Mixed convection boundary-layer on a vertical cylinder embedded in a saturated porous medium

Summary The mixed convection boundary layer on a vertical circular cylinder embedded in a saturated porous medium is considered. It is found that the flow depends on the parameter =R _a /P _e whereR _a andP _e are the Rayleigh number and Peclet number respectively. gives the ratio of the velocity scale for free convection to that for the forced convection. When is small the solution is, to a first approximation, obtained by a known heat conduction problem. The flow near the leading edge is considered and it is shown that a solution is possible only for ₀, ₀–1.354, and that a stable finite-difference solution away from the leading edge can be obtained only if –1; with <–1 there is a region of reversed flow near the cylinder. The finite-difference scheme is unable to give a satisfactory solution at very large distances from the leading edge, and to overcome this difficulty a simple approximate solution is developed. This solution shows that at large distances along the cyclinder, forced convection eventually becomes the dominant mechanism for heat transfer. This is also confirmed by an asymptotic solution of the full boundarylayer problem.Nomenclature a radius of cylinder - g acceleration of gravity - K permeability of the porous medium - N _u non-dimensional Nusselt number - r radial coordinate - non-dimensionalr=r/a - R _a Rayleigh number=(g T)Ka/ - P _e Peclet number=U ₀ a/ - T temperature - T _w temperature of the cylinder (constant) - T ₀ temperature of the ambient fluid (constant) - T temperature difference=T _w –T ₀ - u Darcy's law velocity in thex direction - U ₀ velocity of the outer flow - v Darcy's law velocity in ther-direction - x coordinate measuring distance along the cylinder - X non-dimensionalx,=x(aP _e )^–1 - equivalent thermal diffusivity - coefficient of thermal expansion - ratio of free to forced convection=R _a /P _e - viscosity of the convective fluid - density of the ambient fluid - non-dimensional temperature - stream function With 2 Figures     

Free convection and radiation characteristics for a vertical plate embedded in a porous medium

Steady two‐dimensional free convection flow due to combined effect of radiation and convection through a porous medium bounded by a vertical infinite plate is considered. The behaviour of Darcy and non‐Darcy flow is investigated. The flow of water through different porous media under different environmental conditions is discussed. Effect of four non‐dimensional parameters, i.e. Prandtl number (Pr), modified Grashof number (G), permeability parameter (K) and radiation parameter (N) has been studied. Effect of these parameters on Nusselt number is analysed. Copyright © 2005 John Wiley & Sons, Ltd.     

Free convection flow past a vertical flat plate embedded in a saturated porous medium

A numerical solution for the free convection flow past a vertical semi-infinite flat plate embedded in a highly saturated porous medium by allowing the plate to have a non-uniform temperature or a non-uniform heat flux distributions has been developed. Both local heat transfer rate and excess surface temperature as a function of the distance along the plate are tabulated for a few cases of prescribed wall temperature and heat flux distributions. Such tabulations serve as a reference against which other approximate solutions can be compared in the future.     

Transient free convection about a vertical flat plate embedded in a porous medium

The problems of transient free convection in a porous medium adjacent to a vertical semi-infinite flat plate with a step increase in wall temperature and surface heat flux are considered in this paper. By assuming a temperature profile in each case, the governing equation for the boundary layer thickness is obtained by an integral method. These governing equations are first-order partial differential equations of the hyperbolic type that can be solved exactly by the method of characteristics and approximately by the method of integral relations. The results based on the method of characteristics clearly indicate that during the initial stage when the leading edge effect is not being felt, heat is transferred as if by transient 1-dimensional heat conduction. At a later time, depending on the vertical location, the heat transfer characteristics change from transient 1-dimensional heat conduction to steady 2-dimensional convection. The thickness of the boundary layer is shown to be increasing with time until it reaches steady state where its value remains constant thereafter. The growth rate of the boundary layer thickness exhibits a discontinuity at the end of the transient period and the beginning of the steady state period. On the other hand, the results based on the method of integral relations show that the boundary layer thickness grows continuously with time and approaches the steady state value asymptotically; the growth rate of the boundary layer thickness decreases from a finite value to zero continuously as the steady state is approached. Except between the end of the transient period and the beginning of the steady state period, the results based on the method of integral relations are in good agreement with those based on the method of characteristics.     

Free convection heat and mass transfer along a vertical surface in a porous medium

Summary This study deals with free convection heat and mass transfer from a vertical plate embedded in a fluid saturated porous medium with constant wall temperature and concentration. The temperature and concentration variations across the boundary layer produce a buoyancy effect which gives rise to flow field. Integral method of Von-Karman type is applied to obtain the analytical solution of this fundamental problem.     

