Boundary layer flow and heat transfer over an unsteady stretching vertical surface

The solution to the unsteady mixed convection boundary layer flow and heat transfer problem due to a stretching vertical surface is presented in this paper. The unsteadiness in the flow and temperature fields is caused by the time-dependent of the stretching velocity and the surface temperature. The governing partial differential equations with three independent variables are first transformed into ordinary differential equations, before they are solved numerically by a finite-difference scheme. The effects of the unsteadiness parameter, buoyancy parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. Both assisting and opposing buoyant flows are considered. It is observed that for assisting flow, the solutions exist for all values of buoyancy parameter, whereas for opposing flow, they exist only if the magnitude of the buoyancy parameter is small. Comparison with known results for steady-state flow is excellent.     

Non-darcy double-diffusive mixed convection from heated vertical and horizontal plates in saturated porous media

The non-darcy mixed convection flows from heated vertical and horizontal plates in saturated porous media have been considered using boundary layer approximations. The flows are considered to be driven by multiple buoyancy forces. The similarity solutions for both vertical and horizontal plates have been obtained. The governing equations have been solved numerically using a shooting method. The heat transfer, mass transfer and skin friction are reduced due to inertial forces. Also, they increase with the buoyancy parameter for aiding flow and decrease for the opposing flow. For aiding flow, the heat and mass transfer coefficients are found to approach asymptotically the forced or free convection values as the buoyancy parameter approaches zero or infinity.     

Mixed convection boundary layer flow past a wedge with permeable walls

The effects of suction/injection on steady laminar mixed convection boundary layer flow over a permeable horizontal surface of a wedge in a viscous and incompressible fluid is considered in this paper. The similarity solutions of the governing boundary layer equations are obtained for some values of the suction/injection parameter f ₀, the constant exponent m of the wall temperature as well as the mixed convection parameter λ. The resulting system of nonlinear ordinary differential equations is solved numerically for both assisting and opposing flow regimes using an implicit finite-difference scheme known as the Keller-box method. Numerical results for the reduced skin friction coefficient, the local Nusselt number, and the velocity and temperature profiles are obtained for various values of parameters considered. Dual solutions are found to exist for the case of opposing flow.     

Mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder

The steady mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder in a stream flowing vertically upwards is numerically studied for both cases of heated and cooled cylinders. The governing partial differential equations are transformed into dimensionless forms using an appropriate transformation and then solved numerically using the Keller-box method. The comparison between the solutions obtained and those for a Newtonian fluid is found to be very good. Effects of the mixed convection and elasticity parameters on the skin friction and heat transfer coefficients for a fluid having the Prandtl number equal to one are also discussed. It is found that for some values of the viscoelastic parameter and some negative values of the mixed convection parameter (opposing flow) the boundary layer separates from the cylinder. Heating the cylinder delays separation and can, if the cylinder is warm enough, suppress the separation completely. Similar to the case of a Newtonian fluid, cooling the cylinder brings the separation point nearer to the lower stagnation point. However, for a sufficiently cold cylinder there will not be a boundary layer.     

Flow of couple stress fluid with variable thermal conductivity

The steady flow and heat transfer of a couple stress fluid due to an inclined stretching cylinder are analyzed. The thermal conductivity is assumed to be temperature dependent. The governing equations for the flow and heat transfer are transformed into ordinary differential equations. Series solutions of the resulting problem are computed. The effects of various interested parameters, e.g., the couple stress parameter, the angle of inclination, the mixed convection parameter, the Prandtl number, the Reynolds number, the radiation parameter, and the variable thermal conductivity parameter, are illustrated. The skin friction coefficient and the local Nusselt number are computed and analyzed. It is observed that the heat transfer rate at the surface increases while the velocity and the shear stress decrease when the couple stress parameter and the Reynolds number increase. The temperature increases when the Reynolds number increases.     

