Solutions similaires pour un problème de couches limites en milieux poreux

We consider heat transfer from a surface embedded in an unbounded porous medium, saturated with a fluid at rest. A thermal boundary-layer approximation, based on the assumption that convection takes place in a thin layer around the heating surface, is done. The use of boundary layer technics show that we can find similarity solutions by solving a one-dimensional boundary value problem, involving a third-order nonlinear differential equation depending on a parameter. We prove existence and uniqueness results for some values of the parameter, non-existence for the other ones, and when it is possible, we construct explicit solutions.     

MHD forced convection boundary layer flow with a flat plate and porous substrate

The flow and heat transfer for an electrically conducting fluid with a porous substrate and a flat plate under the influence of magnetic field is considered. The magnetic field is assumed to be uniform and also along normal to the surface. The momentum and energy equations are transformed to ordinary differential equations by using suitable similarity transformation and are solved by standard techniques. But the energy equation is solved by considering two boundary layers, one in the porous substrate and the other above the porous substrate. Numerical results are presented through graphs with various values of magnetic parameter for both velocity and thermal boundary layers along with Nusselt number and for various values of Prandtl number and Eckert number in thermal boundary layer.     

Forced convective flow and heat transfer over a?porous plate in a?Darcy-Forchheimer porous medium in presence of?radiation

A mathematical model is presented for analyzing the boundary layer forced convective flow and heat transfer of an incompressible fluid past a plate embedded in a Darcy-Forchheimer porous medium. Thermal radiation term is considered in the energy equation. The similarity solutions for the problem are obtained and the reduced nonlinear ordinary differential equations are solved numerically. It is noticed that the boundary layer decreases with an increase in the value of inertial parameter and in this case the temperature profile is found to decrease smoothly within the boundary layer. In case of porous plate, fluid velocity increases whereas non-dimensional temperature decreases for increasing values of suction parameter. The rate of heat transfer increases with the increasing values of Prandtl number. The effect of thermal radiation on temperature field is also analyzed.     

A generalized thermal boundary condition for the hyperbolic heat conduction model

 A generalized thermal boundary condition is derived for the hyperbolic heat conduction equation to include all thermal effects of a thin layer, whether solid-skin or fluid film, moving or stationary, in perfect or imperfect thermal contact with an adjacent domain. The thin layer thermal effects include, among others, thermal capacity of the layer, thermal diffusion, enthalpy flow, viscous dissipation within the layer and convective losses from the layer. Six different kinds of thermal boundary conditions can be obtained as special cases of the generalized boundary condition. The importance of the generalized boundary condition is demonstrated comprehensively in an example. The effects of different geometrical and thermophysical properties on the validity of the generalized thermal boundary condition are investigated. Received on 23 May 2001 / Published online: 29 November 2001     

Influence of the Prandtl Number on the Unsteady Thermo-Fluid Dynamic Field Around a Thick Plate

The problem of a semi-infinite “thick” plate impulsively accelerated to a constant speed is discussed, considering a compressible laminar boundary layer when the thermo-fluid dynamic field in the flow is coupled with the thermal field in the solid (conjugated heat transfer). Different cases are defined by the boundary condition for the thermal field on the unwetted plate side, A first-order analytical approximation of a more general method of solution is here proposed, with emphasis to the analysis of the effects of the Prandtl number (Pr), when the temperature on the unwetted side of the plate is assumed constant. The comparison with some reference solution available for the limiting conditions of steady flow and plate of infinite length shows a very satisfactory accuracy of the present results in spite of the first order approximation.     

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.     

Asymptotic analysis of an end-loaded, transversely isotropic, elastic, semi-infinite strip weak in shear

We use linear elasticity to study a transversely isotropic (or specially orthotropic), semiinfinite slab in plane strain, free of traction on its faces and at infinity and subject to edge loads or displacements that produce stresses and displacements that decay in the axial direction. The governing equations (which are identical to those for a strip in plane stress, free of traction on its long sides and at infinity, and subject to tractions or displacements on its short side) are reduced, in the standard way. to a fourth-order partial differential equation with boundary conditions for a dimensionless Airy stress function ƒ. We study the asymptotic solutions to this equation for four sets of end conditions—traction, mixed (two), displacement—as g3, the ratio of the shear modulus to the geometric mean of the axial and transverse extensional moduli, approaches zero. In all cases, the solutions for ƒ consist of a “wide” boundary layer that decays slowly in the axial direction (over a distance that is long compared to the width of the strip) plus a “narrow” boundary layer that decays rapidly in the axial direction (over a distance that is short compared to the width of the strip). Moreover, we find that the narrow boundary layer has a “sinuous” part that varies rapidly in the transverse direction, but which, to lowest order, does not enter the boundary conditions nor affect the transverse normal stress or the displacements. Because the exact biorthogonality condition for the cigenfunctions associated with ƒ can be replaced by simpler orthogonality conditions in the limit as →b 0, we are able to obtain, to lowest order, explicit formulae for the coeflicients in the eigenfunction expansions of ƒ for the four different end conditions.     

