Influence of Variable Permeability on Combined Convection Along a Nonisothermal Wedge in a Saturated Porous Medium

Mixed convection along a vertical nonisothermal wedge embedded in a fluid-saturated porous media incorporating the variation of permeability and thermal conductivity is studied. The surface temperature is assumed to vary as a power of the axial coordinate measured from the leading edge of the plate. A nonsimilar mixed convection parameter and a pseudo-similarity variable are introduced to cast the governing boundary layer equations into a system of dimensionless equations which are solved numerically using finite difference method. The entire mixed convection regime is covered by the single nonsimilarity parameter =[1+(Ra _x /Pe _x )^1/2]^–1 from pure forced convection (=1) to pure free convection (=0). The problem is solved using nonsimilarity solution for the case of variable wall temperature. Velocity and temperature profiles as well as local Nusselt number are presented. The wedge angle geometry parameter is ranged from 0 to 1.     

Nonsimilar Solutions for Mixed Convection in Non‐Newtonian Fluids Along a Vertical Plate in a Porous Medium

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in powerlaw type nonNewtonian fluids along a vertical plate with powerlaw wall temperature distribution. The mixed convection regime is divided into two regions, namely,the forced convection dominated regime and the free convection dominated regime. The two solutions are matched. Numerical results are presented for the details of the velocity and temperature fields. A discussion is provided for the effect of viscosity index on the surface heat transfer rate.     

Application of the Galerkin method to the solution of the problem of combined (forced and free) turbulent convection in a vertical circular channel in a uniform solid mass

We consider steady-state combined (forced and free) turbulent convection in a vertical circular channel in a uniform solid medium for the case in which a constant vertical temperature gradientis maintained in the solid mass, far from the channel. The velocity and temperature distributions are found, and the critical values of the Rayleigh number for axisymmetric and antisymmetric fluid motions are calculated. The problem is solved by the Galerkin method.Notation v⁽⁰⁾ forced convection velocity - v⁽¹⁾ free convection velocity - v velocity with combination of forced and free convection - v average velocity across channel section - T temperature with combined forced and free convection - T_w channel wall temperature - y distance from channel wall - y_* dimensionless distance from wall - r₀ channel radius - r distance from centerline - v_t turbulent viscosity - _t turbulent thermal diffusivity - P₀ averaged pressure corresponding to constant fluid ternperature - z coordinate along channel axis, directed upward - Q quantity of heat released by internal sources per unit fluid volume per unit time - fluid thermal conductivity (_e for the surrounding mass) - R Reynolds number - R^* Rayleigh number - P Prandtl number - G Grashof number - V_* dynamic viscosity     

Free convection around a vertical isothermically heated plate of finite length at large prandtl numbers

A free convection boundary layer arises because of the appearance of viscosity forces near a solid boundary. For high viscosity fluids the viscosity is significant over the whole flow region, and the thermal boundary layer which forms because of the restriction of heat diffusion from a heated wall by convection is characterized by the ratio between the coefficients of viscosity and thermal diffusivity, i.e., the Prandtl number. The divergence between the theoretical [1–4] and experimental data [5, 6] for the velocity profiles of free convective flow around a vertical surface at large Prandtl nunbers is due to an insufficiently clear distinction between the physical laws mentioned. In the present study the form of the velocity and temperature profiles is determined more accurately on the basis of an asymptotic analysis of the complete Navier-Stokes equations and energy equation with Prandtl number Pr and Grashof numbers of the order of unity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 161–165, September–October, 1984.     

The Effect of Time-Periodic Surface Temperature Oscillations on Free Convection from a Vertical Surface in a Porous Medium

In this paper, we consider the unsteady free convection boundary layer flow which is induced by time-periodic variations in the surface temperature of a vertical surface embedded in a porous medium. The basic steady flow is that of a power-law distribution where the surface temperature varies as the nth power of the distance from the leading edge. Small-amplitude time-periodic disturbances are added to this basic distribution. Both the low- and high-frequency limits are considered separately, and these are compared with a full numerical solution obtained by using the Keller-box method. Attention is restricted to the cases n1; when n=1, the flow is locally self-similar for any prescribed frequency of modulation.     

