MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical plate is studied numerically, taking into account the effects of Ohmic heating and viscous dissipation. A uniform magnetic field is applied perpendicular to the plate. The resulting governing equations are transformed into the non-similar boundary layer equations and solved using the Keller box method. Both the aiding-buoyancy mode and the opposing-buoyancy mode of the mixed convection are examined. The velocity and temperature profiles as well as the local skin friction and local heat transfer parameters are determined for different values of the governing parameters, mainly the magnetic parameter, the Richardson number, the Eckert number and the suction/injection parameter, f_w. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase owing to suction of fluid, increasing the Richardson number, Ri (i.e. the mixed convection parameter) or decreasing the Eckert number. This trend reverses for blowing of fluid and decreasing the Richardson number or decreasing the Eckert number. It is disclosed that the value of Ri determines the effect of the magnetic parameter on the momentum and heat transfer.     

Similarity solutions for mixed convection boundary layer flow over a permeable horizontal flat plate

The effects of suction and injection on steady laminar mixed convection boundary layer flow over a permeable horizontal flat plate in a viscous and incompressible fluid is investigated in this paper. The similarity solutions of the governing boundary layer equations are obtained for some values of the suction and injection parameter f₀, the constant exponent n 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 a finite-difference scheme known as the Keller-box method. Numerical results for the reduced skin friction coefficient, the reduced local Nusselt number, and the velocity and temperature profiles are obtained for various values of the parameters considered. Dual solutions are found to exist for the opposing flow.     

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.     

The study of Heat and Mass Transfer in a Visco elastic fluid due to a continuous stretching surface using Homotopy Analysis Method

In this paper, an approximate analytical solution is derived for the flow velocity and temperature due to the laminar, two-dimensional flow of non-Newtonian incompressible visco elastic fluid due to a continuous stretching surface. The surface is stretched with a velocity proportional to the distance $x$ along the surface. The surface is assumed to have either power-law heat flux or power-law temperature distribution. The presence of source/sink and the effect of uniform suction and injection on the flow are considered for analysis. An approximate analytical solution has been obtained using Homotopy Analysis Method(HAM) for various values of visco elastic parameter, suction and injection rates. Optimal values of the convergence control parameters are computed for the flow variables. It was found that the computational time required for averaged residual error calculation is very very small compared to the computation time of exact squared residual errors. The effect of mass transfer parameter, visco elastic parameter, source/sink parameter and the power law index on flow variables such as velocity, temperature profiles, shear stress, heat and mass transfer rates are discussed.     

Effect of suction/injection on free convection along a vertical plate in a nanofluid saturated non-Darcy porous medium with internal heat generation

The effect of internal heat generation on free convection along a vertical plate embedded in a nanofluid saturated non-Darcy porous medium in the presence of suction/injection is analyzed. The non-linear governing equations and their associated boundary conditions are initially cast into dimensionless forms by non-dimensional variables. The resulting equations are solved numerically by an accurate, implicit, iterative finite-difference methodology and the obtained results are compared favorably with previously published work. A parametric study is performed to illustrate influence of the temperature exponent, non-Darcy, suction/injection, Brownian motion and thermophoresis parameters on the profiles of the velocity components, temperature and nanoparticle volume fraction. The numerical data for the heat and nanoparticle mass transfer rates have been tabulated for various parametric conditions.     

热幅射对抽吸或喷射三维Couette流温度分布的影响

在固定的底板上有横向正弦射流,而匀速运动的多孔介质顶板以常速率完全抽出的情况下,理论分析了热幅射对三维Couette流动温度分布的影响.在这种射流速度下,流动呈现三维流动.利用图形分析了Prandtl数、幅射参数和射流参数对传热速率的影响.Prandtl数对温度分布的影响比射流参数或幅射参数大得多.     

Numerical study of mixed convection flow of a micropolar fluid towards permeable vertical plate with convective boundary condition

In this article, the mixed convective flow of a micropolar fluid along a permeable vertical plate under the convective boundary condition is analyzed. The scaling group of transformations is applied to get the similarity representation of the system of partial differential equations of the problem and then the resulting equations are solved by using Spectral Quasi-Linearisation Method. This study reveals that the dual solutions exists for certain values of mixed convection parameter. The outcomes are analyzed with dual solutions in detail. Effects of micropolar parameter, Biot number and suction/injection parameters on different flow profiles are discussed and depicted graphically.     

