Self-similar solution of the unsteady flow in the stagnation point region of a rotating sphere with a magnetic field

The unsteady flow and heat transfer of a viscous incompressible electrically conducting fluid in the forward stagnation point region of a rotating sphere in the presence of a magnetic field are investigated in this study. The unsteadiness in the flow field is caused by the velocity at the edge of the boundary layer and the angular velocity of the rotating sphere, both varying continuously with time. The system of ordinary differential equations governing the flow is solved numerically. For some particular cases, an analytical solution is also obtained. It is found that the surface shear stresses in x- and y-directions and the surface heat transfer increase with the acceleration, the magnetic and the rotation parameters whether the magnetic field is fixed relative to the fluid or body, except that the surface shear stress in x-direction and the surface heat transfer decrease with increasing the magnetic parameter when the magnetic field is fixed relative to the body. For a certain value of the acceleration parameter, the surface shear stress in the x-direction vanishes while the surface shear stress in the y-direction and the surface heat transfer remain finite. Also, below a certain value of the acceleration parameter, reverse flow occurs in the x-component of the velocity profile. Received on 18 May 1998     

Unsteady laminar MHD flow and heat transfer in the stagnation region of an impulsively spinning and translating sphere in the presence of buoyancy forces

 An analysis has been carried out to determine the development of momentum and heat transfer occurring in the laminar boundary layer of an incompressible viscous electrically conducting fluid in the stagnation region of a rotating sphere caused by the impulsive motion of the free stream velocity and the angular velocity of the sphere. At the same time the wall temperature is also suddenly increased. This analysis includes both short and long-time solutions. The partial differential equations governing the flow are solved numerically using an implicit finite-difference scheme. There is a smooth transition from the short-time solution to the long-time solution. The surface shear stresses in the longitudinal and rotating directions and the heat transfer are found to increase with time, magnetic field, buoyancy parameter and the rotation parameter. Received on 27 January 2000     

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.     

Influence of lateral mass flux on mixed convection heat and mass transfer over inclined surfaces in porous media

 The effect of lateral mass flux on mixed convection heat and mass transfer in a saturated porous medium adjacent to an inclined permeable surface is analyzed. A similarity solution is obtained when surface temperature and concentration, free stream velocity and injection/suction velocity of fluid are prescribed as power functions of distance from the leading edge. The cases when the flow and buoyancy forces are in the same and opposite directions are discussed both for aiding and opposing buoyancy effects. The governing parameters are the mixed convection parameter Gr, the Lewis number Le, the buoyancy ratio N, the lateral mass flux parameter f _w, representing the effects of injection or withdrawal of fluid at the wall, and λ which specifies three cases of the inclined plate. The interactive effect of these parameters on heat and mass transfer rates are presented. It is observed that the diffusion ratio (Le) has a more pronounced effect on concentration field than on flow and temperature fields. It is found that the rates of heat and mass transfer increase with suction and decrease with injection of the fluid. Received on 31 August 2000 / Published online: 29 November 2001     

Unsteady mixed convection on the stagnation-point flow adjacent to a vertical plate with a magnetic field

An analysis is performed to study the unsteady combined forced and free convection flow (mixed convection flow) of a viscous incompressible electrically conducting fluid in the vicinity of an axisymmetric stagnation point adjacent to a heated vertical surface. The unsteadiness in the flow and temperature fields is due to the free stream velocity, which varies arbitrarily with time. Both constant wall temperature and constant heat flux conditions are considered in this analysis. By using suitable transformations, the Navier–Stokes and energy equations with four independent variables (x, y, z, t) are reduced to a system of partial differential equations with two independent variables (, ). These transformations also uncouple the momentum and energy equations resulting in a primary axisymmetric flow, in an energy equation dependent on the primary flow and in a buoyancy-induced secondary flow dependent on both primary flow and energy. The resulting system of partial differential equations has been solved numerically by using both implicit finite-difference scheme and differential-difference method. An interesting result is that for a decelerating free stream velocity, flow reversal occurs in the primary flow after certain instant of time and the magnetic field delays or prevents the flow reversal. The surface heat transfer and the surface shear stress in the primary flow increase with the magnetic field, but the surface shear stress in the buoyancy-induced secondary flow decreases. Further the heat transfer increases with the Prandtl number, but the surface shear stress in the secondary flow decreases.     

