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.     

Natural Convection Induced by a Centrifugal Force Field in a Horizontal Annular Porous Layer Saturated with a Binary Fluid

The Darcy Model with the Boussinesq approximation is used to study natural convection in a horizontal annular porous layer filled with a binary fluid, under the influence of a centrifugal force field. Neumann boundary conditions for temperature and concentration are applied on the inner and outer boundary of the enclosure. The governing parameters for the problem are the Rayleigh number, Ra, the Lewis number, Le, the buoyancy ratio, j{\varphi } , the radius ratio of the cavity, R, the normalized porosity, e{\varepsilon } , and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in a thin annular layer (R → 1), analytical solutions for the stream function, temperature and concentration fields are obtained using a concentric flow approximation and an integral form of the energy equation. The critical Rayleigh number for the onset of supercritical convection is predicted explicitly by the present model. Also, results are obtained from the analytical model for finite amplitude convection for which the flow and heat and mass transfer are presented in terms of the governing parameters of the problem. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical model and the numerical simulations.     

Stability of Convection in a Gravity Modulated Porous Layer Heated from Below

The linear stability theory is used to investigate analytically the effects of gravity modulation on convection in a homogenous porous layer heated from below. The gravitational field consists of a constant part and a sinusoidally varying part, which is tantamount to a vertically oscillating porous layer subjected to constant gravity. The linear stability results are presented for the specific case of low amplitude vibration for which it is shown that increasing the frequency of vibration stabilises the convection.     

Convective Roll Instabilities of Vertical Throughflow with Viscous Dissipation in a Horizontal Porous Layer

The vertical throughflow with viscous dissipation in a horizontal porous layer is studied. The horizontal plane boundaries are assumed to be isothermal with unequal temperatures and bottom heating. A basic stationary solution of the governing equations with a uniform vertical velocity field (throughflow) is determined. The temperature field in the basic solution depends only on the vertical coordinate. Departures from the linear heat conduction profile are displayed by the temperature distribution due to the forced convection effect and to the viscous dissipation effect. A linear stability analysis of the basic solution is carried out in order to determine the conditions for the onset of convective rolls. The critical values of the wave number and of the Darcy–Rayleigh number are determined numerically by the fourth-order Runge–Kutta method. It is shown that, although generally weak, the effect of viscous dissipation yields an increase of the critical value of the Darcy–Rayleigh number for downward throughflow and a decrease in the case of upward throughflow. Finally, the limiting case of a vanishing boundary temperature difference is discussed.     

Stability of the Horizontal Throughflow in a Power-Law Fluid Saturated Porous Layer

Transport in Porous Media - Double-diffusive convective instability of horizontal throughflow in a power-law fluid saturated porous layer is investigated. The boundaries of this horizontal porous...     

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.     

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.     

Influence of Chemical Reaction on Stability of Thermo-Solutal Convection of Couple-Stress Fluid in a Horizontal Porous Layer

We consider the onset of thermo-solutal convection in a couple-stress fluid-saturated anisotropic porous medium, where the chemical equilibrium on the bounding surfaces and the solubility of the dissolved components depend on temperature. The entire study has been spilt into two parts: (i) linear stability analysis (ii) weakly non-linear stability analysis. Stationary case of linear stability analysis is discussed for two modes of bounding surfaces (a) realistic bounding surfaces i.e. Rigid-Rigid and Rigid-Free (R/R and R/F), (b) non-realistic bounding surfaces i.e. Free-Free (F/F). Howsoever, investigation of oscillatory state and weakly non-linear stability are restricted to F/F case. Galerkin method is used to solve the eigenvalue problem for R/R and R/F cases, whereas, exact solutions are obtained for F/F case.A comparative study among flow stability for above different cases is made as function of ratio of viscosities ( i.e., couple-stress viscosity to fluid viscosity which is defined as couple-stress parameter, $(C)$ ) and effective chemical reaction (i.e. chemical reaction parameter, $(\chi )$ ). It has been found that increasing viscosity of the couple-stress fluid, in terms of increasing $C$ , increases flow stability in all three cases, but among all cases its stabilization effect for R/R is maximum. However, in the absence of couple-stress parameter the maximum stability of flow is observed for F/F. Apart from this, the chemical reaction stabilizes the flow for all the three cases. Furthermore, stability analysis for F/F case indicates that couple-stress parameter stabilizes the system in all modes (stationary, oscillatory and finite amplitude) of convection.Damköhler number $(\chi )$ is found to delay the stationary convection, however, it speeds up the onset of oscillatory convection. The non-linear theory based on truncated representation of Fourier series method predicts the occurrence of sub-critical instability in the form of finite amplitude motion. The effect of $C$ and $\chi $ on heat and mass transfer is also examined.     