Non-Darcy mixed convection flow with thermal dispersion on a vertical cylinder in a saturated porous medium

Summary The non-Darcy mixed convection flow on a vertical cylinder embedded in a saturated porous medium has been studied taking into account the effect of thermal dispersion. Both forced flow and buoyancy force dominated cases with constant wall temperature condition have been considered. The governing partial differential equations have been solved numerically using the Keller box method. The results are presented for the buoyancy parameter which cover the entire regime of mixed convection flow ranging from pure forced convection to pure free convection. The effect of thermal dispersion is found to be more pronounced on the heat transfer than on the skin friction and it enhances the heat transfer but reduces the skin friction.     

Free convection from a vertical cooling fibre

We study the free-convective boundary-layer flow that is induced when a slender circular cylinder emerges from an orifice and moves vertically downwards. We demonstrate, by numerical solution of the equations, that the boundary-layer solution develops a singularity at a finite point, where the limiting form of solution is as predicted by Kuiken [3] for an analogous two-dimensional flow.     

Natural convection on a thin vertical cylinder moving in a high-porosity ambient medium

An analysis has been performed to study the natural convection flow over a thin vertical cylinder which is moving with a constant velocity in a non-Darcy high-porosity ambient medium. Both constant wall temperature and constant heat flux conditions have been considered. The coupled non-linear parabolic partial differential equations have been solved numerically by using an implicit finite-difference scheme. The heat transfer is found to be significantly affected by the inertia and porosity parameters, and the Prandtl number, whereas the skin friction is weakly affected. The heat transfer for the constant heat flux case is more than that of the constant wall temperature case and this difference increases with the Prandtl number. The heat transfer increases with the buoyancy force, but the skin friction is slightly reduced.     

Unconditional nonlinear stability for convection in a porous medium with vertical throughflow

Summary Linear and nonlinear stability analyses of vertical throughflow in a fluid saturated porous layer, which is modelled using a cubic Forchheimer model, are studied. To ensure unconditional nonlinear results are obtainable, and to avoid the loss of key terms, a weighted functional is used in the energy analysis. The linear instability and nonlinear stability thresholds show considerable agreement when the vertical throughflow is small, although there is substantial deterioration of this agreement as the vertical throughflow increases.     

Mixed convection flow in a porous medium bounded by two vertical walls

The aim of the investigation is to study the fully developed mixed convection flow in a porous medium bounded by two vertical walls having a linear axial temperature variation. Solution of the governing second order simultaneous differential equations, derived by non-dimensionalising the momentum and energy equations, has been obtained by converting them into a fourth order differential equation. It has been found that the fourth order differential equation has five different solutions depending upon the values of the Darcy number, Rayleigh number and ratio of the effective viscosity of the porous domain to the viscosity of the fluid. Graphical representation of the analytical solution shows that the flow is of parabolic type only for higher Darcy number whereas there is a reverse type of motion for lower Darcy numbers. The effect of the ratio of the effective viscosity of the porous matrix to the viscosity of the fluid is to decrease the velocity field.     

Free convection in h horizontal porous layer with a partly heated or cooled wall

The natural convective flow in a fluid-saturated porous medium is considered for an infinite horizontal channel with the bottom wall being partially heated or cooled. The flow and heat transfer are analysed for a range of values of the two non-dimensional parameters which define the problem, namely the Rayleigh number Ra and aspect ratio . Numerical solutions are obtained for = 1 and = 0.1 for both heated and cooled cases and for a range of values of Ra. In the heated case, the nature of the flow is seen to change from unicellular for smaller values of Ra to multicellular for large Ra, with the value of Ra at this changeover being decreased as is decreased. Also, for this case, a range of values of Ra is found over which both unicellular and multicellular flows are possible. For the cooled case, a boundary layer is seen to develop on the bottom wall as Ra is increased for both the values of taken. Finally, a solution for 1 is obtained and is compared with the numerical solutions for = 0.1.     

Surface wave propagation in a fluid-saturated incompressible porous medium

A study of surface wave propagation in a fluid-saturated incompressible porous half-space lying under a uniform layer of liquid is presented. The dispersion relation connecting the phase velocity with wave number is derived. The variation of phase velocity and attenuation coefficients with wave number is presented graphically and discussed. As a particular case, the propagation of Rayleigh type surface waves at the free surface of an incompressible porous half-space is also deduced and discussed.     