Unsteady compressible boundary layer flow over a circular cone near a plane of symmetry

An analysis has been performed to study the unsteady laminar compressible boundary layer governing the hypersonic flow over a circular cone at an angle of attack near a plane of symmetry with either inflow or outflow in the presence of suction. The flow is assumed to be steady at time t=0 and at t>0 it becomes unsteady due to the time-dependent free stream velocity which varies arbitrarily with time. The nonlinear coupled parabolic partial differential equations under boundary layer approximations have been solved by using an implicit finite-difference method. It is found that suction plays an important role in stabilising the fluid motion and in obtaining unique solution of the problem. The effect of the cross flow parameter is found to be more pronounced on the cross flow surface shear stress than on the streamwise surface shear stress and surface heat transfer. Beyond a certain value of the cross flow parameter overshoot in the cross flow velocity occurs and the magnitude of this overshoot increases with the cross flow parameter. The time variation of the streamwise surface shear stress is more significant than that of the cross flow surface shear stress and surface heat transfer. The suction and the total enthalpy at the wall exert strong influence on the streamwise and cross flow surface shear stresses and the surface heat transfer except that the effect of suction on the cross flow surface shear stress is small.     

Laminar Mixed Convection Adjacent to Vertical Continuously Stretching Sheets With Variable Viscosity and Variable Thermal Diffusivity

The paper presents a study of the laminar mixed convection adjacent to vertical continuously stretching sheets, taking into account the effects of variable viscosity and variable thermal diffusivity. The similarity solutions are reported for isothermal sheet moving with a velocity of the form u_w=Bx^0.5 and a continuous linearly stretching sheet with a linear surface temperature distribution. The equations of conservation of mass, momentum and energy, which govern the flow and heat transfer, are solved numerically by using the shooting method. The numerical results obtained for the flow and heat transfer characteristics reveal many interesting behaviors. The numerical results show that, variable viscosity, variable thermal diffusivity, the velocity exponent parameter, the temperature exponent parameter and the buoyancy force parameter have significant influences on the velocity and temperature profiles, shear stress and Nusselt number in two cases air and water.     

Three dimensional free convection flow and heat transfer along a porous vertical plate

The free convection flow along a vertical porous plate with transverse sinusoidal suction velocity distribution is investigated. Due to this type of suction velocity at the plate the flow becomes three dimensional one. For the asymptotic flow condition, the wall shear stress in the direction of main flow for different values of buoyancy parameter G is obtained. For G=0, the skin friction in the direction of free stream and the rate of heat transfer from the plate to the fluid are given. It is found that these results differ from those obtained by Gersten and Gross.     

Development of mixed convection flow over a vertical plate due to an impulsive motion

The development of the mixed convection flow of an incompressible laminar viscous fluid over a semi-infinite vertical plate has been investigated when the fluid in the external stream is set into motion impulsively, and at the same the surface temperature is suddenly raised from its ambient temperature. The problem is formulated in such a way that at time t = 0, it reduces to Rayleigh type of equation and as time t , it tends to Blasius type of equation. The scale of time has been selected such that the traditional infinite region of integration becomes finite which significantly reduces the computational time. The nonlinear coupled singular parabolic partial differential equations governing the unsteady mixed convection flow have been solved numerically by using an implicit finite-difference scheme. The surface shear stress and the heat transfer increase or decrease with time when the buoyancy parameter is greater or less than a certain valve. There is a smooth transition from the initial steady state to the final steady state. The skin friction and heat transfer for the constant heat flux case are more than those of the constant wall temperature case. Also they increase with the buoyancy force.     