Effect of Rotation on the Stability of Advective Flow in a Horizontal Fluid Layer at a Small Prandtl Number

The stability of advective flow in a rotating infinite horizontal fluid layer with rigid bound-aries is investigated for a small Prandtl number Pr = 0.1 and various Taylor numbers for perturbations of the hydrodynamic type. Within the framework of the linear theory of stability, neutral curves describing the dependence of the critical Grashof number on the wave number are obtained. The behavior of finite-amplitude perturbations beyond the stability threshold is studied numerically.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 29–38.Original Russian Text Copyright © 2005 by Schwarz.     

Non-linear Convective Transport in a Binary Nanofluid Saturated Porous Layer

In this article, we study double-diffusive convection in a horizontal porous medium saturated by a nanofluid, for the case when the base fluid of the nanofluid is itself a binary fluid such as salty water. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis, while the Darcy model is used for the porous medium. The thermal energy equations include the diffusion and cross-diffusion terms. The linear stability is studied using normal mode technique and for non-linear analysis, a minimal representation of the truncated Fourier series analysis involving only two terms has been used. For linear theory analysis, critical Rayleigh number has been obtained, while non-linear analysis has been done in terms of the Nusselt numbers.     

Weak Non-Linear Analysis of Convection in a Gravity Modulated Porous Layer

We investigate the convection amplitude in an infinite porous layer subjected to a vibration body force that is collinear with the gravitational acceleration. The analysis shows that increasing the vibration frequency causes the convection amplitude to approach zero, i.e., increasing the vibration frequency stabilizes the convection.     

Weak 2D Convective Plumes in a Sloping Permeable Layer

This is a study of thermal plumes in a permeable fluid-saturated slab of a porous medium (of finite uniform thickness but otherwise infinite extent) that is heated by either an instantaneous or a steady line source embedded in the medium. The slab, which may be horizontal or sloping, is initially at ambient conditions; the impervious upper surface remains at the ambient temperature, while the impervious base is thermally insulated. The transient temperature distribution, the surface heat flux and the convective flow velocity are calculated for small instantaneous line heat energy sources. They show how the flow develops, spreads and slows as time progresses, and an estimate of the time to decay is given. Steady-state temperature and velocity profiles are calculated for embedded line sources that provide heat energy at a small constant rate, and the surface heat flux distribution is calculated.     

The Onset of Bioconvection in a Horizontal Porous-Medium Layer

In this note the problem of the onset of bioconvection in a horizontal layer occupied by a saturated porous medium is analyzed. Gyrotactic effects are incorporated in the analysis. The Darcy flow model is employed, and it is assumed that the bioconvection Péclet number is not greater than unity. Critical values of the bioconvection Rayleigh number and the corresponding critical Rayleigh number are obtained for various values of the bioconvection Péclet number, the gyrotaxis number and the cell eccentricity.     

Influence of the Skin-Effect on the Convective Stability of a Binary Mixture in a Porous Layer with Modulation of the Boundary Temperature

The influence of the skin-effect on the convective stability of a binary mixture in a horizontal porous layer with respect to modulation of the temperature on one of the boundaries is studied. The absence of an admixture is considered as a particular case.     

Stability of Flow with Viscous Dissipation in a Horizontal Porous Layer with an Open Boundary Having a Prescribed Temperature Gradient

A buoyancy-induced stationary flow with viscous dissipation in a horizontal porous layer is investigated. The lower boundary surface is impermeable and subject to a uniform heat flux. The upper open boundary has a prescribed, linearly varying, temperature distribution. The buoyancy-induced basic velocity profile is parallel and non-uniform. The linear stability of this basic solution is analysed numerically by solving the disturbance equations for oblique rolls arbitrarily oriented with respect to the basic velocity field. The onset conditions of thermal instability are governed by the Rayleigh number associated with the prescribed wall heat flux at the lower boundary, by the horizontal Rayleigh number associated with the imposed temperature gradient on the upper open boundary, and by the Gebhart number associated with the effect of viscous dissipation. The critical value of the Rayleigh number for the onset of the thermal instability is evaluated as a function of the horizontal Rayleigh number and of the Gebhart number. It is shown that the longitudinal rolls, having axis parallel to the basic velocity, are the most unstable in all the cases examined. Moreover, the imposed horizontal temperature gradient tends to stabilise the basic flow, while the viscous dissipation turns out to have a destabilising effect.     

