Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition

The steady laminar boundary layer flow over a permeable flat plate in a uniform free stream, with the bottom surface of the plate is heated by convection from a hot fluid is considered. Similarity solutions for the flow and thermal fields are possible if the mass transpiration rate at the surface and the convective heat transfer from the hot fluid on the lower surface of the plate vary like x^−1/2, where x is the distance from the leading edge of the solid surface. The governing partial differential equations are first transformed into ordinary differential equations, before being solved numerically. The effects of the governing parameters on the flow and thermal fields are thoroughly examined and discussed.     

Effects of thermal radiation and magnetic field on unsteady mixed convection flow and heat transfer over an exponentially stretching surface with suction in the presence of internal heat generation/absorption

In this paper, the problem of unsteady laminar two-dimensional boundary layer flow and heat transfer of an incompressible viscous fluid in the presence of thermal radiation, internal heat generation or absorption, and magnetic field over an exponentially stretching surface subjected to suction with an exponential temperature distribution is discussed numerically. The governing boundary layer equations are reduced to a system of ordinary differential equations. New numerical method using Mathematica has been used to solve such system after obtaining the missed initial conditions. Comparison of obtained numerical results is made with previously published results in some special cases, and found to be in a good agreement.     

Analytical solutions of a coupled nonlinear system arising in a flow between stretching disks

In this paper, we consider the axi-symmetric flow between two infinite stretching disks. By using a similarity transformation, we reduce the governing Navier-Stokes equations to a system of nonlinear ordinary differential equations. We first obtain analytical solutions via a four-term perturbation method for small and large values of the Reynolds number R. Also, we apply the Homotopy Analysis Method (which may be used for all values of R) to obtain analytical solutions. These solutions converge over a larger range of values of the Reynolds number than the perturbation solutions. Our results agree well with the numerical results of Fang and Zhang [22]. Furthermore, we obtain the analytical solutions valid for moderate values of R by use of Homotopy Analysis.     

Stagnation slip flow and heat transfer over a nonlinear stretching sheet

Stagnation slip flow and heat transfer characteristics of a viscous fluid over a nonlinear stretching surface has been investigated. The governing partial differential equations are transformed to nonlinear ordinary differential equations using similarity transformations. The analytical solution of the nonlinear system is obtained in series form using the very efficient homotopy analysis method (HAM). Convergence of the series solution is shown explicitly. Important features of flow and heat transfer characteristics are plotted and discussed. Comparison is made with existing numerical results when the stagnation‐point and slip effects are excluded. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Flow,heat, and species transfer over a stretching plate considering coupled Stefan blowing effects from species transfer

In this paper, we investigate the flow, heat and mass transfer of a viscous fluid flow over a stretching sheet by including the blowing effects of mass transfer under high flux conditions. Mass transfer in this work means species transfer and is different from mass transpiration for permeable walls. The new contribution from this work is, for the first time, to consider the coupled blowing effects from massive species transfer on flow, heat, and species transfer for a stretching plate. Based on the exact solutions of the momentum equations, which are valid for the whole Navier–Stokes equations, the energy and mass transfer equations are solved exactly and the effects of the blowing parameter, the Schmidt number, and the Prandtl number on the flow, heat and mass transfer are presented and discussed. The solution is given in terms of an incomplete Gamma function. It is found the coupled blowing effects due to mass transfer can have significant influences on velocity profiles, drag, heat flux, as well as temperature and concentration profiles. These solutions provide rare results with closed form analytical expressions and can be used as benchmark problem for numerical code validation.     

MHD flow and heat transfer of a micropolar fluid over a stretching surface with heat generation (absorption) and slip velocity

In this work, the effects of slip velocity on the flow and heat transfer for an electrically conducting micropolar fluid over a permeable stretching surface with variable heat flux in the presence of heat generation (absorption) and a transverse magnetic field are investigated. The governing partial differential equations describing the problem are converted to a system of non-linear ordinary differential equations by using the similarity transformation, which is solved numerically using the Chebyshev spectral method. The effects of the slip parameter on the flow, micro-rotation and temperature profiles as well as on the local skin-friction coefficient, the wall couple stress and the local Nusselt number are presented graphically. The numerical results of the local skin-friction coefficient, the wall couple stress and the local Nusselt number are given in a tabular form and discussed.     

Similarity solution of boundary layer stagnation-point flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption/generation and suction/blowing: A Lie group analysis

In this paper, heat and mass transfer analysis for boundary layer stagnation-point flow over a stretching sheet in a porous medium saturated by a nanofluid with internal heat generation/absorption and suction/blowing is investigated. The governing partial differential equation and auxiliary conditions are converted to ordinary differential equations with the corresponding auxiliary conditions via Lie group analysis. The boundary layer temperature, concentration and nanoparticle volume fraction profiles are then determined numerically. The influences of various relevant parameters, namely, thermophoresis parameter Nt, Brownian motion parameter Nb, Lewis number Le, suction/injection parameter S, permeability parameter k₁, source/sink parameter λ and Prandtl parameter Pr on temperature and concentration as well as wall heat flux and wall mass flux are discussed. Comparison with published results is presented.     

