the hydromagnetic natural convection boundary layer flow past an infinite vertical flat plate under the influence of a transverse magnetic field with magnetic induction effects included. The transformed ordinary differential equations are solved exactly, under physically appropriate boundary conditions. Closed-form expressions are obtained for the non-dimensional velocity (u), non-dimensional induced magnetic field component (B _x ) and wall frictional shearing stress i.e. skin friction function (τ _x ) as functions of dimensionless transverse coordinate (η), Grashof free convection number (G _r ) and the Hartmann number (M). The bulk temperature in the boundary layer (Θ) is also evaluated and shown to be purely a function of M. The Rayleigh flow distribution (R) is derived and found to be a function of both Hartmann number (M) and the buoyant diffusivity parameter (ϑ ^*). The influence of Grashof number on velocity, induced magnetic field and wall shear stress profiles is computed. The response of Rayleigh flow distribution to Grashof numbers ranging from 2 to 200 is also discussed as is the influence of Hartmann number on the bulk temperature. Rayleigh flow is demonstrated to become stable with respect to the width of the boundary layer region and intensifies with greater magnetic field i.e. larger Hartman number M, for constant buoyant diffusivity parameter ϑ ^*. The induced magnetic field (B _x ), is elevated in the vicinity of the plate surface with a rise in free convection (buoyancy) parameter G _r , but is reduced over the central zone of the boundary layer regime. Applications of the study include laminar magneto-aerodynamics, materials processing and MHD propulsion thermo-fluid dynamics.     

Onset of unsteady axi-symmetric laminar natural convection in a vertical cylindrical enclosure heated at the wall

In the present study laminar transition to oscillatory convection of fluids having different Prandtl numbers in a laterally heated vertical cylindrical enclosure for different aspect ratios (melt height to crucible radius) of 2–4 is investigated numerically for 0.01 ≤ Pr ≤ 10. Numerical solution to two-dimensional axisymmetric transient Navier Stokes equations and energy equation were solved by finite volume method using SIMPLE algorithm. Numerical results illustrate that there exists a critical Rayleigh number for each Prandtl number beyond which sustained laminar oscillatory flow sets in. The oscillatory regime was characterised by the oscillation of the average kinetic energy and average thermal energy of the melt. For a given aspect ratio, critical Rayleigh number increases with Pr upto 1 and then flattens. It was observed that for low Prandtl number fluids, Pr < 1.0, critical Rayleigh number is found to increase with increase in aspect ratio while for high Prandtl number fluids, Pr ≥ 1.0, it is found to decrease with increase in aspect ratio. The influence of aspect ratio on the transient behaviour of the melt volume below and above the critical Rayleigh number was studied.     

Variable permeability effect on convection in binary mixtures saturating a porous layer

The Darcy Model with the Boussinesq approximation is used to study natural convection in a shallow porous layer, with variable permeability, filled with a binary fluid. The permeability of the medium is assumed to vary exponentially with the depth of the layer. The two horizontal walls of the cavity are subject to constant fluxes of heat and solute while the two vertical ones are impermeable and adiabatic. The governing parameters for the problem are the thermal Rayleigh number, R _T, the Lewis number, Le, the buoyancy ratio, φ, the aspect ratio of the cavity, A, the normalized porosity, ε, the variable permeability constant, c, and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in an infinite layer, an analytical solution of the steady form of the governing equations is obtained on the basis of the parallel flow approximation. The onset of supercritical convection, or subcritical, convection are predicted by the present theory. A linear stability analysis of the parallel flow model is conducted and the critical Rayleigh number for the onset of Hopf’s bifurcation is predicted numerically. Numerical solutions of the full governing equations are found to be in excellent agreement with the analytical predictions.     

Immiscible Newtonian displacement by a viscoplastic material in a capillary plane channel

The immiscible displacement in a capillary plane channel of a Newtonian liquid by a viscoplastic one that obeys a Papanastasiou’s constitutive equation is numerically analyzed. An elliptic mesh generation technique, coupled with the Galerkin finite element method is used to determine the velocity field and the configuration of the interface between the two materials. We investigate the displacement efficiency and the flow patterns of the problem as functions of the dimensionless parameters that govern the problem: the capillary number (Ca), the viscosity ratio of the two fluids (N _η ) and the yield number, (τ′₀). The numerical results showed that for a fixed viscosity ratio, the fraction of mass attached to the wall is a decreasing function of τ′₀. We constructed maps of streamlines in the Cartesian space defined by τ′₀ and Ca for fixed viscosity ratios in order to capture the rough location of bypass and recirculating flow regimes. Higher yield number values induce bypass flow regimes, especially for high Ca. The dimensionless forms of the momentum conservation equation and the force balance at the interface were essential for the understanding of the role played by the dimensionless numbers that govern the problem.     

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.     