Mixed convection cooling of a heated, continuously stretching surface

 An numerical study of the flow and heat transfer characteristics associated with a heated, continuously stretching surface being cooled by a mixed convection flow has been carried out. The relevant heat transfer mechanisms are of interest in a wide variety of practical applications, such as hot rolling, continuous casting, extrusion, and drawing. The surface velocity of the continuously stretching sheet was assumed to vary according to a power-law form, that is, u _w (x)=Cx ^p . Two conditions of surface heating were considered, a variable wall temperature (VWT) in the form T _w (x)−T _∞=Ax ⁿ and a variable surface heat flux (VHF) in the form q _w (x)=Bx ^m . The governing differential equations are transformed by introducing proper nonsimilarity variables and solved numerically using a procedure based on finite difference approximations. Results for the local Nusselt number and the local friction coefficient are obtained for a wide range of governing parameters, such as the surface velocity parameter p, the wall temperature exponent n, the surface heat flux exponent m, the buoyancy force parameters (ξ for the VWT case and χ for the VHF case), and Prandtl number of the fluid. It is found that the local Nusselt number is increased with increasing the velocity exponent parameter p for the VWT case, while the opposite trend is observed for the VHF case. The local friction coefficient is increased for a decelerated stretching surface, while it is decreased for an accelerated stretching surface. Also, appreciable effects of the buoyancy force on the local Nusselt number and the local friction coefficient are observed for both VWT and VHF cases, as expected. Received on 11 January 1999     

Combined forced and free convection in an axisymmetric stagnation flow on a vertical non-isothermal cylinder

Boundary layer solutions are presented to study the effects of buoyancy on forced convection in an axisymmetric stagnation flow over a vertical cylinder with arbitrary surface heat flux variations. Numerical solutions are given for the governing momentum and energy equations. Two flow regions, namely, the buoyancy-assisted and buoyancy-opposed cases are analyzed. It is observed that the wall shear stress and surface heat transfer rate increase or decrease with the buoyancy force parameter depending upon the flow regime being buoyancy-assisted or buoyancy-opposed.     

Thermal dispersion effects on non-Darcy natural convection over horizontal plate with surface mass flux

Summary The effect of surface mass flux on the non-Darcy natural convection over a horizontal flat plate in a saturated porous medium is studied using similarity solution technique. Forchheimer extension is considered in the flow equations. The suction/injection velocity distribution has been assumed to have power function form Bx ^l , similar to that of the wall temperature distribution Ax ⁿ , where x is the distance from the leading edge. The thermal diffusivity coefficient has been assumed to be the sum of the molecular diffusivity and the dynamic diffusivity due to mechanical dispersion. The dynamic diffusivity is assumed to vary linearly with the velocity component in the x direction, i.e. along the hot wall. For the problem of constant heat flux from the surface (n=1/2), similarity solution is possible when the exponent l takes the value −1/2. Results indicate that the boundary layer thickness decreases whereas the heat transfer rate increases as the mass flux parameter passes from the injection domain to the suction domain. The increase in the thermal dispersion parameter is observed to favor the heat transfer by reducing the boundary layer thickness. The combined effect of thermal dispersion and fluid suction/injection on the heat transfer rate is discussed. Received 7 December 1995; accepted for publication 7 January 1997     

Numerical solutions of free convection boundary layer flow on a solid sphere with Newtonian heating in a micropolar fluid

In this paper, the problem of free convection boundary layer flow on a solid sphere in a micropolar fluid with Newtonian heating, in which the heat transfer from the surface is proportional to the local surface temperature, is considered. The transformed boundary layer equations in the form of partial differential equations are solved numerically using an implicit finite-difference scheme. Numerical solutions are obtained for the local wall temperature, the local skin friction coefficient, as well as the velocity, angular velocity and temperature profiles. The features of the flow and heat transfer characteristics for different values of the material or micropolar parameter K, the Prandtl number Pr and the conjugate parameter γ are analyzed and discussed.     

Combined heat and mass transfer along a vertical moving cylinder with a free stream

The mixed convection flow over a continuous moving vertical slender cylinder under the combined buoyancy effect of thermal and mass diffusion has been studied. Both uniform wall temperature (concentration) and uniform heat (mass) flux cases are included in the analysis. The problem is formulated in such a manner that when the ratio λ(= u _w/(u _w + u _∞), where u _w and u _∞ are the wall and free stream velocities, is zero, the problem reduces to the flow over a stationary cylinder, and when λ = 1 it reduces to the flow over a moving cylinder in an ambient fluid. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. We have also obtained the solution using a perturbation technique with Shanks transformation. This transformation has been used to increase the range of the validity of the solution. For some particular cases closed form solutions are obtained. The surface skin friction, heat transfer and mass transfer increase with the buoyancy forces. The buoyancy forces cause considerable overshoot in the velocity profiles. The Prandtl number and the Schmidt number strongly affect the surface heat transfer and the mass transfer, respectively. The surface skin friction decreases as the relative velocity between the surface and free stream decreases. Received on 17 May 1999     

