Instability of the equilibrium state of a liquid with a free surface in the presence of volume heat sources

The problem of the convection of a weakly compressible fluid is considered. In the free convection equations a heat source function is taken into account. The stability of the equilibrium state of a horizontal layer relative to small perturbations is studied using the linearization method. On the basis of numerical calculations it is shown that the mechanical equilibrium state of the fluid is unstable. The neutral curves are plotted and the critical Rayleigh numbers are found. In the calculations values of the physical parameters typical of Lake Baikal were used.     

Combined free and forced convection in a porous medium between two vertical walls with viscous dissipation

Viscous dissipation effects in the problem of a fully-developed combined free and forced convection flow between two symmetrically and asymmetrically heated vertical parallel walls filled with a porous medium is analyzed. The equation of motion contains the modified Rayleigh number for a porous medium and the small-order viscous dissipation parameter. Particular attention is given to the solutions near the critical Rayleigh numbers at which infinite flow rates are predicted. Information concerning the multiplicity of solutions at critical Rayleigh numbers is also deduced from perturbation solutions of the governing equation.     

The Prandtl number effect near the onset of Bénard convection in a porous medium

This note focuses on Kladias and Prasad's claim that the critical Rayleigh number for the onset of Bénard convection in an infinite horizontal porous layer increases as the Prandtl number decreases, and argues that the critical Rayleigh number (Ra_c) depends only on the Darcy number (Da), as linear stability analysis indicates. The two-dimensional steady-convection problem is then solved numerically to document the convection heat transfer effect of the Rayleigh number, Darcy number, Prandtl number, and porosity. The note concludes with an empirical correlation for the overall Nusselt number, which shows the effect of Prandtl number at above-critical Rayleigh numbers. The correlation is consistent with the corresponding correlation known for Bénard convection in a pure fluid.     

Parametric excitation of Rayleigh convection in the presence of a variable heat flux on the free surface

The onset of Rayleigh convection in a semi-infinite fluid layer is investigated for a heat flux harmonically modulated along the normal to the surface of the fluid. The problem of the evolution of the velocity and temperature perturbations is solved numerically by means of a finite-difference method. The stability limits and the characteristics of the critical perturbations are determined as functions of the Prandtl numbers. The behavior of the critical Rayleigh number is studied for finite layer depths.Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 40–44, November–December, 1996.     

Patterned ground formation and convection in porous media with a phase change

The influence of a water-ice phase change on the onset of natural convection is examined for a saturated porous layer overlying a frozen region. Darcy's law is used and a parabolic equation of state is assumed for water. From a linear instability analysis we obtain predictions for the onset of convection in the melted region and the corresponding criticial wavenumber. The critical numbers are calculated by employing finite differences and solving the associated generalized eigenvalue problem. We study the effect varying thermal conductivities and boundary conditions have on these predictions. The analysis is applied to the formation of patterned ground, a geophysical phenomenon of stone borders forming regular hexagonal patterns. The theoretical model for patterned ground is based on natural convection in saturated soil below which is a cold permafrost layer.     

Free-convection Flow of a Binary Mixture in a Thin Porous Ring

The problem of natural convection of a binary mixture in a thin porous ring is considered. In the simplified formulation steady-state solutions of the problem are obtained. The stability of these solutions is investigated and a stability map is plotted in the plane of the Rayleigh numbers calculated from the temperature and concentration. It is shown that an auto-oscillation convection regime is established in the ring under certain conditions. It is also found that there is a region of variation of the seepage and diffusion-seepage Rayleigh numbers in which three steady-state solutions are stable.     

Numerical study of convection of a liquid heated from below

The equilibrium of a liquid heated from below is stable only for small values of the vertical temperature gradient. With increase of the temperature gradient a critical equilibrium situation occurs, as a result of which convection develops. If the liquid fills a closed cavity, then there is a discrete sequence of critical temperature gradients (Rayleigh numbers) for which the equilibrium loses stability with respect to small characteristic disturbances. This sequence of critical gradients and motions may be found from the solution of the linear problem of equilibrium stability relative to small disturbances. If the temperature gradient exceeds the lower critical value, then (for steady-state heating conditions) there is established in the liquid a steady convective motion of a definite amplitude which depends on the magnitude of the temperature gradient. Naturally, the amplitude of the steady convective motion cannot be determined from linear stability theory; to find this amplitude we must solve the problem of convection with heating from below in the nonlinear formulation. A nonlinear study of the steady motion of a liquid in a closed cavity with heating from below was made in [1]. In that study it was shown that for Rayleigh numbers R which are less than the lower critical value R_c steady-state motions of the liquid are not possible. With R>R_c a steady convection arises, whose amplitude near the threshold is small and proportional to (R–R_c)^1/2 (the so-called soft instability)-this is in complete agreement with the results of the phenom-enological theory of Landau [2, 3].Primarily, various versions of the method of expansion in powers of the amplitude [4–8] have been used, and, consequently, the results obtained in those studies are valid only for values of R which are close to R_c, i. e., near the convection threshold.It is apparent that the study of developed convective motion far from the threshold can be carried out only numerically, with the use of digital computers. In [9, 10] the numerical methods have been successfully used for the study of developed convection in an infinite plane horizontal liquid layer.The present paper undertakes the numerical study of plane convective motions of a liquid in a closed cavity of square section. The complete nonlinear system of convection equations is solved by the method of finite differences on a digital computer for various values of the Rayleigh number, the maximal value exceeding by a factor of 40 the minimal critical value R_c. The numerical solution permits following the development of the steady motion which arises with R>R_c in the course of increase of the Rayleigh number and permits study of the oscillatory motions which occur at some value of the parameter R. The heat transfer through the cavity is studied. The corresponding linear problem on equilibrium stability is solved approximately by the Galerkin method.     