Linear Stability and Convection in a Gravity Modulated Porous Layer Heated from Below: Transition from Synchronous to Subharmonic Solutions

The linear stability theory is used to investigate analytically the effects of gravity modulation on convection in a homogenous porous layer heated from below. The linear stability results are presented for both the synchronous and subharmonic solutions and the exact point for the transition from synchronous to subharmonic solutions is computed. It is also demonstrated that increasing the excitation frequency rapidly stabilizes the convection up to the transition point from synchronous to subharmonic convection. Beyond the transition point, the effect of increasing the frequency is to slowly destabilize the convection.     

Influence of Vertical Vibrations on the Stability of a Binary Mixture in a Horizontal Porous Layer Subjected to a Vertical Heat Flux

Macroscale three-dimensional modeling of fluid flow in a thin porous layer under unsaturated conditions is a challenging task. One major issue is that such layers do not satisfy the representative elementary volume length-scale requirement. Recently, a new approach, called reduced continua model (RCM), has been developed to describe multiphase fluid flow in a stack of thin porous layers. In that approach, flow equations are formulated in terms of thickness-averaged variables and properties. In this work, we have performed a set of experiments, where a wet \(260\hbox {-}\upmu \hbox {m}\)-thin porous layer was placed on top of a dry layer of the same material. We measured the change of average saturation with time using a single-sided low-field nuclear magnetic resonance device known as NMR-MOUSE. We have employed both RCM and the traditional Richards equation-based models to simulate our experimental results. We found that the traditional unsaturated flow model cannot simulate experimental results satisfactorily. Very close agreement was obtained by including the dynamic capillary term as postulated by Hassanizadeh and Gray in the traditional equations. The reduced continua model was found to be in good agreement with the experimental result without adding dynamic capillarity term. Moreover, the computational effort needed for RCM simulations was one order of magnitude less than that of traditional models.     

Free Convection Boundary Layer Flow from a Heated Upward Facing Horizontal Flat Plate Embedded in a Porous Medium Filled by a Nanofluid with Convective Boundary Condition

The steady laminar incompressible free convective flow of a nanofluid over a permeable upward facing horizontal plate located in porous medium taking into account the thermal convective boundary condition is studied numerically. The nanofluid model used involves the effect of Brownian motion and the thermophoresis. Using similarity transformations the continuity, the momentum, the energy, and the nanoparticle volume fraction equations are transformed into a set of coupled similarity equations, before being solved numerically, by an implicit finite difference numerical method. Our analysis reveals that for a true similarity solution, the convective heat transfer coefficient related with the hot fluid and the mass transfer velocity must be proportional to x ^−2/3, where x is the horizontal distance along the plate from the origin. Effects of the various parameters on the dimensionless longitudinal velocity, the temperature, the nanoparticle volume fraction, as well as on the rate of heat transfer and the rate of nanoparticle volume fraction have been presented graphically and discussed. It is found that Lewis number, the Brownian motion, and the convective heat transfer parameters increase the heat transfer rate whilst the thermophoresis decreases the heat transfer rate. It is also found that Lewis number and the convective heat transfer parameter enhance the nanoparticle volume fraction rate whilst the thermophoresis parameter decreases nanoparticle volume fraction rate. A very good agreement is found between numerical results of the present article for special case and published results. This close agreement supports the validity of our analysis and the accuracy of the numerical computations.     

对流边界层湍流特性的数值研究

采用大涡模拟方法研究了存在逆温层的情况下大气对流边界层的湍流特性。实际大气边界层中出现逆温层是较常见的，逆温层会导致大气边界层湍流结构的变化，从而影响大气的湍流扩散和输运特性。本文比较了不同逆温梯度的工况，着重分析了逆温层对边界层中热量逆梯度输运（counter gradient heat transportation，CGHT）的影响。计算结果表明：逆温梯度越大，对流边界层的发展越受到抑制；逆温层高度降低会影响整个对流边界层的温度抬升；逆温梯度越大，垂直速度方差越小；在逆温梯度较大的情况下，其逆梯度输运区域要略微低一些，初步分析认为是由于逆温层对热对流的抑制造成的；对于逆温层高度不同的情况，高度越低的逆温层对逆梯度输运的抑制作用更明显。