Der Einflu\ der freien Konvektion in laminaren und turbulenten Grenzschichtströmungen mit Stoffaustausch

The paper presents numerical solutions of the boundary layer equations for forced convection flows in which buoyancy effects caused by temperature and concentration gradients are important. Critical parameters for the onset of the free convection effects and for boundary layer separation were determined when the forced and free convection act in opposite directions. Results are given for laminar boundary layers and boundary layers in the transition regime with mass transfer.     

Gravity-driven laminar film flow of power-law fluids along vertical walls

A theoretical analysis is presented which brings steady laminar film flow of power-law fluids within the framework of classical boundary layer theory. The upper part of the film, which consists of a developing viscous boundary layer and an external inviscid freestream, is treated separately from the viscous dominated part of the flow, thereby taking advantage of the distinguishing features of each flow region. It is demonstrated that the film boundary layer developing along a vertical wall can be described by a generalized Falkner-Skan type equation originally developed for wedge flow. An exact similarity solution for the velocity field in the film boundary layer is thus made available.Downstream of the boundary layer flow regime the fluid flow is completely dominated by the action of viscous shear, and fairly accurate solutions are obtained by the Von Karman integral method approach. A new form of the velocity profile is assumed, which reduces to the exact analytic solution for the fully-developed film. By matching the downstream integral method solution to the upstream generalized Falkner-Skan similarity solution, accurate estimates for the hydrodynamic entrance length are obtained. It is also shown that the flow development in the upstream region predicted by the approximate integral method closely corresponds to the exact similarity solution for that flow regime. An analytical solution of the resulting integral equation for the Newtonian case is compared with previously published results.     

Mixed convection in non-Newtonian fluids along a vertical plate with variable surface heat flux in a porous medium

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in power-law type non-Newtonian fluids along a vertical plate with power-law surface heat flux 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. Received on 13 October 1998     

Comprehensive correlations for laminar mixed convection on vertical and horizontal flat plates

This paper presents comprehensive correlation equations of the local Nusselt numbers and surface shear stresses for laminar forced convection, natural convection, and mixed convection on vertical and horizontal flat plates which are maintained with uniform wall temperature or uniform surface heat flux. The correlation for pure forced convection and pure natural convection are very accurate for any Prandtl number between 0.001 and infinity. The correlations for mixed convection coincide very well with the numerical results over the entire regimes of mixed convection intensity and Prandtl number for the eight cases of the two plates with distinct thermal boundary conditions and buoyancy-assisting and-opposing flow conditions.     

Mixed convection in non-Newtonian fluids along nonisothermal horizontal surfaces in porous media

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in power-law type non-Newtonian fluids along horizontal surfaces with variable 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. Received on 11 March 1997     

Nonsimilar Solutions for Mixed Convection in Non-Newtonian Fluids Along Horizontal Surfaces in Porous Media

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in power-law type non-Newtonian fluids along horizontal surfaces with variable heat flux 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.     

A general treatment for non-Darcy film condensation within a porous medium in the presence of gravity and forced flow

Non-Darcy film condensation over a vertical flat plate within a porous medium is considered. The Forchheimer extended Darcy model is adopted to account for the non-Darcy effects on film condensation in the presence of both gravity and externally forced flow. A general similarity transformation is proposed upon introducing a modified Peclet number based on the total velocity of condensate, resulting from both gravitational force and externally forced flow. This general treatment makes it possible to obtain all possible similarity solutions including the asymptotic results in the four different limiting regimes, namely, Darcy forced convection regime, Forchheimer forced convection regime, Darcy body force predominant regime and Forchheimer body force predominant regime. Appropriate dimensionless groups for distinguishing these asymptotic regimes are found to be the micro-scale Grashof and Reynolds numbers based on the square root of the permeability of the porous medium. Correspondingly, the non-Darcy effect on the heat transfer rate are investigated in terms of these micro-scale dimensionless numbers.     