The effects of temperature-dependent fluid properties on free convective flow along a semi-infinite vertical plate by the presence of radiation

The effects of temperature-dependent density, viscosity and thermal conductivity on the free convective steady laminar boundary layer flow by the presence of radiation for large temperature differences, are studied. The fluid density and the thermal conductivity are assumed to vary linearly with temperature. The fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature. The usual Boussinesq approximation is neglected due to the large temperature difference between the plate and the fluid. The nonlinear boundary layer equations, governing the problem under consideration, are solved numerically by applying an efficient numerical technique based on the shooting method. The effects of the density/temperature parameter n, the thermal conductivity parameter , the viscosity/temperature parameter _r and the radiation parameter F are examined on the velocity and temperature fields as well as the coefficient of heat flux and the shearing stress at the plate.     

On the convective instability of a thermal boundary layer

When the surface temperature of a liquid is a harmonic function of time with a frequency, a temperature wave propagates into the liquid. The amplitude of this wave decreases exponentially with distance from the surface. The temperature oscillation is essentially concentrated in a layer of the order of (2/)^1/2, where x is the thermal conductivity of the liquid (thermal boundary layer). Depending on the phase, at certain positions below the surface the temperature gradient is directed downwards and if its magnitude is sufficiently large (the magnitude is a function of the amplitude and frequency of the surface oscillations) the liquid can become unstable with respect to the onset of convection. In that case the convective motion may spread beyond the initial unstable layer. For low frequencies the stability condition can be derived from the usual static Rayleigh criterion, on the basis of the Rayleigh number and the average temperature gradient of the unstable layer. This quasi-static approach, used by Sal'nikov [1], is appropriate to those cases in which the period of the temperature oscillations is much larger than the characteristic time of the perturbations. But when these times are of the same order, the problem must be analyzed in dynamic terms. The stability problem must then be formulated as a problem of parametricresonance excitation of velocity oscillations due to the action of a variable parameter-the temperature gradient.In an earlier work [2] we considered the problem of the stability of a horizontal layer of liquid with a periodically varying temperature gradient. It was assumed that the thickness of the layer was much smaller than the penetration depth of the thermal wave, so that the temperature gradient could be assumed to be independent of position. In the present work we consider the opposite case, in which the liquid layer is assumed to be much larger than the penetration depth, i. e., a thermal boundary layer can be defined. The temperature gradient at equilibrium, which is a parameter in the equations determining the onset of perturbations, is here a periodic function of time and a relatively complicated function of the depth coordinate z. The periodic oscillations are solved by the Fourier method; the equations for the amplitudes are solved by the approximate method of KarmanPohlhausen.The authors are grateful to L. G. Loitsyanskii for helpful criticism.     

Binary laminar boundary layer on a vertical surface in the presence of combined free and forced convection

Heat and mass transfer at a vertical surface is examined in the case of combined free and forced convection. The boundary layer equations, transformed to ordinary differential equations, contain a parameter that determines the effect of free convection on the forced motion. Criteria are offered for differentiating the free-convection, forced-convection, and combined regimes.Notation x, y coordinates - u, v velocity components - g acceleration of gravity - T temperature - kinematic viscosity - coefficient of thermal expansion - a thermal diffusivity - ₁ partial vapor density - D diffusion coefficient - W₂ mass velocity of air - independent variable - _w shear stress at wall - thermal conductivity - r latent heat of phase transition - , dimensionless temperature and partial vapor density - m^* the complex (m ₁–m _1w )/(1–m(_1w ) - c_p specific heat at constant pressure - G Grashof number - R Reynolds number - P Prandtl number - S Schmidt number     

Combined natural and forced convection in a laminar wall jet along a vertical plate with uniform surface heat flux

The steady state flow and heat transfer characteristics of the combined natural and forced convection in a two dimensional, laminar, incompressible wall jet over a vertical wall are obtained for constant wall heat flux boundary condition. The velocity and temperature distribution are assumed to be power series, where the zeroth term corresponds to that for a plane wall jet in the absence of buoyancy effects. Numerical results for the momentum and thermal series functions are presented for a Prandtl number of 0.73. Wall values of the momentum and thermal series functions are presented for Prandtl numbers ranging from 0.01 to 1000.Nomenclature Gr* modified Grashof number - k thermal conductivity - Nu Nusselt number - Pr Prandtl number - q _w heat flux at the wall - Re Reynolds number - T temperature - u velocity component in x-direction - v velocity component in y-direction - x co-ordinate along the plane wall - y co-ordinate normal to the wall - () gamma function - non-dimensional co-ordinate defined in (6) - non-dimensional temperature - dynamic viscosity - kinematic viscosity - non-dimensional co-ordinate defined in (6) - density - w values at the wall - values at large distances away from the wall     