Locally similar solutions for hydromagnetic and thermal slip flow boundary layers over a flat plate with variable fluid properties and convective surface boundary condition

This paper presents heat transfer process in a two-dimensional steady hydromagnetic convective flow of an electrically conducting fluid over a flat plate with partial slip at the surface of the boundary subjected to the convective surface heat flux at the boundary. The analysis accounts for both temperature-dependent viscosity and temperature dependent thermal conductivity. The local similarity equations are derived and solved numerically using the Nachtsheim-Swigert iteration procedure. Results for the dimensionless velocity, temperature and ambient Prandtl number within the boundary layer are displayed graphically delineating the effect of various parameters characterizing the flow. The results show that momentum boundary layer thickness significantly depends on the surface convection parameter, Hartmann number and on the sign of the variable viscosity parameter. The results also show that plate surface temperature is higher when there is no slip at the plate compared to its presence. For both slip and no-slip cases surface temperature of the plate can be controlled by controlling the strength of the applied magnetic field. In modelling the thermal boundary layer flow with variable viscosity and variable thermal conductivity, the Prandtl number must be treated as a variable irrespective of flow conditions whether there is slip or no-slip at the boundary to obtain realistic results.     

Similarity solutions of Prandtl mixing length modelled two dimensional turbulent boundary layer equations

The exact similarity solutions of two dimensional laminar boundary layer were obtained by Blasius in 1908, however, for two dimensional turbulent boundary layers, no Blasius type similarity solutions (special exact solutions) have ever been found. In the light of Blasius’ pioneer works, we extend Blasius similarity transformation to the two dimensional turbulent boundary layers, and for a special case of flow modelled by Prandtl mixing-length, we successfully transform the two dimensional turbulent boundary layers partial differential equations into a single ordinary differential equation. The ordinary differential equation is numerically solved and some useful quantities are produced. For numerical calculations, a complete Maple code is provided.     

Nonequivalent similarity reductions of steady thermal boundary layer equations for an incompressible laminar flow over a continuous moving hot surface

This paper uses symmetry methods to obtain nonequivalent similarity reductions of the steady two-dimensional thermal boundary layer equations of an incompressible laminar flow. New similarity reductions are uncovered and results obtained through the classical dimensional analysis are recovered.     

Heat and mass transfer on a MHD third grade fluid with partial slip flow past an infinite vertical insulated porous plate in a porous medium

The influence of third grade, partial slip and other thermophysical parameters on the steady flow, heat and mass transfer of viscoelastic third grade fluid past an infinite vertical insulated plate subject to suction across the boundary layer has been investigated. The space occupying the fluid is porous. The momentum equation is characterized by a highly nonlinear boundary value problem in which the order of the differential equation exceeds the number of available boundary conditions. An efficient numerical scheme of midpoint technique with Richardson’s extrapolation is employed to solve the governing system of coupled nonlinear equations of momentum, energy and concentration. Numerical calculations were carried out for different values of various interesting non-dimensional quantities in the slip flow regime with heat and mass transfer and were shown with the aid of figures. The values of the wall shear stress, the local rate of heat and mass transfers were obtained and tabulated. The analysis shows that as the fluid becomes more shear thickening, the momentum boundary layer decreases but the thermal boundary layer increases; the magnetic field strength is found to decrease with an increasing temperature distribution when the porous plate is insulated. The consequences of increasing the permeability parameter and Schmidt number decrease both the momentum and concentration boundary layer thicknesses respectively whereas an increase in the thermal Grashof number gives rise to the thermal boundary layer thickness.     

Analytical study of heat transfer from circular cylinder in liquid metals

In this study the influence of a thin hydrodynamic boundary layer on the heat transfer from a single circular cylinder in liquid metals having low Prandtl number (0.004–0.03) is investigated under isothermal and isoflux boundary conditions. Two separate analytical heat transfer models, viscous and inviscid, are developed to clarify the discrepancy between previous results. For both models, integral approach of the boundary layer analysis is employed to derive closed form expressions for the calculation of the average heat transfer coefficients. For an inviscid model, the energy equation is solved using potential flow velocity only whereas for a viscous model, a fourth-order velocity profile is used in the hydrodynamic boundary layer and potential flow velocity is used outside the boundary layer. The third-order temperature profile is used inside the thermal boundary layer for both models. It is shown that the inviscid model gives higher heat transfer coefficients whereas viscous flow model gives heat transfer results in a fairly good agreement with the previous experimental/numerical results.     

MHD effect of mixed convection boundary-layer flow of Powell-Eyring fluid past nonlinear stretching surface

Sufficient conditions are found for the existence of similar solutions of the mixed convection flow of a Powell-Eyring fluid over a nonlinear stretching permeable sur- face in the presence of magnetic field. To achieve this, one parameter linear group trans- formation is applied. The governing momentum and energy equations are transformed to nonlinear ordinary differential equations by use of a similarity transformation. These equations are solved by the homotopy analysis method （HAM） to obtain the approximate solutions. The effects of magnetic field, suction, and buoyancy on the Powell-Eyring fluid flow with heat transfer inside the boundary layer are analyzed. The effects of the non- Newtonian fluid （Powell-Eyring model） parameters ε and δon the skin friction and local heat transfer coefficients for the cases of aiding and opposite flows are investigated and discussed. It is observed that the momentum boundary layer thickness increases and the thermal boundary layer thickness decreases with the increase in ε whereas the momentum boundary layer thickness decreases and thermal boundary layer thickness increases with the increase in δ for both the aiding and opposing mixed convection flows.     