Forced and mixed convection boundary layer flow along a flat plate with variable viscosity and variable Prandtl number: new results

The steady laminar boundary layer flow along a flat plate is studied taking into account the variation of fluid viscosity and fluid Prandtl number with temperature. In the forced convection case the plate moves with constant velocity and its temperature varies in power law with x. In the mixed convection case the plate temperature is constant and the fluid moves upwards due to an external free stream and due to buoyancy forces. The results are obtained with the direct numerical solution of the boundary layer equations. The study concerns the wall heat transfer, the wall shear stress and velocity and temperature profiles across the boundary layer. The results of the present work are different from those existing in the literature, which have been obtained with the assumption of constant Pr number.     

Mixed Convection-Radiation in Power-Law Fluids Along a NOn-Isothermal Wedge in a Porous Medium with Variable Permeability

A boundary layer analysis has been presented for the interaction of mixed convection with thermal radiation in laminar boundary flow from a vertical wedge in a porous medium saturated with a power-law type non-Newtonian incorporating the variation of permeability and thermal conductivity. The transformed conservation laws are solved numerically for the case of variable surface temperature conditions. The combined convection non-similar parameter we note that =0 and 1 correspond to pure free and forced convection cases. The Rosseland approximation is used to describe the radiative heat flux in energy equation. Velocity and temperature profiles as well as the local Nusselt number are presented.     

Experimental studies on mixed convection heat transfer in laminar flow through a plain square duct

This paper reports the findings of experimental studies on combined free and forced convection through a plain square duct in laminar region. The test fluid flows through an inner square duct, hot water at high flow rate circulated through a annular channel formed between square duct and circular tube, in counter current fashion to attain a nearly uniform wall temperature conditions. The importance of mixed convection is judged by the value of the Richardson number (Ri). It was observed that at low Reynolds number, heat transfer was mainly governed by mixed convection. However at higher values of Reynolds number, heat transfer was significantly dominated by forced convection. It was found that Reynolds number higher than 1050 for water and 480 for ethylene glycol resulted in laminar forced convention heat transfer. The empirical correlation developed for Nusselt number in terms of Grashoff number and Graez number, was found to fit with experimental Nusselt number within ±11 and ±12?% for water and ethylene glycol respectively.     

Mixed convection boundary layer flow along vertical moving thin needles with variable heat flux

The problem of steady laminar mixed convection boundary layer flow of an incompressible viscous fluid along vertical moving thin needles with variable heat flux for both assisting and opposing flow cases is theoretically considered in this paper. The governing boundary layer equations are first transformed into non-dimensional forms. The curvature effects are incorporated into the analysis whereas the pressure variation in the axial direction has been neglected. These equations are then transformed into similarity equations using the similarity variables, which are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The solutions are obtained for a blunt-nosed needle (m = 0). Numerical calculations are carried out for various values of the dimensionless parameters of the problem, which include the mixed convection parameter λ, the Prandtl number Pr and the parameter a representing the needle size. It is shown from the numerical results that the skin friction coefficient, the surface (wall) temperature and the velocity and temperature profiles are significantly influenced by these parameters. The results are presented in graphical form and are discussed in detail.     

Radiation effect on mixed convection along a vertical plate with uniform surface temperature

This paper investigates the effect of radiation on the forced and free convection flow of an optically dense viscous incompressible fluid along a heated vertical flat plate with uniform free stream and uniform surface temperature with Rosseland diffusion approximation. With appropriate transformations, the boundary layer equations governing the flow are reduced to local nonsimilarity equations valid in the forced convection regime as well as in the free convection regime. A group of transformation is, also, introduced to reduce the boundary layer equations to a set of local nonsimilarity equations valid in both the forced and free convection regimes. Solutions of the governing equations are obtained by employing the implicit finite difference methods together with Keller box scheme and are expressed in terms of local shear stress and local rate of heat transfer for a range of values of the pertinent parameters.     

Mixed Convection in Water Near the Density Extremum Along a Vertical Plate with Sinusoidal Surface Temperature Variation Embedded in a Porous Medium

A steady laminar boundary layer flowing along a vertical plate immersed in a Darcy–Brinkman porous medium saturated with water at 4°C is studied. The plate temperature varies sinusoidally along the plate between 0 and 8°C where the density of water varies parabolically and is almost symmetrical at about 4°C. Except for the existence of the buoyancy force, it is assumed that either the plate moves upwards or the ambient water moves upwards (moving stream). The results are obtained with the direct numerical solution of the boundary layer equations taking into account the temperature dependence of water thermophysical properties (ρ, μ and c _p). Results are presented for the wall temperature gradient and the wall shear stress along the plate for free convection and mixed convection. Temperature and velocity profiles are also presented.     

