Transient free convection from a vertical plate subjected to a change in surface heat flux in porous media

In this paper we examine the general transient natural convection response arising due to a sudden change of the level of uniform flux dissipation rate from a vertical surface which is embedded in a porous medium. From an analytical investigation of the governing boundary-layer equations both a series solution which is valid at small values of the non-dimensional time and a solution which is valid at large times, when the transport of energy is steady, are derived. A numerical, transient formulation of the full unsteady boundary-layer equations is developed using an explicit finite-difference scheme. The numerical temperature profiles are observed to closely follow the small time solution initially and evolve along a curve which approaches the steady-state solution asymptotically. Results are presented to illustrate the occurrence of transients from both an increase and a decrease in the levels of existing energy inputs.     

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.     

Mixed convection boundary-layer flow over a vertical surface embedded in a porous medium

In this paper we have numerically investigated the existence and uniqueness of a vertically flowing fluid passed a model of a thin vertical fin in a saturated porous media. We have assumed the two-dimensional mixed convection from a fin, which is modelled as a fixed, semi-infinite vertical surface, embedded in a fluid-saturated porous media under the boundary-layer approximation. We have taken the temperature, in excess of the constant temperature in the ambient fluid on the fin, to vary as  , where is measured from the leading edge of the plate and λ is a fixed constant. The Rayleigh number is assumed to be large so that the boundary-layer approximation may be made and the fluid velocity at the edge of the boundary-layer is assumed to vary as . The problem then depends on two parameters, namely λ and , the ratio of the Rayleigh to Péclet numbers. It is found that when λ>0 (<0) there are (is) dual (unique) solution(s) when is grater than some negative values of (which depends on λ). When λ<0 there is a range of negative value of (which depends on λ) for which dual solutions exist and for both λ>0 and λ<0 there is a negative value of (which depends on λ) for which there is no solution. Finally, solutions for 0<1 and 1 have been obtained.     

Transient free convection about a horizontal circular cylinder in a porous medium

The method of matched asymptotic expansions is employed for investigating the growth of the free convection boundary-layer on a horizontal circular cylinder which is embedded in a porous medium. It is assumed that the Rayleigh number is large, but finite, and the time of investigation is short. It is shown that the solution contains terms that are absent from the solution based on the boundary-layer approximation and that vortices form at both sides of the cylinder. The development of the plume region near the top of the cylinder, as well as the local and average Nusselt numbers, are evaluated and presented in graphical form.     

Transient boundary-layer heat transfer from a flat plate subjected to a sudden change in heat flux

We examine the transient forced convection heat transfer from a fixed, semi-infinite, flat plate situated in a fluid which, at large distances, is moving with a constant velocity parallel to the plate. Both the fluid and the plate are initially at a constant temperature and the transients are initiated when the zero heat flux at the plate is suddenly changed to a constant value. The thermal boundary-layer equations are solved using numerical techniques to extend a series which is valid for small times and describe fully the development from the initial unsteady state solution (small times) to the ultimate steady state solution (large time).     

Surface activity and bubble motion

This paper reviews recent progress in the theories of the surface boundary conditions of adsorbed solutes in liquids, and of the effects of those solutes on the steady motion of a bubble or drop in the liquid. Both singular perturbation theory and numerical solutions have useful roles in this problem, and their relationship is explored. In addition, analytical solutions are given to two problems concerning a spherical bubble rising steadily at low Reynolds number in a viscous fluid. One of these is displacement of the internal vortex centre from its position in the absence of surface activity when there is a small stagnant cap of surfactant at the rear. The results agree with experimental data in the direction of that displacement but give only about half its amount. The other problem is the velocity perturbation all round the surface caused by a very dilute solution of a weak surfactant at high Péclet number. This compares quite well with the numerical solution for a Péclet number of 60, having relative errors of order (60)^–1/2 as would be expected.     