Heat transfer in a liquid film over an unsteady stretching surface with viscous dissipation in presence of external magnetic field

This paper presents a mathematical analysis of MHD flow and heat transfer to a laminar liquid film from a horizontal stretching surface. The flow of a thin fluid film and subsequent heat transfer from the stretching surface is investigated with the aid of similarity transformation. The transformation enables to reduce the unsteady boundary layer equations to a system of non-linear ordinary differential equations. Numerical solution of resulting non-linear differential equations is found by using efficient shooting technique. Boundary layer thickness is explored numerically for some typical values of the unsteadiness parameter S and Prandtl number Pr, Eckert number Ec and Magnetic parameter Mn. Present analysis shows that the combined effect of magnetic field and viscous dissipation is to enhance the thermal boundary layer thickness.     

Hall effect on unsteady MHD Couette flow and heat transfer of a Bingham fluid with suction and injection

The unsteady magnetohydrodynamic flow of an electrically conducting viscous incompressible non-Newtonian Bingham fluid bounded by two parallel non-conducting porous plates is studied with heat transfer considering the Hall effect. An external uniform magnetic field is applied perpendicular to the plates and the fluid motion is subjected to a uniform suction and injection. The lower plate is stationary and the upper plate moves with a constant velocity and the two plates are kept at different but constant temperatures. Numerical solutions are obtained for the governing momentum and energy equations taking the Joule and viscous dissipations into consideration. The effect of the Hall term, the parameter describing the non-Newtonian behavior, and the velocity of suction and injection on both the velocity and temperature distributions are studied.     

Three-dimensional flow over a stretching surface in a viscoelastic fluid

This article looks at the hydrodynamic elastico-viscous fluid over a stretching surface. The equations governing the flow are reduced to ordinary differential equations, which are analytically solved by applying an efficient technique namely the homotopy analysis method (HAM). The solutions for the velocity components are computed. The numerical values of wall skin friction coefficients are also tabulated. The present HAM solution is compared with the known exact solution for the two-dimensional flow and an excellent agreement is found.     

Momentum and heat transfer over a continuously moving surface with a parallel free stream in a viscoelastic fluid

The flow and heat transfer characteristics for a continuous moving surface in a viscoelastic fluid are investigated. Constitutive equations of viscoelastic fluid obey the second‐grade model. Analytic expressions to velocity and temperature have been developed by employing homotopy analysis method. The criterion to the convergence of the solution is properly discussed. Furthermore, the values of skin friction coefficient and the local Nusselt number have been computed and discussed. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Heat transfer in a visco-elastic fluid past a stretching sheet with viscous dissipation and internal heat generation

This paper deals with the study of heat transfer characteristics in the laminar boundary layer flow of a visco-elastic fluid over a linearly stretching continuous surface with variable wall temperature subjected to suction or blowing. The study considers the effects of frictional heating (viscous dissipation) and internal heat generation or absorption. An analysis has been carried out for two different cases of heating processes namely: (i) Prescribed surface temperature (PST) and (ii) Prescribed wall heat flux (PHF) to get the effect of visco-elastic parameter for various situations. Further increase of visco-elastic parameter is to decrease the skin friction on the sheet. The solutions for the temperature and the heat transfer characteristics are obtained in terms of Kummers function. Received: June 16, 2004; revised: February 8, 2005     

Hall current effect on MHD mixed convection flow from an inclined continuously stretching surface with blowing/suction and internal heat generation/absorption

Hydromagnetic heat transfer by mixed convection along an inclined continuously stretching surface, with power-law variation in the surface temperature or heat flux, in the presence of Hall current and internal heat generation/absorption has been studied. The surface is considered to be permeable to allow fluid suction or blowing, and stretching with a surface velocity varied according to a power-law. Two cases of the temperature boundary conditions were considered at the surface. The governing equations have been transformed into non-similar partial differential equations which have been integrated by the forth-order Runge–Kutta method. The effect of Hall parameter, magnetic parameter, dimensionless blowing/suction parameter, space and temperature dependent internal heat generation/absorption parameters and buoyancy force parameters on the temperature, primary and secondary flow velocity have been studied parametrically. All parameters involved in the problem affect the flow and thermal distributions except the temperature-dependent internal heat generation/absorption in the case of prescribed heat flux (PHF). Numerical values of the local skin-friction and the local Nusselt numbers for various parametric conditions have been tabulated.     