On the Convection in a Porous Medium with Inclined Temperature Gradient and Vertical Throughflow. Part I. Normal Modes

In this second part of our analysis of the destabilization of transverse modes in an extended horizontal layer of a saturated porous medium with inclined temperature gradient and vertical throughflow, we apply the mathematical formalism of absolute and convective instabilities to studying the nature of the transition to instability of such modes by assuming on physical grounds that the transition is triggered by growing localized wavepackets. It is revealed that in most of the parameter cases treated in the first part of the analysis (Brevdo and Ruderman 2009), at the transition point the evolving instability is convective. Only in the cases of zero horizontal thermal gradient, and in the cases of zero vertical throughflow and the horizontal Rayleigh number R _h < 49, the instability is absolute implying that, as the vertical Rayleigh number, R _v, increases passing through its critical value, R _vc, the destabilization tends to affect the base state throughout and eventually destroys it at every point in space. For the parameter values considered, for which the destabilization has the nature of convective instability, we found that, as R _v, increases beyond the critical value, while the horizontal Rayleigh number, R _h, and the Péclet number, Q _v, are kept fixed, the flow experiences a transition from convective to absolute instability. The values of the vertical Rayleigh number, R _v, at the transition from convective to absolute instability are computed. For convectively unstable, but absolutely stable cases, the spatially amplifying responses to localized oscillatory perturbations, i.e., signaling, are treated and it is found that the amplification is always in the direction of the applied horizontal thermal gradient.     

Onset of Convection in a Porous Layer with Continuous Periodic Horizontal Stratification. Part I. Two-Dimensional Convection

The onset of convection in a horizontal porous layer is investigated theoretically. The permeability of the porous medium is a continuous periodic function of the horizontal x coordinate. Floquet theory has been employed to determine the favoured two-dimensional mode of convection. For a wide range of periods of the permeability variation, a matrix eigenvalue technique with eighth order accuracy has been employed to find the critical Darcy– Rayleigh number. This is supplemented by a multiple-scales analysis of the large-period limit, and a brief consideration of the anisotropic limit for very short periods.     

Transition from Convective to Absolute Instability in a Porous Layer with Either Horizontal or Vertical Solutal and Inclined Thermal Gradients,and Horizontal Throughflow

Recently, in Diaz and Brevdo (J Fluid Mech 681: 567–596, 2011), further in the text referred to as D&B, we found an absolute/convective instability dichotomy at the onset of convection in a flow in a saturated porous layer with either horizontal or vertical solutal and inclined thermal gradients, and horizontal throughflow. The control parameter in D&B triggering the destabilization is the vertical thermal Rayleigh number, R _v. In this article, we treat the parameter cases considered in D&B in which the onset of convection has the character of convective instability and occurs through longitudinal modes. By increasing the vertical thermal Rayleigh number starting from its critical value, R _vc, we determine the value R _vt of R _v at which the transition from convective to absolute instability takes place and compute the physical characteristics of the emerging absolutely unstable wave packet. In some cases, the value of the transitional vertical thermal Rayleigh number, R _vt, is only slightly greater than the critical value, R _vc, meaning that at the onset of convection the base convectively unstable state can be viewed as marginally absolutely unstable. However, in several cases considered, the value of R _vt is significantly greater than the critical value, R _vc, implying that the base state is not marginally but essentially absolutely stable at the point of destabilization.     

Squeezed flow and heat transfer over a porous surface for viscous fluid

Flow and heat transfer over a permeable sensor surface placed in a squeezing channel is analyzed. A constant transpiration through the sensor surface is assumed. Locally non-similar momentum and energy equations are solved by three different methods, against the transpiration parameter τ, for different values of the squeezing parameter b, and Prandtl number Pr. From the investigation, it is found that when the channel being squeezed, the skin-friction reduces but the heat transfer coefficient increases. Increase in the value of the squeezing parameter onsets reverse flow at the sensor surface when fluid is being injected and the affect is enhanced with the increase of injection through the surface. It is further observed that increase of suction of fluid through the sensor thins the thermal and the momentum boundary layer regions, whereas injection of fluid leads to thickening of both the thermal and the momentum boundary layer regions. Heat transfer from the surface of the sensor increases with the increase of the value of Pr for the entire range of surface mass-flux parameter τ. M. A. Hossain is on leave of absence from University of Dhaka.     

Heat transfer across convecting porous layers with flux boundaries

The ‘power integral method’ of calculating heat transfer across a convecting porous layer is extended to flux and porous boundaries. Convection starts at lower Rayleigh numbers for constant flux than for isothermal impervious boundaries and the flux is much greater. At higher Rayleigh numbers, as more of the higher modes contribute to the flux, the type of boundary has less influence on the heat transfer across the layer. For constant flux boundaries, simplified equations are developed to determine critical values for the second and higher modes and these values can be related simply to those for isothermal impervious boundaries.     

Axial annular flow of plastic fluids: dead zones and plug-free flow

In axial annular flow, the shear stress decreases from its value τ(κR) at the inner cylinder to 0 at r = λR and increases from then on to τ(R) at the outer cylinder. For plastic fluids with a yield stress τ _c, λ will be such that flow commences when τ(κR) = τ(R) = τ _c. For fluids with position-dependent yield stresses (electro- and magnetorheological fluids are examples), the situation is more complex. While it is possible that yielding and flow occur everywhere, it is also possible that flow occurs only in parts of the fluid-filled space, and a dead zone (region in which the fluid is at rest) close to one of the walls exists. In that case, the fluid will flow no matter how small the applied pressure difference is. If P is large enough, the dead zone ceases to exist and flow without any plug is possible. The fluid flows as if no yield stress exists.