Steady laminar mixed convection stagnation-point flow of a nanofluid over a vertical permeable surface in the presence of a magnetic field

A similarity solution for a steady laminar mixed convection boundary layer flow of a nanofluid near the stagnation point on a vertical permeable plate with a magnetic field and a buoyancy force is obtained by solving a system of nonlinear ordinary differential equations. These equations are solved analytically by using a new kind of a powerful analytic technique for nonlinear problems, namely, the homotopy analysis method (HAM). Three different types of nanoparticles, namely, copper (Cu), alumina (Al₂O₃), and titanium oxide (TiO₂), with water as the base fluid are considered. The influence of the volume fraction of nanoparticles, permeability parameter, magnetic parameter, and mixed convection parameter on the surface shear stress and surface heat transfer, as well as on the velocity and temperature profiles, is considered. It is observed that the skin friction coefficient and the local Nusselt number increase with the nanoparticle volume fraction for all types of nanoparticles considered in this study. The greatest values of the skin friction coefficient and the local Nusselt number are obtained for Cu nanoparticles.     

Mixed convection boundary layer flow over a horizontal circular cylinder with Newtonian heating

The steady mixed convection boundary layer flow over a horizontal circular cylinder, generated by Newtonian heating in which the heat transfer from the surface is proportional to the local surface temperature, is considered in this study. The governing boundary layer equations are first transformed into a system of non-dimensional equations via the non-dimensional variables, and then into non-similar equations before they are solved numerically using a numerical scheme known as the Keller-box method. Numerical solutions are obtained for the skin friction coefficient Re ^1/2 C _f and the local wall temperature θ _w (x) as well as the velocity and temperature profiles with two parameters, namely the mixed convection parameter λ and the Prandtl number Pr.     

Steady mixed convection flow of Maxwell fluid over an exponentially stretching vertical surface with magnetic field and viscous dissipation

The steady mixed convection flow and heat transfer from an exponentially stretching vertical surface in a quiescent Maxwell fluid in the presence of magnetic field, viscous dissipation and Joule heating have been studied. The stretching velocity, surface temperature and magnetic field are assumed to have specific exponential function forms for the existence of the local similarity solution. The coupled nonlinear ordinary differential equations governing the local similarity flow and heat transfer have been solved numerically by Chebyshev finite difference method. The influence of the buoyancy parameter, viscous dissipation, relaxation parameter of Maxwell fluid, magnetic field and Prandtl number on the flow and heat transfer has been considered in detail. The Nusselt number increases significantly with the Prandtl number, but the skin friction coefficient decreases. The Nusselt number slightly decreases with increasing viscous dissipation parameter, but the skin friction coefficient slightly increases. Maxwell fluid reduces both skin friction coefficient and Nusselt number, whereas buoyancy force enhances them.     

Entropy analysis in electrical magnetohydrodynamic(MHD) flow of nanofluid with effects of thermal radiation,viscous dissipation,and chemical reaction

The unsteady mixed convection flow of electrical conducting nanofluid and heat transfer due to a permeable linear stretching sheet with the combined effects of an electric field, magnetic field, thermal radiation, viscous dissipation, and chemical reaction have been investigated. A similarity transformation is used to transform the constitutive equations into a system of nonlinear ordinary differential equations.The resultant system of equations is then solved numerically using implicit finite difference method.The velocity, temperature, concentration, entropy generation, and Bejan number are obtained with the dependence of different emerging parameters examined. It is noticed that the velocity is more sensible with high values of electric field and diminished with a magnetic field. The radiative heat transfer and viscous dissipation enhance the heat conduction in the system. Moreover, the impact of mixed convection parameter and Buoyancy ratio parameter on Bejan number profile has reverse effects. A chemical reaction reduced the nanoparticle concentration for higher values.     