Asymptotic theory for calculating heat fluxes near the corner of a body

A method is presented for calculating the distribution of the thermal fluxes, friction stresses, and pressure near the corner point of a body contour in whose vicinity the outer supersonic flow passes through an expansion wave. The method is based on a study of the asymptotic solutions of the Navier-Stokes equations as the Reynolds number R approaches infinity for the flow region in which the longitudinal gradients of the flow functions are large, invalidating conventional boundary layer theory. This problem was examined in part in [1], in which the distribution of the friction and pressure in a region with length on the order of a few thicknesses of the approaching boundary layer was obtained in the first approximation. The leading term of the expansion for the thermal flux to the surface of the body vanishes for a value of the Prandtl number equal to unity and for other values of the Prandtl number does not match directly with its value in the undisturbed boundary layer.The thermal-flux distribution is obtained for values of the Prandtl number approaching unity. For this purpose it was necessary to consider a more general double passage to the limit as 1 and 0 for a finite value of the parameter B=[(–1)/] [–ln ^1/4/]^1/4 characterizing the ratio of the effects of thermal conduction, viscous dissipation, and convection. The solution obtained previously [1] corresponds to the particular case B and therefore for actual values of R=10⁴–10⁶, ~ 0.7 overestimates considerably the effect of the dissipative term on heat transfer, although even in first approximation it describes the pressure distribution well and the friction distribution satisfactorily. For smooth matching of the solutions with the corresponding flow functions in the undisturbed boundary layer it was necessary to introduce a flow region with free interaction for the expansion flow. Equations and boundary conditions which describe the flow as a whole are presented. Examples are given of numerical calculations and comparison with experiment.     

Transient non-Darcy free convection between parallel vertical plates in a fluid saturated porous medium

Transient non-Darcy free convection between two parallel vertical plates in a fluid saturated porous medium is investigated using the generalized momentum equation proposed by Vafai and Tien. The effects of porous inertia and solid boundary are considered in addition to the Darcy flow resistance. Exact solutions are found for the asymptotic states at small and large times. The large time solutions reveal that the velocity profiles are rather sensitive to the Darcy number Da when Da<1. It has also been found that boundary friction alters the velocity distribution near the wall, considerably. Finite difference calculations have also been carried out to investigate the transient behaviour at the intermediate times in which no similarity solutions are possible. This analytical and numerical study reveals that the transient free convection between the parallel plates may well be described by matching the two distinct asymptotic solutions obtained at small and large times.Nomenclature C empirical constant for the Forchheimer term - f velocity function for the small time solution - F velocity function for the large time solution - g acceleration due to gravity - Gr* micro-scale Grashof number - H a half distance between two infinite plates - K permeability - Nu Nusselt number - Pr Prandtl number - t time - T temperature - u, v Darcian velocity components - x, y Cartesian coordinates - effective thermal diffusivity - coefficient of thermal expansion - porosity - dimensionless time - similarity variable - dimensionless temperature - viscosity - kinematic viscosity - density - the ratio of heat capacities     

Thermal convection at a melting benzene surface

Summary Heat transfer by thermal free convection at the surface of a sphere has been studied experimentally by melting a sphere of solid benzene in a large quantity of liquid benzene of homogeneous temperature. The influence of cold liquid produced by the melting process is taken into account to yield results that are representative for the pure effect of heat transfer without melting. In the general formula for heat transfer by thermal convection, =C(GrPr)^1/4, we found C=0.525.     

Compressibility effects in laminar free convection from a vertical cone

In this paper an analysis of the laminar compressible natural convection about a vertical cone is presented. The governing boundary layer equations are suitably transformed and studied in terms of a compressibility variable expansion. The numerical solutions for the first three terms have been obtained for the ratio of specific heats=1.4, and a Prandtl number Pr=0.72, as characteristics of a diatomic gas, and for a range of values of the wall-ambient temperature difference ofA=0.2 to 1000. The numerical results obtained allow an evaluation of the skin friction and heat transfer at the surface of the cone. It is found that the effect of the compressibility is to increase the heat transfer rate at the wall for small values of the wall-ambient temperature difference parameter and to decrease it for large values of this parameter. The skin friction, however, is always reduced.     

First Fundamental Axisymmetric Problem of Thermoelasticity for a Compressed Spheroid with a Concentric Spherical Cavity

An axisymmetric boundaryvalue problem of thermoelasticity for a compressed spheroid with a concentric spherical cavity is studied by the generalized Fourier method. The problem is reduced to an infinite system of linear algebraic equations with the Fredholm operator under the condition that the boundary surfaces are not crossed. Results of a numerical analysis of stresses in the case of loadfree boundary surfaces in the presence of a temperature field caused by a constant temperature distribution on the boundary surfaces are presented.     