Operating regimes of a chemical reactor with longitudinal and transverse mixing in the case of large Péclet numbers

A two-dimensional model of a chemical reactor with longitudinal and transverse mixing is investigated in the case of large Péclet numbers calculated from the effective thermal conductivity in the transverse direction. For this model the existence of at least one steady-state regime has been demonstrated [1], sufficient criteria of its uniqueness have been determined, an asymptotic expansion of the solution has been constructed in the case of small Péclet numbers, and the critical ignition and quenching parameters have been found. In this paper the other limiting case of the model, in which heat is propagated in the transverse direction much more slowly than it is transported by the flow along the reactor (large Péclet numbers), is analyzed in detail. An asymptotic expansion of the solution which closely coincides with the data of numerical calculations is constructed. The critical quenching and ignition conditions of the process are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 120–127, January–February, 1987.     

Steady thermocapillary motion in a horizontal layer of liquid metal locally heated from above

The article considers stationary thermocapillary convection in a thin horizontal layer of fluid with Prandtl number Pr < 1 when it is being locally heated from above in conditions in which the curvature of the free surface is small. It is shown that the motion has a cellular structure. The size of the convective cell is determined from the solution to the spectral problem to which the integration of the free convection system of equations reduces. If the Maragoni (Péclet) number is sufficiently high, the length of the convective cell turns out to be large in comparison with the thickness of the layer. The convection picture is considered and an expression obtained for the velocity of the developing flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 146–152, November–December, 1984.     

Convective heat transfer and temperature stratification in a sphere completely filled with a liquid,with a given heat flux

A number of articles have been devoted to the theoretical and experimental investigation of natural convection in spherical vessels completely filled with a liquid [1–6]. Analytical solutions are known, obtained by the expansion of the sought function in series in powers of the Rayleigh number (see, for example, [1]), valid for very small values of this number. A numerical solution of the nonlinear Boussinesq equations can be used to obtain solutions with larger Rayleigh numbers, but the existing data for spherical regions [2, 3] embrace a relatively narrow range of Rayleigh numbers. The experimental data with a given heat flux, published in [4–6], were obtained with relatively large Rayleigh numbers (Ra*=10⁹?10¹¹) and Prandtl numbers (P= 3?1500). Data on the characteristics of convection in spherical vessels are still not very numerous and, in a number of cases, contradictory. This relates, in particular, to the boundaries of unsteady-state conditions. The present article, continuing [7–9], expounds a method and gives the results of a calculation of convection in a sphere with a thinwalled shell, in a range of Rayleigh and Fourier numbers embracing the principal conditions of unsteady-state laminar convection with a given heat flux.     

Closed streamline flows past small rotating particles: Heat transfer at high péclet numbers

A heat transfer problem is solved, first for an infinitely long heated cylinder and then for a small heated sphere, each freely suspended in a general linear flow at Reynolds numbers $\text{Re ? 1}$. Asymptotic solutions to the convection problem are developed for very large values of the Péclet number $\text{Pe}$, and expressions are obtained for the asymptotic Nusselt number for two-dimensional flows ranging from solid body rotation to hyperbolic flow. Since the objects in these cases are surrounded by a region of effectively isothermal closed streamlines, the asymptotic Nusselt number becomes independent of the Péclet number in the limit $\text{Pe → ∞}$.     

Mass transfer problem for particles,drops and bubbles in a shear flow

A numerical solution of the axisymmetric steady heat and mass transfer problem for spherical particles, drops and bubbles in a linear Stokes shear flow is obtained for the entire range of Péclet numbers. Simple approximate expressions for the average Sherwood number in good agreement with the results of the numerical calculations are proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–141, July–August, 1990.     