Effect of Surface Corrugation on Convection in a Three-Dimensional Finite Box of Fluid Saturated Porous Material

The problem of finite amplitude thermal convection in a three-dimensional finite box of fluid saturated porous material is investigated, when the lower boundary of the fluid is corrugated. The nonlinear problem of three-dimensional convection in the box for the values of the Rayleigh number close to the classical critical value and for small values of the amplitude of the corrugations is solved by a perturbation technique. The preferred mode of convection is determined by stability analysis. In the absence of corrugation three-dimensional modes of convection can be either stable or unstable depending on the values of the aspect ratios of the box, while two-dimensional rolls are always stable, provided that the box aspect ratios allow the existence of such modes of convection. In the presence of boundary corrugation with the appropriate form, different three-dimensional or two-dimensional modes of corrugation can be stable or unstable. For a rough boundary with local roughness sites, the location, size, and number of the roughness elements plus the wave numbers of the convection modes and the box aspect ratios can all play a role leading to either stable or unstable particular three- or two-dimensional flow patterns. For a wavy boundary, resonant wave-vector excitation can lead to the preference of stable two- or three-dimensional flow patterns whose wave vectors are in a subset of those due to the wavy boundary, while nonresonant wave-vector excitation can lead to the preference of stable flow patterns whose wave vectors are not generally in a subset of those due to the wavy boundary. Heat transported by convection can either be enhanced or be reduced by certain proper forms of the corrugations and by appropriate values of the box aspect ratios. Due to the surface corrugation highly subcritical modes of convection are stable, while highly supercritical modes of convection are unstable. Received 24 July 1998 and accepted 11 April 1999     

Integral equation formulation for buoyancy-driven convection problems

An integral equation formulation for buoyancy-driven convection problems is developed and illustrated. Buoyancy-driven convection in a bounded cylindrical geometry with a free surface is studied for a range of aspect ratios and Nusselt numbers. The critical Rayleigh number, the nature of the cellular motion, and the heat transfer enhancement are computed using linear theory. Green's functions are used to convert the linear problem into linear Fredholm integral equations. Theorems are proved which establish the properties of the eigenvalues and eigenfunctions of the linear integral operator which appears in these equations.     

Stability of a layer of viscoelastic fluid heated from below

The polymerization of methyl methacrylate is accompanied by liberation of heat; this results in overheating of the reaction mass during production of plastics. The temperature distribution in the polymerizing layer is complicated by convection, which disrupts the natural temperature field. Thus, in addition to the stress along the sheet, local internal stresses appear that show up in operation of the product. Product quality and intensification of the polymerization process depend on the critical temperature gradient, which determines the stability threshold of the layer of polymerizing methyl methacrylate. The Rayleigh-Jeffrey problem is considered for a weak viscoelastic fluid described by an integral rheological constitutive relationship. The critical Rayleigh numbers are determined for stationary and oscillatory instabilities with free and ideally heat-conducting rigid boundaries.     

Stability and slightly supercritical oscillatory regimes of natural convection in a 8:1 cavity: solution of the benchmark problem by a global Galerkin method

The global Galerkin method is applied to the benchmark problem that considers an oscillatory regime of convection of air in a tall two‐dimensional rectangular cavity. The three most unstable modes of the linearized system of the Boussinesq equations are studied. The converged values of the critical Rayleigh numbers together with the corresponding oscillation frequencies are calculated for each mode. The oscillatory flow regimes corresponding to each of the three modes are approximated asymptotically. No direct time integration is applied. Good agreement with the previously published results obtained by solution of the time‐dependent Boussinesq equations is reported. Copyright © 2004 John Wiley & Sons, Ltd.     

Küppers-Lortz instability in rotating Rayleigh-Bénard convection in a porous medium

The Darcy-Lapwood-Brinkman model with the Boussinesq approximation is used to study Küppers-Lortz (KL) instability in the nonlinear regime of rotating Rayleigh-Bénard convection in a sparsely packed porous medium near the onset of stationary convection. The threshold Taylor numbers and critical angles for the onset of KL instability are obtained for different values of Λ, M for finite Prandtl numbers (1.5≤Pr≤100). Heat transfer is studied from Nusselt number at the onset of stationary convection.     