Combined convection along a non-isothermal wedge in a porous medium

A boundary layer analysis has been presented for the combined convection along a vertical non-isothermal wedge embedded in a fluid-saturated porous medium. The transformed conservation laws are solved numerically for the case of variable surface temperature. Results are presented for the details of the velocity and temperature fields as well as the Nusselt number. The wedge angle geometry parameter m ranged from 0 to 1.     

Free Convection from a Vertical Plate in a Porous Medium Subjected to a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

An analysis is made of the transient free convection from a vertical flat plate which is embedded in a fluid-saturated porous medium. It is assumed that for time a steady state temperature or velocity has been obtained in the boundary-layer which occurs due to a uniform flux dissipation rate . Then at time the heat flux on the plate is suddenly changed to and maintained at this value for 0$$ " align="middle" border="0"> . An analytical solution has been obtained for the temperature/velocity field for small times in which the transport effects are confined within an inner layer adjacent to the plate. These effects cause a new steady boundary layer. A numerical solution of the full boundary-layer equations is then obtained for the whole transient from to the steady state, firstly by means of a step-by-step method and then by a matching technique. The transition between the two distinct solution methods is always observed to occur very near to the turning point of the plate surface temperature, a time at which the fluid temperature is close to its steady state profile. The solution obtained using the step-by-step method shows excellent agreement with the small time analytical solution. Results are presented to illustrate the occurrence of transients from both small and large increases and decreases in the levels of existing energy inputs.     

On the Temperature Slip Boundary Condition in a Mixed Convection Boundary-Layer Flow in a Porous Medium

A problem derived previously (Rohni et?al., Transp Porous Media 92:1?C14, 2012) for unsteady mixed convection flow in a porous medium involving a ??temperature slip?? boundary condition and fluid transfer through the boundary is considered. It is shown that the solution to this problem can be directly related to the solution of the corresponding problem for a prescribed surface temperature, involving a mixed convection parameter ??, an unsteadiness parameter A and transpiration parameter s. This latter problem is discussed in detail, particular attention being given to the steady analogue, A?=?0, allowing for fluid transfer through the surface, and to the unsteady problem, A?>?0, for an impermeable surface, s?=?0. Asymptotic results are obtained for large fluid transfer rates, ${s \gg 1}$ and ${s <0 , |s| \gg 1}$ and for large A. Particular attention is given to deriving asymptotic results for the critical points which determine the range of existence of solutions.     

Heat transfer in a thin layer of a viscous heavy noncompressible liquid under wavy flow conditions

The article describes a method for calculating the flow of heat through a wavy boundary separating a layer of liquid from a layer of gas, under the assumption that the viscosity and heat-transfer coefficients are constant, and that a constant temperature of the fixed wall and a constant temperature of the gas flow are given. A study is made of the equations of motion and thermal conductivity (without taking the dissipation energy into account) in the approximations of the theory of the boundary layer; the left-hand sides of these equations are replaced by their averaged values over the layer. These equations, after linearization, are used to determine the velocity and temperature distributions. The qualitative aspect of heat transfer in a thin layer of viscous liquid, under regular-wavy flow conditions, is examined. Particular attention is paid to the effect of the surface tension coefficient on the flow of heat through the interface.Notation x, y coordinates of a liquid particle - t time - v and u coordinates of the velocity vector of the liquid - p pressure in the liquid - c_v, , T,, andv heat capacity, thermal conductivity coefficient, temperature, density, and viscosity of the liquid, respectively - g acceleration due to gravity - surface-tension coefficient - c phase velocity of the waves at the interface - T_w wall temperature - h₀ thickness of the liquid layer - u₀ velocity of the liquid over the layer Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 147–151, July–August, 1970.     

Unsteady Thermo-Gravitational Convection Effects in a Side-Heated or Cooled Near-Critical Fluid

Unsteady thermo-gravitational convection of a near-critical fluid in an enclosed cavity, whose side wall temperature increases or decreases while the other boundaries are thermally insulated, is considered. An effective numerical method based on a two-scale pressure representation is used for solving the complete Navier-Stokes equations with a Van-der-Waals equation of state. For the neighborhood of the critical point, a transformation of the similarity parameters, which allows the introduction of effective values of these parameters, is found. The characteristic times of rapid temperature equalization due to adiabatic compression (piston effect), heat conduction, and thermo-gravitational convection are compared. The reasons why, in an unsteady convective jet, the temperature of the near-critical fluid is higher than the fixed side-wall temperature are analyzed.