Soret and Dufour effects on heat and mass transfer by mixed convection over a vertical surface saturated porous medium with temperature dependent viscosity

The diffusion‐thermo and thermal‐diffusion effects on heat and mass transfer by mixed convection boundary layer flow over a vertical isothermal permeable surface embedded in a porous medium were studied numerically in the presence of chemical reaction with temperature‐dependent viscosity. The governing nonlinear partial differential equations are transformed into a set of coupled ordinary differential equations, which are solved numerically by using Runge–Kutta method with shooting technique. Numerical results are obtained for the velocity, temperature and concentration distributions, and the local skin friction coefficient, local Nusselt number and local Sherwood number for several values of the parameters, namely, the variable viscosity parameter, suction/injection parameter, Darcy number, chemical reaction parameter, and Dufour and Soret numbers. The obtained results are presented graphically and in tabulated form, and the physical aspects of the problem are discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

Three-dimensional mixed convection squeezing flow

The unsteady mixed convection squeezing flow of an incompressible Newtonian fluid between two vertical parallel planes is discussed. The fluid is electrically conducting. The governing equations are transformed into ordinary differential equations (ODEs) by appropriate transformations. The transformed equations are solved successfully by a modern and powerful technique. The effects of the emerging parameters on the flow and heat transfer characteristics are studied and examined. The values of the skin friction coefficient and the local Nusselt number are tabulated and analyzed.     

Mixed convection boundary layers in the stagnation-point flow toward a stretching vertical sheet

An analysis is made for the steady mixed convection boundary layer flow near the two-dimensional stagnation-point flow of an incompressible viscous fluid over a stretching vertical sheet in its own plane. The stretching velocity and the surface temperature are assumed to vary linearly with the distance from the stagnation-point. Two equal and opposite forces are impulsively applied along the x-axis so that the wall is stretched, keeping the origin fixed in a viscous fluid of constant ambient temperature. The transformed ordinary differential equations are solved numerically for some values of the parameters involved using a very efficient numerical scheme known as the Keller-box method. The features of the flow and heat transfer characteristics are analyzed and discussed in detail. Both cases of assisting and opposing flows are considered. It is observed that, for assisting flow, both the skin friction coefficient and the local Nusselt number increase as the buoyancy parameter increases, while only the local Nusselt number increases but the skin friction coefficient decreases as the Prandtl number increases. For opposing flow, both the skin friction coefficient and the local Nusselt number decrease as the buoyancy parameter increases, but both increase as Pr increases. Comparison with known results is excellent.     

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     

Mixed convection boundary-layer flow on a horizontal flat surface with a convective boundary condition

The steady mixed convection boundary-layer flow on an upward facing horizontal surface heated convectively is considered. The problem is reduced to similarity form, a necessary requirement for which is that the outer flow and surface heat transfer coefficient are spatially dependent. The resulting similarity equations involve, apart from the Prandtl number, two dimensionless parameters, a measure of the relative strength of the outer flow M and a heat transfer coefficient γ. The free convection, M=0, case is considered with the asymptotic limits of large and small γ being derived. Results for the general, M>0, case are presented and the asymptotic limit of large M being treated.     

Slip effects on a mixed convection flow of a third-grade fluid near the orthogonal stagnation point on a vertical surface

A mixed convection flow of a third-grade fluid near the orthogonal stagnation point on a vertical surface with slip and viscous dissipation effects is investigated. The governing partial differential equations for the third-grade fluid are converted into a system of nonlinear ordinary differential equations by using a similarity transformation. The effects of various parameters, including the Weissenberg number, third-grade parameter, local Reynolds number, Prandtl number, Eckert number, mixed convection parameter, velocity slip, and thermal slip on the velocity and temperature profiles, local skin friction coefficient, and local Nusselt number are discussed.     

Mixed convection heat transfer over a non-linear stretching surface with variable fluid properties