Marangoni convection in a cylinder of finite size

This paper considers a liquid in a finite-size cylinder in which Marangoni instability occurs. The upper boundary of the liquid is free and deformable. The problem of the occurrence of convection in a cylindrical container is solved using the method of separation of variables. A homogeneous differential equation of the sixth order with constant coefficients and complex boundary conditions is obtained. An analytical expression for critical Marangoni numbers is derived for the case of monotonic perturbations. The case is considered where the liquid in the cylinder is weightless.     

On the stopping of thermal convection in viscoplastic liquid

The thermal convection in a square cavity filled in with a viscoplastic liquid is considered as a model example to illustrate the mechanism of convection termination. It is shown that at low Rayleigh numbers, the stopping of convection corresponds to a limit point in the parameter space. Using this observation, we propose a heuristic numerical approach to calculate the critical Rayleigh numbers.     

Stability of the equilibrium state in a convection model with nonlinear temperature and pressure dependences of density

The convection of a heat-conducting viscous liquid is considered. It is assumed that the liquid density depends quadratically on the temperature and pressure. The instability of the equilibrium state of a free-boundary horizontal layer with respect to small perturbations is studied using a linearization method. It is found that the state of mechanical equilibrium is unstable. Neutral curves are constructed and the critical Rayleigh numbers are found. The results are compared with the well-known solution of the same problem for the limiting case where the density is a quadratic function of temperature and does not depend on pressure. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 66–74, March–April, 2007.     

Effect of the adsorption-desorption process intensity on solutal convection near a drop in a horizontal channel

The interaction between gravity convection and Marangoni convection in a horizontal rectangular channel filled with a liquid containing a surfactant and a drop of another liquid is numerically investigated. For large Schmidt numbers the occurring oscillatory regime of solutal convection is analyzed. In the model with a surface phase the effect of the adsorption and desorption processes on the convective flow structure is determined. The corresponding initial and boundary value problem is solved using a difference method.     

Modelling of the free surface with heat and mass transfer considerations on a liquid filling a porous slab heated from its sides

The position of the free surface is calculated numerically for a porous slab which is partly filled with a liquid and differentially heated from its sides. A coordinate transformation is used to transform the original problem from a physical coordinate system to a non‐orthogonal system where the free surface becomes a fixed straightline. The transformed problem is then solved using a finite difference method. Results are obtained for Rayleigh numbers up to 1000. The Nusselt numbers increase slightly with medium Rayleigh numbers (convection‐dominated region) as expected. Since at low Ra the conduction is dominant and at high Ra radiation is dominant. Hadizadeh and Tien (Int. J. Heat Mass Transfer 2004; 17 (6):799–804) studied the forced convection on the surface of porous layer. In that paper they dealt with in detail the boundary regime of liquid in the channel and modelled the flow and heat transfer. Copyright © 2007 John Wiley & Sons, Ltd.     

Application of a non-linear local analysis method for the problem of mixed convection instability

We consider the problem of laminar mixed convection flow between parallel, vertical and uniformly heated plates where the governing dimensionless parameters are the Prandtl, Rayleigh and Reynolds numbers. Using the method based on the centre manifold theorem which was derived from the general theory of dynamical systems, we reduce a three-dimensional simplified model of ordinary differential amplitude equations emanating from the original Navier-Stokes system of the problem in the vicinity of a trivial stationary solution. We have found that when the forcing parameter, the Rayleigh number, increases beyond the critical value Ra_s, the stationary solution is a pitchfork bifurcation point of the system.     

Linear and nonlinear stability analysis of binary Maxwell fluid convection in a porous medium

The stability analysis of the quiescent state in a Maxwell fluid-saturated densely packed porous medium subject to vertical concentration and temperature gradients is presented. A single phase model with local thermal equilibrium between the porous matrix and the Maxwell fluid is assumed. The critical Darcy–Rayleigh numbers and the corresponding wave numbers for the onset of stationary and oscillatory convection are determined. A Lorenz like system is obtained for weakly nonlinear stability analysis.     

Effect of Thermal Modulation on the Onset of Convection in a Porous Medium Layer Saturated by a Nanofluid

The effect of time-periodic temperature modulation at the onset of convection in a Boussinesq porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion. Three types of boundary temperature modulations are considered namely, symmetric, asymmetric, and only the lower wall temperature is modulated while the upper wall is held at constant temperature. The perturbation method is applied for computing the critical Rayleigh and wave numbers for small amplitude temperature modulation. The shift in the critical Rayleigh number is calculated as a function of frequency of modulation, concentration Rayleigh number, porosity, Lewis number, and thermal capacity ratio. It has been shown that it is possible to advance or delay the onset of convection by time-periodic modulation of the wall temperature. The nanofluid is found to have more stabilizing effect when compared to regular fluid. Low frequency is destabilizing, while high frequency is always stabilizing for symmetric modulation. Asymmetric modulation and only lower wall temperature modulation is stabilizing for all frequencies when concentration Rayleigh number is greater than one.