Stationary Transverse Diffusion in a Horizontal Porous Medium Boundary-Layer Flow with Concentration-Dependent Fluid Density and Viscosity

Stable transport of high-concentrated solute is considered in horizontal boundary-layer flows above a wall of constant concentration. Mixing is accomplished by advection and molecular diffusion only. The utilized boundary-layer approximation allows to investigate the exclusive influence of gravity on vertical diffusion. The hydrodynamic dispersion mechanism was disregarded in the present study which confines its applicabilty to flows with small molecular Péclet numbers. A linear variability of both the fluid's density and viscosity with changing concentration is taken into account as well as the complete set of mass-fraction based balance equations. Steady-state concentration and velocity distributions above the horizontal wall have been obtained using the series truncation method which recently had proven successful to solve the corresponding problem using the Boussinesq assumption. The impact of the latter on these distributions is discussed by what has been additionally-facilitated by the existence of an exact analytical solution for the simpler Boussinesq case. Whereas no density variability influence exists with use of the Boussinesq assumption the complete system of mass-fraction based equations predicts opposing effects of density and viscosity differences between oncoming and near-wall fluids on concentration distributions. Larger density differences narrow the transition zone between both fluids, larger viscosity differences widen it. Thus, a compensation of both effects can be observed for individual fluids and for certain regions of the flow field.     

Buoyancy-Opposed Darcy’s Flow in a Vertical Circular Duct with Uniform Wall Heat Flux: A Stability Analysis

An analytical and numerical study is presented to show that buoyancy-opposed mixed convection in a vertical porous duct with circular cross-section is unstable. The duct wall is assumed to be impermeable and subject to a uniform heat flux. A stationary and parallel Darcy’s flow with a non-uniform radial velocity profile is taken as a basic state. Stability to small-amplitude perturbations is investigated by adopting the method of normal modes. It is proved that buoyancy-opposed mixed convection is linearly unstable, for every value of the Darcy–Rayleigh number, associated with the wall heat flux, and for every mass flow rate parametrised by the Péclet number. Axially invariant perturbation modes and general three-dimensional modes are investigated. The stability analysis of the former modes is carried out analytically, while general three-dimensional modes are studied numerically. An asymptotic analytical solution is found, suitable for three-dimensional modes with sufficiently small wave number and/or Péclet number. The general conclusion is that the onset of instability selects the axially invariant modes. Among them, the radially invariant and azimuthally invariant mode turns out to be the most unstable for all possible buoyancy-opposed flows.     

Heat flow in a channel with discontinuity of the temperature on the wall

We study the temperature field in the flow of a viscous fluid in a circular tube when there is an abrupt change in the boundary condition for the temperature on the walls at a section of the channel. Following the classical studies [1, 2], this problem has often been considered (for example, in [3, 4, 5]) under different assumptions about the type of flow, the form of the boundary conditions, and the values of the Péclet number. The solutions hitherto obtained are frequently cumbersome and do not exhaust all situations of physical interest. In the present paper, we find the solution to the problem for the case of Poiseuille flow, boundary conditions of the first kind for the temperature, and arbitrary values of the Péclet number. We establish an expression that determines the Nusselt number at different sections of the channel. The results of calculations based on the obtained formulas are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Shidkosti i Gaza, No. 5, pp. 194–198, September–October, 1979.     

Asymptotic Analysis of the Joining of an Ice-Rock Body in Flow Through Porous Media

The problem of determining the equilibrium configuration of a plane, doubly connected ice-rock body formed about a system of two freezing columns traversing a flow through a porous medium is asymptotically analyzed in the limit of small Péclet numbers. Two terms of the asymptotic expansion are retained. It is shown that in this approximation the criterion of joining of the doubly connected body coincides with the criterion of non-disjoining of the simply connected body. However, the solution structure is such that taking the third asymptotic term into account can lead to a second solution when the ice-rock body is close to joining. This means that the size of the joining-disjoining hysteresis loop is of at least the second order in the Péclet number.     