Analytical solution of hydromagnetic flow with Hall effect over a surface stretching with a power‐law velocity

The nonlinear magnetohydrodynamic (MHD) flow problem with Hall current caused by stretching surface having power law velocity distribution is solved by employing homotopy analysis method (HAM). Perturbation solution of stream function, the expression of skin friction coefficient and graphical results in absence of Hall current (Chiam, Int J Eng Sci 33 (1995), 429) are recovered as the limiting cases. It is found that unlike the solution obtained by Chiam (1995), the present results are valid for weak and large magnetic parameters. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 937–959, 2011     

Unsteady heat and mass transfer over a vertical stretching sheet in a parallel free stream with variable wall temperature and concentration

Our aim in this article is to investigate numerically the unsteady two‐dimensional mixed convection flow along a vertical semi‐infinite stretching sheet in a parallel free stream with a power‐law wall temperature and concentration distributions of the form T _w (x) = T_∞ + Ax^2m?1 and C_w (x) = C_∞ + Bx^2m?1, where A, B and m are constants. The unsteadiness in the flow is caused by the time dependent stretching sheet as well as by the free stream velocity. The governing nonlinear partial differential equations in the velocity, temperature and concentration fields are written in nondimensional form using suitable transformations. The final set of resulting coupled nonlinear partial differential equations is solved using an implicit finite‐difference scheme in combination with a quasi‐linearization technique. The effects of various governing parameters on the velocity, temperature and concentration profiles as well as on the skin friction coefficient, local Nusseltnumber and local Sherwood number are presented and discussed in details. The computed numerically results are compared with previously reported work and are found to be in excellent agreement. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Unsteady flow of a Maxwell fluid over a stretching surface in presence of chemical reaction

An analysis is presented for unsteady two-dimensional flow of a Maxwell fluid over a stretching surface in presence of a first order constructive/destructive chemical reaction. Using suitable transformations, the governing partial differential equations are converted to ordinary one and are then solved numerically by shooting method. The flow fields and mass transfer are significantly influenced by the governing parameters. Fluid velocity initially decreases with increasing unsteadiness parameter and concentration decreases significantly due to unsteadiness. The effect of increasing values of the Maxwell parameter is to suppress the velocity field. But the concentration is enhanced with increasing Maxwell parameter.     

Effect of Hall currents and chemical reaction on hydromagnetic flow of a stretching vertical surface with internal heat generation/absorption

The problem of steady, laminar, hydromagnetic, simultaneous heat and mass transfer by laminar flow of a Newtonian, viscous, electrically conducting and heat generating/absorbing fluid over a continuously stretching surface in the presence of the combined effect of Hall currents and mass diffusion of chemical species with first and higher order reactions is investigated. The fluid is permeated by a strong transverse magnetic field imposed perpendicularly to the plate on the assumption of a small magnetic Reynolds number. Certain transformations are employed to transform the governing differential equations to a local similarity form which are solved numerically. Comparisons with previously published work have been conducted and the results are found to be in good agreement. A parametric study is performed to illustrate the influence of the magnetic field parameter, Hall parameter, the coefficients of space-dependent and temperature-dependent internal heat generation/absorption, the chemical reaction parameter and order of reaction on the fluid velocity, temperature and concentration distributions. Numerical data for the local skin-friction coefficient, the local Nusselt number and the local Sherwood number have been tabulated for various values of parametric conditions.     

Series solution for unsteady axisymmetric flow and heat transfer over a radially stretching sheet

This paper deals with the unsteady axisymmetric flow and heat transfer of a viscous fluid over a radially stretching sheet. The heat is prescribed at the surface. The modelled non-linear partial differential equations are solved using an analytic approach namely the homotopy analysis method. Unlike perturbation technique, this approach gives accurate analytic approximation uniformly valid for all dimensionless time. The explicit expressions for velocity, temperature and skin friction coefficient are developed. The influence of time on the velocity, temperature and skin friction coefficient is discussed.     

A general class of coupled nonlinear differential equations arising in self-similar solutions of convective heat transfer problems

We establish existence and uniqueness results for a general class of coupled nonlinear third order differential equations arising in flow and heat transfer problems. We consider solutions over the semi-infinite interval. As special cases, we recover the existence and uniqueness results of solutions for the following physically meaningful scenarios (among others): (i) flow and heat transfer over a stretching sheet, (ii) flow and heat transfer over a nonlinearly stretching porous sheet, (iii) linear convective flow and heat transfer over a porous nonlinearly stretching sheet and (iv) nonlinear convective heat transfer over a porous nonlinearly stretching sheet. In all the cases the effects of viscous dissipation and the internal heat generation/absorption on the flow and heat transfer characteristics are included. Moreover, the obtained results are applicable to several problems dealing with flow and heat transfer phenomena.     

The effect of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet

An analysis has been carried out to study the momentum and heat transfer characteristics in an incompressible electrically conducting non-Newtonian boundary layer flow of a viscoelastic fluid over a stretching sheet. The partial differential equations governing the flow and heat transfer characteristics are converted into highly non-linear coupled ordinary differential equations by similarity transformations. The effect of variable fluid viscosity, Magnetic parameter, Prandtl number, variable thermal conductivity, heat source/sink parameter and thermal radiation parameter are analyzed for velocity, temperature fields, and wall temperature gradient. The resultant coupled highly non-linear ordinary differential equations are solved numerically by employing a shooting technique with fourth order Runge–Kutta integration scheme. The fluid viscosity and thermal conductivity, respectively, assumed to vary as an inverse and linear function of temperature. The analysis reveals that the wall temperature profile decreases significantly due to increase in magnetic field parameter. Further, it is noticed that the skin friction of the sheet decreases due to increase in the Magnetic parameter of the flow characteristics.