Dual solutions in mixed convection boundary-layer flow with suction or injection

The effect of suction/injection on the laminar mixed convectionboundary-layer flow about a vertical wall in an incompressibleviscous fluid is considered. The similarity solutions are obtainedfor some values of the suction/injection parameter as well asthe mixed convection parameter for three particular cases: uniformtemperature, uniform heat flux and stagnation flow. The resultingsystem of non-linear ordinary differential equations is solvednumerically for both assisting and opposing flow regimes usinga finite-difference scheme known as the Keller box method. Numericalresults are obtained for the skin friction coefficient and localNusselt number as well as velocity and temperature profiles.The effects of the involved parameters on the skin frictioncoefficient and the local Nusselt number characteristics arediscussed. It is found that dual solutions exist for assistingflow, besides that usually reported in the literature for opposingflow.     

Effects of chemical reaction and variable viscosity on hydromagnetic mixed convection heat and mass transfer for Hiemenz flow through porous media with radiation

The effect of chemical reaction and variable viscosity on hydromagnetic mixed convection heat and mass transfer for Hiemenz flow through porous media has been studied in the presence of radiation and magnetic field. The plate surface is embedded in a uniform Darcian porous medium in order to allow for possible fluid wall suction or blowing and has a power-law variation of both the wall temperature and concentration. The similarity solution is used to transform the system of partial differential equations, describing the problem under consideration, into a boundary value problem of coupled ordinary differential equations, and an efficient numerical technique is implemented to solve the reduced system. Numerical calculations are carried out, for various values of the dimensionless parameters of the problem, which include a variable viscosity, chemical reactions, radiation, magnetic field, porous medium and power index of the wall temperature parameters. Comparisons with previously published works are performed and excellent agreement between the results is obtained. The results are presented graphically and the conclusion is drawn that the flow field and other quantities of physical interest are significantly influenced by these parameters.     

Heat and mass transfer in MHD free convection from a moving permeable vertical surface by a perturbation technique

Numerical results are presented for heat and mass transfer effect on hydromagnetic flow of a moving permeable vertical surface. An analysis is performed to study the momentum, heat and mass transfer characteristics of MHD natural convection flow over a moving permeable surface. The surface is maintained at linear temperature and concentration variations. The non-linear coupled boundary layer equations were transformed and the resulting ordinary differential equations were solved by perturbation technique [Aziz A, Na TY. Perturbation methods in heat transfer. Berlin: Springer-Verlag; 1984. p. 1–184; Kennet Cramer R, Shih-I Pai. Magneto fluid dynamics for engineers and applied physicists 1973;166–7]. The solution is found to be dependent on several governing parameter, including the magnetic field strength parameter, Prandtl number, Schmidt number, buoyancy ratio and suction/blowing parameter, a parametric study of all the governing parameters is carried out and representative results are illustrated to reveal a typical tendency of the solutions. Numerical results for the dimensionless velocity profiles, the temperature profiles, the concentration profiles, the local friction coefficient and the local Nusselt number are presented for various combinations of parameters.     

流经有热辐射的无限垂直多孔平板时非稳定MHD流动的Soret和Dufour效应

不可压缩粘性导电流体,流经无限垂直多孔平板,平板存在振荡吸入速度和热辐射时,研究流动参数对自由对流和传质的非稳定磁流体动力学流动的Dufour(扩散热)和Soret(热扩散)效应.应用有限单元法,数值求解该问题的速度、温度和浓度场,还得到了表面摩擦、传热传质率的表达式.数值结果以图表方式给出,对外表致冷的平板(Gr0)和外表加热的平板(Gr0),给出了该方程中所遇参数的影响.     

传导-辐射和磁流体组合对磁化可渗透平板自然对流影响的数值解法

研究粘性、不可压缩、导电流体,在磁化可渗透竖直平板上作自然对流时,数值地分析辐射和磁流体组合的影响.采用两种方法数值地求解非相似的控制方程:1)对所有吸入参数值ξ,采用有限差分法;2)为小数值和大数值的吸入参数值ξ,分别建立起级数的渐近解.用图形和列表形式,给出Prandtl数Pr,磁Prandtl数Pm,磁力参数S,辐射参数Rd,壁面温度θw的变化,对壁面摩擦因数、热交换率和电流密度的影响.最后分析上述物理参数对速度分布、温度分布和磁场横向分量的影响.     

Spatial Stability for Mixed Convection Boundary Layer over a Heated Horizontal Plate

The spatial stability properties of a mixed convection boundary layer developing over a heated horizontal plate is studied here under linear and quasi-parallel flow assumption. The main aim of the present work is to find out if there is a critical buoyancy parameter that would indicate the importance of heat transfer in destabilizing mixed convection boundary layers, when the buoyancy effect is given by Boussinesq approximation. The undisturbed flow used here is that given by the similarity solution of [ 1 ] that implies the wall temperature to vary as the inverse square root of the distance from the leading edge of the plate. The stability of this flow has been investigated by using the compound matrix method (CMM)—that allows finding all the modes in the chosen range in the complex wave number plane for spatial stability analysis. Presented neutral curves for mixed convection boundary layer show the existence of two types of disturbances present simultaneously, for large buoyancy parameter. One notices very unstable high-frequency mode when the buoyancy parameter exceeds the above-mentioned critical value. This unstable thermal mode is in addition to the hydrodynamic mode of isothermal flow given by corresponding similarity profile. The calculated critical buoyancy parameter is shown to qualitatively match with experimental results.     