Route to Chaos for Moderate Prandtl Number Convection in a Porous Layer Heated from Below

The route to chaos for moderate Prandtl number gravity driven convection in porous media is analysed by using Adomian's decomposition method which provides an accurate analytical solution in terms of infinite power series. The practical need to evaluate numerical values from the infinite power series, the consequent series truncation, and the practical procedure to accomplish this task, transform the otherwise analytical results into a computational solution achieved up to a desired but finite accuracy. The solution shows a transition to chaos via a period doubling sequence of bifurcations at a Rayleigh number value far beyond the critical value associated with the loss of stability of the convection steady solution. This result is extremely distinct from the sequence of events leading to chaos in low Prandtl number convection in porous media, where a sudden transition from steady convection to chaos associated with an homoclinic explosion occurs in the neighbourhood of the critical Rayleigh number (unless mentioned otherwise by 'the critical Rayleigh number' we mean the value associated with the loss of stability of the convection steady solution). In the present case of moderate Prandtl number convection the homoclinic explosion leads to a transition from steady convection to a period-2 periodic solution in the neighbourhood of the critical Rayleigh number. This occurs at a slightly sub-critical value of Rayleigh number via a transition associated with a period-1 limit cycle which seem to belong to the sub-critical Hopf bifurcation around the point where the convection steady solution looses its stability. The different regimes are analysed and periodic windows within the chaotic regime are identified. The significance of including a time derivative term in Darcy's equation when wave phenomena are being investigated becomes evident from the results.     

多孔介质中热对流的分叉机理研究

本文利用解析分析方法研究了数值模拟发现的多孔介质层中出现的对流分叉机理，指出控制方程中的Ｒａｙｌｅｉｇｈ数，是决定流动的特征参数。当Ｒａｙｌｅｉｇｈ数小于临界数值时，多孔介质内流动处于静止传热状态，并且这种状态是稳定的。如果Ｒａｙｌｅｉｇｈ数大于临界数值，非线性方程出现分叉解，文中指出，存在多个使平凡解失稳而分叉的临界Ｒａｙｌｅｉｇｈ数，当Ｒａｙｌｅｉｇｈ数由小到大经历这些临界数值时，其由平凡解发展起来的分叉解的流态，依次由单回流区转变为双回流区及三回流区。理论分析给出了分叉解和分叉解的振幅方程，阐明了分叉的机理，其结论和数值结果定性一致．     

Convective heat loss of water in a circular pipe cooled at constant rate

A numerical study is made of the heat loss by natural convection of water within a horizontal circular cylinder with wall temperature decreasing at a constant rate. The particular situation of water with maximum density at 4 °C is formulated in dimensionless relations based on a linear relationship between the water thermal expansion coefficient and the temperature. Such an approach leads to an exhaustive solution in terms of a non linear Rayleigh number. The link is also established with the standard situation where the hypothesis of a linear relationship between density and temperature is applicable. In particular it is shown that the quasi steady state results obtained for a standard situation become equilibrium curves to which the system tends with increasing difference between the temperature of the boundary and 4 °C. A complete numerical solution is obtained for non linear Rayleigh numbers ranging between 0 and 10⁷. Previous numerical and experimental results on the horizontal circular cylinder are also discussed and recast in terms of the present dimensional approach.     

Mass transfer to particle clusters and bubbles in a fluidized bed at low Reynolds numbers

The process of mass transfer to a particle cluster or bubble rising in a developed fluidized bed rapidly enough for a region of closed circulation of the fluidizing agent (cloud) to be formed is investigated in the Stokes approximation on the basis of a model of the steady-state motion of the fluid and solid phases near the cluster or bubble [1]. Within the cloud surroundinga local inhomogeneity of the fluidized bed intense mixing of the fluid phase takes place and the mass transfer between the cloud and the surrounding medium is determined by diffusion. The method of matched asymptotic expansions is used to obtain an analytic solution of the problem of the concentration field and the diffusion mass flux to the surface of the cloud at small and large values of the Péclet number. The latter is determined from the relative velocity of the cluster, the radius of the cloud, and the effective diffusion coefficient. In the limiting case of zero concentration of the solid phase within the cluster the solution obtained describes the mass transfer to a bubble in the fluidized bed. A comparison is made with the corresponding results previously obtained within the framework of a model of the solid phase as an inviscid fluid [2]. It is shown that the effect of viscosity on the mass transfer to the bubble is most important at large Péclet numbers, and that the correction to the total diffusion flux to the surface of the closed circulation zone due to viscosity effects may reach 40%.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 60–67, July–August, 1986.