Mixed convection heat transfer from a horizontal plate with variable surface heat flux in a porous medium

In this article nonsimilarity solution for mixed convection from a horizontal surface in a saturated porous medium was obtained for the case of variable surface heat flux. The entire mixed convection regime, ranging from pure forced convection to pure free convection, is considered by introducing a single nonsimilarity parameter. Heat transfer results are predicted by employing four different flow models, namely, Darcy's law, the Ergun model, and the Brinkman-Forchheimer-extended Darcy model with constant and variable porosity. The variable porosity effect is approximated by an exponential function. Effects of transverse thermal dispersion are taken into consideration in the energy equation, along with variable stagnant thermal conductivities. The formulation of the present problem shows that the flow and heat transfer characteristics depend on five parameters, that is, the power in the variation of surface heat flux, the nonsimilarity mixed-convection parameter, the inertia effect parameter, the boundary effect parameter, and the ratio of thermal conductivity of the fluid phase to that of the solid phase. Numerical results for the local Nusselt number variations, based on the various flow models, are presented for the entire mixed convection regime. The impacts␣of different governing parameters on the heat transfer results are thoroughly investigated. Received on 7 August 1997     

Natural Convection Induced by a Heated Vertical Plate Embedded in a Porous Medium with Transpiration: Local Thermal Non-equilibrium Similarity Solutions

This paper is concerned with the thermal non-equilibrium free convection boundary layer, which is induced by a vertical heated plate embedded in a saturated porous medium. The effect of suction or injection on the free convection boundary layer is also studied. The plate is assumed to have a linear temperature distribution, which yields a boundary layer of constant thickness. On assuming Darcy flow, similarity solutions are obtained for governing the steady laminar boundary layer equations. The reduced Nusselt numbers for both the solid and fluid phases are calculated for a wide range of parameters, and compared with asymptotic analyses.     

Marangoni Mixed Convection Boundary Layer Flow

The paper deals with a steady coupled dissipative layer, called Marangoni mixed convection boundary layer, which can be formed along the interface of two immiscible fluids, in surface driven flows. The mixed convection boundary layer is generated when besides the Marangoni effects there are also buoyancy effects due to gravity and external pressure gradient effects. We shall use a model proposed by Golia and Viviani (L’ Aerotecnica missili e Spazio 64 (1985) 29–35, Meccanica 21 (1986) 200–204) wherein the Marangoni coupling condition has been included into the boundary conditions at the interface. The similarity equations are first determined, and the pertinent equations are solved numerically for some values of the governing parameters and the features of the flow and temperature fields as well as the interface velocity and heat transfer at the interface are analysed and discussed.     

The effects of transpiration on the boundary layer flow and heat transfer over a vertical slender cylinder

This paper deals with a theoretical (numerical) analysis of the effects that blowing/injection and suction have on the steady mixed convection or combined forced and free convection boundary layer flows over a vertical slender cylinder with a mainstream velocity and a wall surface temperature proportional to the axial distance along the surface of the cylinder. Both cases of buoyancy forces aid and oppose the development of the boundary layer are considered. Similarity equations are derived and their solutions are dependent upon the mixed convection parameter, the non-dimensional transpiration parameter and the curvature parameter, as well as of the Prandtl number. Dual solutions for the previously studied mixed convection boundary layer flows over an impermeable surface of the cylinder are shown to exist also in the present problem for aiding and opposing flow situations.     

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.     

Self-similar thermal boundary layers on plane and axisymmetric bodies

This paper proposes an approximate solution procedure for the prediction of the forced convection heat transfer through self-similar laminar boundary layers. The differential equations governing the viscous and thermal boundary layers have been reduced to a pair of algebraic equations for the boundary layer shape factor and the boundary layer thickness ratio. The local Nusselt number predicted under various pressure gradients turns out to be in excellent agreement with that of the exact solution over a wide range of the Prandtl number.