Effect of pressure work on free convection flow along a vertical circular cone with suction and variable surface temperature

Free convection laminar flow from a vertical circular cone maintained a variable surface temperature with suction and pressure work effects has been investigated. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equations are reduced to local non-similarity equations. The governing non-similarity equations are then solved numerically by implicit finite difference method together with Keller box scheme. Numerical results are presented in terms of velocity and temperature profiles of the fluid as well as the local skin-friction coefficients and the local heat transfer rate for different values of Prandtl number Pr, suction parameter ξ, temperature gradient parameter n and the pressure work parameter ∈. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convective heat and mass transfer in three-dimensional mixed convection flow of viscoelastic fluid in presence of chemical reaction and heat source/sink

Heat and mass transfer effects in the three-dimensional mixed convection flow of a viscoelastic fluid with internal heat source/sink and chemical reaction have been investigated in the present work. The flow generation is because of an exponentially stretching surface. Magnetic field normal to the direction of flow is considered. Convective conditions at the surface are also encountered. Appropriate similarity transformations are utilized to reduce the boundary layer partial differential equations into the ordinary differential equations. The homotopy analysis method is used to develop the solution expressions. Impacts of different controlling parameters such as ratio parameter, Hartman number, internal heat source/sink, chemical reaction, mixed convection, concentration buoyancy parameter and Biot numbers on the velocity, temperature and concentration profiles are analyzed. The local Nusselt and Sherwood numbers are sketched and examined.     

Heat transfer over an unsteady stretching permeable surface with prescribed wall temperature

The unsteady laminar boundary layer flow over a continuously stretching permeable surface is investigated. The unsteadiness in the flow and temperature fields is caused by the time-dependence of the stretching velocity and the surface temperature. Effects of the unsteadiness parameter, suction/injection parameter and Prandtl number on the heat transfer characteristics are thoroughly examined.     

Unsteady mixed convection heat transfer along a vertical stretching surface with variable viscosity and viscous dissipation

The effect of variable viscosity on laminar mixed convection flow and heat transfer along a semi-infinite unsteady stretching sheet taking into account the effect of viscous dissipation is studied. The flow of the fluid and subsequent heat transfer from the stretching surface is investigated with the aid of suitable transformation variables. Solutions for the velocity and temperature fields are obtained for some representative values of the unsteadiness parameter, variable viscosity parameter, mixed convection parameter and Eckert number. Typical velocity and temperature profiles, the local skin friction coefficient and the local heat transfer rate are presented at selected controlling parameters.     

Effect of sudden suction on the hydromagnetic natural convection from a vertical plate

The effects of step function change in suction velocity are investigated for a natural convection flow from a vertical porous flat plate of infinite length in the presence of transverse magnetic field for two cases namely (i) when the plate is suddenly raised to a uniform higher temperature, (ii) when the plate suddenly begins to generate a uniform heat flux at its surface. In (i) the coefficient of heat transfer becomes independent of the Hartmann numberM. In either case for a fixed time t except for correct steady state t→ ∞, the effect of the Hartmann number is to decrease both the velocity of the fluid and the skin-friction at the plate. In the correct steady state the velocity field and the skin-fiiction become independent of the Hartmann number.     

Mixed convection of a viscous dissipating fluid about a vertical flat plate

In this study, the effect of the viscous dissipation in steady, laminar mixed convection heat transfer from a heated/cooled vertical flat plate is investigated in both aiding and opposing buoyancy situations. The external flow field is assumed to be uniform. The governing systems of partial differential equations are solved numerically using the finite difference method. A parametric study is performed in order to illustrate the interactive influences of the governing parameters, mainly, the Richardson number, Ri (also known as the mixed convection parameter) and the Eckert number, Ec on the velocity and temperature profiles as well as the friction and heat transfer coefficients. Based on the facts the free stream is either in parallel or reverse to the gravity direction and the plate is heated or cooled, different flow situations are identified. The influence of the viscous dissipation on the heat transfer varied according to the situation. For some limiting cases, the obtained results are validated by comparing with those available from the existing literature. An expression correlating Nu in terms of Pr, Ri and Ec is developed.     

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.