Natural convection flow from a continuously moving vertical surface immersed in a thermally stratified medium

 Natural convection boundary layer flow over a continuously moving isothermal vertical surface immersed in a thermally stratified medium has been investigated here. The non-linear coupled partial differential equations governing the non-similar flow have been solved numerically using an implicit finite difference scheme. For small values of the streamwise distance the partial differential equations are solved by using a perturbation expansion procedure and also using the Shanks transformation. The results indicate that the thermal stratification significantly affects both the surface shear stress and the surface heat transfer. The buoyancy parameter and the Prandtl number increase significantly, both the surface shear stress and heat transfer. Also the buoyancy force gives rise to an overshoot in the velocity profile. Received on 1 February 2000     

Simultaneous effect of rotation and natural convection on the flow about a liquid sphere

The problem of mixed convection around a liquid sphere that experiences a rotation about its axis parallel to the free stream is studied numerically using a finite- difference technique. The coupled boundary-layer energy and momentum equations are numerically solved over a wide range of Grashof number that represents the cases of aiding and opposing free convection and for wide range of the spin parameter Ta/Re². The surface of the sphere also rotates as a result of the shear stress exerted from the external flow of air. The effect of both parameters on the velocity components as well as the temperature within the thermal boundary-layer is presented. Results show that increasing the aiding free convection and the spin parameter cause increases in the shear stress and the local heat transfer coefficient.     

Investigation of a hypersonic viscous shock layer on rotating axisymmetric bodies in the presence of blowing

The hypersonic flow of a laminar stream of viscous compressible gas past blunt axisyrametric bodies rotating about the longitudinal axis is considered. It is assumed that gas blows from the surface of the body. The solution of the problem is obtained by a finite-difference method in a wide range of Reynolds numbers and blowing and rotation parameters. Some results of the calculations characterizing the effect of the rotation on the velocity and temperature profiles across the shock layer, on the friction and heat transfer coefficients, and the shock wave separation are given for the neighborhood of the stagnation point. For large Reynolds numbers and strong blowing an analytic solution of the problem is found in an approximation of two inviscid layers separated by a contact surface. The calculations are made for the flow past a sphere and a paraboloid and it is shown that in the presence of rotation the maximum of the heat flux is shifted from the stagnation point onto the side surface of the body. The dependence of the pressure distribution, the heat flux, and the friction coefficient is investigated for cases of constant and variable blowing over the contour of the body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 106–114, January–February, 1986.     

Mixed Convection Stagnation-Point Flow Over a Vertical Plate with Prescribed Heat Flux Embedded in a Porous Medium: Brinkman-Extended Darcy Formulation

This article considers the problem of mixed convection stagnation-point flow towards a vertical plate embedded in a porous medium with prescribed surface heat flux. It is assumed that the free stream velocity and the surface heat flux vary linearly from the stagnation point. Using a similarity transformation, the governing system of partial differential equations is transformed into a system of ordinary differential equations, before being solved numerically by a finite-difference method. The features of the flow and the heat transfer characteristics are analyzed and discussed. It is found that dual solutions exist for both buoyancy assisting and opposing flows.     

A similarity solution for the flow and heat transfer over a moving permeable flat plate in a parallel free stream

This paper presents both a numerical and analytical study in connection with the steady boundary layer flow and heat transfer induced by a moving permeable semi-infinite flat plate in a parallel free stream. Both the velocities of the flat plate and the free stream are proportional to x ^1/3. The surface temperature is assumed to be constant. The governing partial differential equations are converted into ordinary differential equations by a new similarity transformation. Numerical results for the flow and heat transfer characteristics are obtained for various values of the moving parameter, transpiration parameter and the Prandtl number. Approximate analytical solutions are also obtained when the suction or injection parameter is very large. It is found that dual solutions exist for the case when the fluid and the plate move in the opposite directions.     

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.