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.     

MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel

Two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell fluid is investigated in a channel. The walls of the channel are taken as porous. Using the similarity transformations and boundary layer approximations, the nonlinear partial differential equations are reduced to an ordinary differential equation. The developed nonlinear equation is solved analytically using the homotopy analysis method. An expression for the analytic solution is derived in the form of a series. The convergence of the obtained series is shown. The effects of the Reynolds number Re, Deborah number De and Hartman number M are shown through graphs and discussed for both the suction and injection cases.     

Synthetic method in thermal boundary layer transition

The problem of a general incompressible viscous fluid flow past a flat plate with heat transfer due to forced convection is considered in this paper. The synthetic method developed by Seth is applied to the Navier-Stokes equations and the equation of energy governing the flow to obtain the dynamic and thermal boundary layer solutions as asymptotic limits of an extended field. As a result, new formulas are derived for both the dynamic and thermal boundary layer thicknesses. Also, algorithms for estimating all the parameters involved in the analysis are provided and boundary layer functions based on the new solutions are determined.     

Exact solutions to an evolution equation of plastic layer flow on a plane

The flow of a thin plastic layer between two rigid plates approaching each other in the normal direction is considered. The kinematics of plastic layer flow is studied. An evolution equation describing the free boundary of the flow region is derived. The similarity solutions to this equation are analyzed. It is shown that the evolution equation can be reduced to a particular case of the nonlinear heat conduction equation. New exact particular solutions to the evolution equation are obtained using the variable separation method and the method of self-similar transformations.     

The influence of thermal radiation on hydromagnetic Darcy-Forchheimer mixed convection flow past a stretching sheet embedded in a porous medium

A boundary layer analysis is performed to study the influence of thermal radiation and buoyancy force on two-dimensional magnetohydrodynamic flow of an incompressible viscous and electrically conducting fluid over a vertical stretching sheet embedded in a porous medium in the presence of inertia effect. The governing system of partial differential equations is first transformed into system of ordinary differential equations using self-similarity transformation. A special form for magnetic field is chosen to obtain the similarity solution. The transformed boundary layer equations are solved numerically for some important values of the physical parameters. The present results are compared with the previously published papers and the results are found to be in excellent agreement. The important features of the flow, heat and mass transfer characteristics for different values of thermal radiation, porous permeability, magnetic field and buoyancy parameters are analyzed and discussed. The effects of various physical parameters on the skin friction coefficient, local Nusselt number and local Sherwood number are also presented. It is found that increase in the value of thermal radiation parameter R ₁ increases the skin friction coefficient and Sherwood number whereas reverse trend is seen for the local Nusselt number.     

Influence of turbulent Prandtl number on heat transfer of a flat plate

In computations involving heat transfer in turbulent flow past bodies it is necessary to assume turbulent Prandtl number distribution across the boundary layer. A review and comparison of results obtained by different authors are given, e.g., in [1–5]. Unfortunately, the existing data are so contradictory that, at present, it does not appear to be possible to establish reliably a function that determines turbulent Prandtl number distribution across the boundary layer. The absence of sufficiently reliable and general results on the distribution of turbulent Prandtl number led to the result that in the majority of studies conducted in earlier years its value was assumed a constant and either close to or equal to one. The effect of turbulent Prandtl number on the intensity of heat transfer from a flat plate is numerically investigated in the present paper. The thermal, turbulent boundary layer equation is integrated for this purpose at different values of turbulent Prandtl number and results are compared with experimental data. Results from [6], where the thermal boundary layer was numerically integrated with Pr_t=1 and compared with experimental data, were used for comparison in the present paper. The same numerical integration procedure as in [6] was used here.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 81–85, July–August, 1984.     

Scaling group transformation for MHD boundary layer flow over permeable stretching sheet in presence of slip flow with Newtonian heating effects

Taking into account the slip flow effects, Newtonian heating, and thermal radiation, two-dimensional magnetohydrodynamic （MHD） flows and heat transfer past a permeable stretching sheet are investigated numerically. We use one parameter group transformation to develop similarity transformation. By using the similarity transformation, we transform the governing boundary layer equations along with the boundary conditions into ordinary differential equations with relevant boundary conditions. The obtained ordinary differential equations are solved with the fourth-fifth order Runge-Kutta- Fehlberg method using MAPLE 13. The present paper is compared with a published one. Good agreement is obtained. Numerical results for dimensionless velocity, temperature distributions, skin friction factor, and heat transfer rates are discussed for various values of controlling parameters.