Free convection from a torsionally oscillating vertical plane

Free convective flow of a viscous incompressible fluid from a uniformly heated vertical plane lamina, undergoing small-amplitude sinosoidal torsional oscillations, is investigated. The non-axisymmetric fluid motion consists of the primary Rosenblat flow and the secondary buoyancyinduced cross-flow. Numerical-analytical and asymptotic solutions of the energy equation, covering the whole range of the values of the Prandtl number of the fluid, are derived for their subsequent use in the analysis of the unsteady cross-flow. The mean cross-flow is found to dominate the steady components of the primary flow.     

Analysis of steady free convection from an axisymmetric heat-generating body embedded in a semi-infinite porous medium

An analysis is presented for the steady state free convective flow and heat transfer from an axisymmetric heat-generating body that is embedded in a fluid-saturated, semi-infinite, porous medium. The porous medium is assumed to be rigid, homogeneous and isotropic, and be in thermal equilibrium with the fluid. The fluid is assumed to be incompressible, with the density changes contributing only towards the buoyancy forces via the Boussinesq approximation. The governing equations for the fluid consist of the equation of continuity, Darcy's law and the equation of energy. After introducing the stream function concept, the equations governing the stream function and pressure are derived. Using the non-dimensional variables, the non-dimensional equations governing the non-dimensional forms of the temperature, stream function and pressure are dervied and the appropriate boundary conditions are stated. The mathematical formulation contains two parameters; D, the non-dimensional depth of the body from the surface of the porous medium, and a product Raθ_s of Rayleigh number (Ra) and the non-dimensional surface temperature of the body (θ_s). The Galerkin finite element method, with linear, isoparametric, quadrilateral elements, is used to reduce the mathematical formulation into a set of algebraic equations. The expressions to calculate the non-dimensional surface temperature and Nusselt number of the body, and the non-dimensional velocity of the fluid, are derived. A computer code has been developed to solve the algebraic equations, using Gauss elimination procedure, in a banded matrix form. The computer code, in addition to the non-dimensional temperature, stream function and pressure, calculates the isothermal lines, non-dimensional surface temperature of the body, Nusselt number of the body, velocity field and isobars. To demonstrate the application of the code, a spherical heat-generating body is considered as an example. Numerical results are obtained for D = 3 and 6, and Raθ_s = 0.001, 0.1, 1 and 5, and presented.     

Mixed convection with viscous dissipation in a vertical channel filled with a porous medium

Summary The fully developed laminar mixed convection flow in a vertical plane parallel channel filled with a porous medium and subject to isoflux ÷ isothermal wall conditions is investigated assuming that (i) the Darcy law and the Boussinesq approximation hold, (ii) the effect of viscous dissipation is significant, and (iii) the average flow velocity U _m (as an experimentally accessible quantity) is prescribed. It is shown that under these conditions both upward (U _m > 0) and downward (U _m < 0) laminar flow solutions may exist as long as U _m does not exceed a maximum value U _{m, max}. The velocity field can either be unidirectional or bidirectional. Moreover, bidirectional flow configurations are possible also for U _m = 0. A remarkable feature of the problem is that for U _m < U _{m, max} even two solution branches (dual solutions) exist, which merge when U _m approaches its maximum value U _{m, max}. The mechanical and thermal characteristics of the flow configurations associated with the dual solutions are investigated in the paper analytically and numerically in detail.     

Free convection on a heated vertical plate: The solution for small Prandtl number

The solution of the equations for the free-convection boundary-layer flow on a vertical plate with a prescribed power-law heating is considered for small values of the Prandtl number σ. It is shown that the boundary layer divides up into two regions. There is a thin inner region, of thickness O{ie273-1}, in which, to leading order, the temperature is a constant, but which is not determined from the inner solution. This gives rise to a large temperature on the plate of O{ie273-2}. This inner region drives a flow in a much thicker inviscid outer region, of thickness of O{ie273-3}. At the outer edge of this outer region the ambient conditions are attained, and it is the matching between the two regions which determines the plate temperature.     

Mixed convection boundary layer flow on a vertical surface in a saturated porous medium

Summary The flow of a uniform stream past an impermeable vertical surface embedded in a saturated porous medium and which is supplying heat to the porous medium at a constant rate is considered. The cases when the flow and the buoyancy forces are in the same direction and when they are in opposite direction are discussed. In the former case, the flow develops from mainly forced convection near the leading edge to mainly free convection far downstream. Series solutions are derived in both cases and a numerical solution of the equations is used to describe the flow in the intermediate region. In the latter case, the numerical solution indicates that the flow separates downstream of the leading edge and the nature of the solution near this separation point is discussed.