This article presents a numerical solution for the steady two-dimensional mixed convection MHD flow of an electrically conducting viscous fluid over a vertical stretching sheet, in its own plane. The stretching velocity and the transverse magnetic field are assumed to vary as a power function of the distance from the origin. The temperature dependent fluid properties, namely, the fluid viscosity and the thermal conductivity are assumed to vary, respectively, as an inverse function of the temperature and a linear function of the temperature. A generalized similarity transformation is introduced to study the influence of temperature dependent fluid properties. The transformed boundary layer equations are solved numerically, using a finite difference scheme known as Keller Box method, for several sets of values of the physical parameters, namely, the stretching parameter, the temperature dependent viscosity parameter, the magnetic parameter, the mixed convection parameter, the temperature dependent thermal conductivity parameter and the Prandtl number. The numerical results thus obtained for the flow and heat transfer characteristics reveal many interesting behaviors. These behaviors warrant further study of the effects of the physical parameters on the flow and heat transfer characteristics. Here it may be noted that, in the case of the classical Navier-Stokes fluid flowing past a horizontal stretching sheet, McLeod and Rajagopal (1987) [42] showed that there exist an unique solution to the problem. This may not be true in the present case. Hence we would like to explore the non-uniqueness of the solution and present the findings in the subsequent paper.     

Mixed convection stagnation-point flow on vertical stretching sheet with external magnetic field

The problem of steady laminar magnetohydrodynamic （MHD） mixed con- vection stagnation-point flow of an incompressible viscous fluid over a vertical stretch- ing sheet is studied. The effect of an externally magnetic field is taken into account. The transformed boundary layer equations are solved numerically by using an implicit finite-difference scheme. Numerical results are obtained for various values of the mixed convection parameter, Hartmann number, and Prandtl number. The effects of an exter- nally magnetic field on the skin friction coefficient, local Nusselt number, velocity, and temperature profiles for both A 〉 1 and A ~ 1, where A is the velocity ratio parameter, are presented graphically and discussed in detail. Both assisting and opposing flows are considered, and it is found that dual solutions exist for the opposing flow.     

Mixed convection adjacent to inclined flat surfaces embedded in a porous medium

An analysis is performed to study the flow and heat transfer characteristics of laminar mixed convection boundary layer flows from inclined (including horizontal and vertical) surfaces embedded in a saturated porous medium with constant aiding external flows and uniform surface temperature. Both the streamwise and normal components of the buoyancy forces are retained in the momentum equations. Nondimensionalization of the boundary layer equations results in the following three governing parameter: (1)Gr/Re, the ratio of the Grashof number to the Reynolds number; (2)Pe _x =Re _x Pr, the Peclet number; (3) φ, the angle of inclination from the horizontal. The resulting nonsimilar equations are solved by an efficient implicit finite-difference scheme. Numerical results are presented for flows with different values ofGr/Re in the range of 0 to 50, over a wide range of the Peclet numbersPe _x, and various values of φ ranging from 0 to 90 degrees. It is found that the local surface heat transfer rate increases with increasing the local Peclet number. In addition, as the plate is tilted from the horizontal to the vertical orientation, the local Nusselt number increases for a given Peclet number and the effect of the buoyancy force on the surface heat transfer rate increases.     

Radiation effect on mixed convection from a horizontal surface in a porous medium

The effect of thermal radiation on the non-Darcy mixed convection flow over a non-isothermal horizontal surface immersed in a saturated porous medium has been studied. The wall temperature is assumed to have a power-law variation with the distance measured from the leading edge of the plate. The non-linear coupled parabolic partial differential equations governing the flow have been solved numerically using a finite-difference scheme. For some particular cases, the self-similar solution has also been obtained. The heat transfer is found to be strongly influenced by the radiative flux number, buoyancy parameter, variation of wall temperature, non-Darcy parameter and the nature of the free stream velocity.     

Free convection from an arbitrarily inclined plate in a porous medium

A mathematical model for the flow and heat transfer in the free convection from an arbitrary inclined isothermal flat plate embedded in a porous medium is presented, in which the Darcy–Boussinesq approximation is adopted to account for bouyancy force. A novel inclination parameter ξ is proposed such that all cases of the horizontal, inclined and vertical plates can be described by a single set of transformed boundary layer equations. Moreover, the similarity equations for the limiting cases of the horizontal and vertical plates are recovered from the transformed equations by setting ξ=0 and ξ=1, respectively. Detailed results for the skin friction coefficient and Nusselt number as well as for the dimensionless velocity and temperature profiles are presented for a wide range of the parameter ξ. A comparison with similarity solution shows excellent agreement.