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.     

Mixed convection flow of a micropolar fluid from an unsteady stretching surface with viscous dissipation

An analysis is performed to study the unsteady mixed convection flow of a viscous incompressible micropolar fluid adjacent to a heated vertical surface in the presence of viscous dissipation when the buoyancy force assists or opposes the flow. The flow of the fluid and subsequent heat transfer from the stretching surface is investigated with the aid of appropriate transformation variables. The effect of the governing parameters on the flow and heat transfer characteristics as well as the local skin friction coefficient, wall couple stress and the heat transfer rate are thoroughly examined.     

Mixed convection of heat generating nanofluid in a lid-driven cavity with uniform and non-uniform heating of bottom wall

Steady state two-dimensional mixed convection in a lid-driven square cavity filled with Cu  –water nanofluid is investigated numerically in the presence of internal heat generation. In the present investigation, bottom wall is uniformly and non-uniformly heated while two vertical walls are fixed and they are thermally insulated. The top wall is moving from left to right at a constant speed. The governing equations are normalized and solved numerically with boundary conditions by finite volume approach using third order accurate upwinding scheme (deferred QUICK). Effects of the pertinent physical parameters are investigated in terms of the flow and temperature fields, as well as Nusselt number distributions. The presented results show that the solid volume fraction plays a significant role on the flow and thermal fields and the Nusselt number distributions for different flow configurations. It is found that Richardson number strongly affect the fluid flow and heat transfer in the cavity. For Ri<1$\mathit{Ri}<1$, the forced convection becomes dominant in the entire cavity, the natural convection relatively weak.     

MHD-mixed convection from a vertical slender cylinder

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical slender cylinder is studied numerically. A uniform magnetic field is applied perpendicular to the cylinder. The resulting governing equations are transformed into the non-similar boundary layer equations and solved using the Keller box method. The velocity and temperature profiles as well as the local skin friction and the local heat transfer parameters are determined for different values of the governing parameters, mainly the transverse curvature parameter, the magnetic parameter, the electric field parameter and the Richardson number. 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, increasing the Richardson number, Ri (i.e. the mixed convection parameter), electric field parameter E₁ and magnetic parameter Mn.     

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.     

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.     

Non-Darcy unsteady mixed convection flow near the stagnation point on a heated vertical surface embedded in a porous medium with thermal radiation and variable viscosity

The unsteady mixed convection boundary layer flow near the stagnation point on a heated vertical plate embedded in a fluid saturated porous medium is studied. It is assumed that the unsteadiness is caused by the impulsive motion of the free stream velocity and by sudden increase in the surface temperature. Both the buoyancy assisting and the buoyancy opposing flow situations are considered with combined effects of the first and second order resistance of solid matrix of non-Darcy porous medium, variable viscosity and radiation. The problem is reduced to a system of non-dimensional partial differential equations, which is solved numerically using the Keller-box method. The features of the flow and the heat transfer characteristics for different values of the governing parameters are analyzed and discussed. The surface shear stress and the heat transfer of the present study are compared with the available results and a good agreement is found.     

Numerical study of viscous dissipation effect on free convection heat and mass transfer of MHD non-Newtonian fluid flow through a porous medium

The problem of free convection heat with mass transfer for MHD non-Newtonian Eyring–Powell flow through a porous medium, over an infinite vertical plate is studied. Taking into account the effects of both viscous dissipation and heat source. The temperature and concentration are of periodic variation. The governing non-linear partial differential equations of this phenomenon are transformed into non-linear algebraic system utilizing finite difference method. Numerical results for the velocity, temperature and concentration distributions as well as the skin friction, heat and mass transfer are obtained and reported in tabular form and graphically for different values of physical parameters of the problem. Also, the stability condition is studied.     

Temperature distribution in Couette flow past a permeable bed

The temperature distribution in a steady plane Couette flow having one permeable bounding wall is investigated in the presence of buoyancy forceN ₀ whenN ₀>0, it is shown that heat is transported both by convection and diffusion. The effect of convection is to increase the magnitude of the temperature distribution both in the free and Darcy flows. In particular, it is shown that the wall shear has no significant effect on the temperature distribution. The rate of heat transfer between the fluid and the surface is also calculated and it is shown that, it increases with the porous parameterσ. Although the viscous dissipation has very little effect on the temperature distribution yet its effect is significant on 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.     

Nonsimilar solutions for unsteady mixed convection from a moving vertical plate

An analysis is performed to obtain the nonsimilar solution of an unsteady laminar mixed convection on a continuously moving vertical plate by taking into account the effect of viscous dissipation. The unsteadiness is introduced by the time-dependent velocity of the vertical plate as well as that of the free stream. The governing boundary layer equations along with the boundary conditions are first cast into a dimensionless form by a nonsimilar transformation and the resulting system of nonlinear coupled partial differential equations is then solved by an implicit finite difference scheme in combination with the quasilinearization technique. The effects of various parameters on velocity and temperature profiles and on skin friction coefficients and heat transfer rate at the wall are reported in the present study.     

竖板表面温度的变化对磁流体动力学流动的影响

在横向磁场作用下,不可压缩的粘性导电流体,流经一个半无限的竖板,完成了壁面温度变化对磁流体动力学流动的分析.假定由粘性耗散和感应磁场产生的热量可以忽略不计.无量纲的控制方程为二维非稳态耦合的非线性方程.结果显示,磁场参数对空气和水的速度有着抑制作用.     

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.     

Nonsimilar incompressible laminar boundary layers with magnetic field

The solution of the steady laminar incompressible nonsimilar magneto-hydrodynamic boundary layer flow and heat transfer problem with viscous dissipation for electrically conducting fluids over two-dimensional and axisymmetric bodies with pressure gradient and magnetic field has been presented. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. The computations have been carried out for flow over a cylinder and a sphere. The results indicate that the magnetic field tends to delay or prevent separation. The heat transfer strongly depends on the viscous dissipation parameter. When the dissipation parameter is positive (i.e. when the temperature of the wall is greater than the freestream temperature) and exceeds a certain value, the hot wall ceases to be cooled by the stream of cooler air because the ‘heat cushion’ provided by the frictional heat prevents cooling whereas the effect of the magnetic field is to remove the ‘heat cushion’ so that the wall continues to be cooled. The results are found to be in good agreement with those of the local similarity and local nonsimilarity methods except near the point of separation, but they are in excellent agreement with those of the difference-differential technique even near the point of separation.     

The effect of conjugate heat transfer on MHD mixed convection about a vertical slender hollow cylinder

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical slender hollow cylinder is studied numerically, under the effect of wall conduction. A uniform magnetic field is applied perpendicular to the cylinder. The non-similar solutions using the Keller box method are obtained. The wall conduction parameter, the magnetic parameter and the Richardson number are the main parameters. For various values of these parameters the local skin friction and local heat transfer parameters are determined. The validity of the methodology is checked by comparing the results with those available in the open literature and a fairly good agreement is observed. Finally, it is determined that the local skin friction and the local heat transfer coefficients increase with an increase the magnetic parameter Mn and buoyancy parameter Ri and decrease with conjugate heat transfer parameter p.     

Viscous dissipation and Joule heating effects on MHD-free convection from a vertical plate with power-law variation in surface temperature in the presence of Hall and ion-slip currents

An analysis is presented to study the effects of viscous dissipation and Joule heating on MHD-free convection flow past a semi-infinite vertical flat plate in the presence of the combined effect of Hall and ion-slip currents for the case of power-law variation of the wall temperature. The fluid is permeated by a strong transverse magnetic field imposed perpendicularly to the plate on the assumption of a small magnetic Reynolds number. The governing differential equations are transformed by introducing proper non-similarity variables and solved numerically. The effects of various parameters on the velocity and temperature profiles as well as the local wall shear stresses and the local Nusselt number are presented graphically and in tabular form. It is found that the magnetic field acts as a retarding force on the tangential flow but has a propelling effect on the induced lateral flow. The skin-friction factor for the tangential flow and the local Nusselt number decrease but the skin-friction factor for the lateral flow increases as the magnetic field increases. The skin-friction factor for the tangential and lateral flows are increased while the local Nusselt number is decreased if the effect of viscous dissipation, Joule heating and heat generation are considered. The opposite trend was observed as the temperature power coefficient n is increased. Also, the skin-friction factor for the tangential flow and the local Nusselt number are increased due to the Hall and ion-slip currents, whereas the skin-friction factor for the tangential flow increases when Hall values increase to one and decreases for values of Hall greater than one, but reduces by rising ion-slip values.     

Effect of viscous dissipation on natural convection flow between vertical parallel plates with time-periodic boundary conditions

This article investigates the natural convection flow of viscous incompressible fluid in a channel formed by two infinite vertical parallel plates. Fully developed laminar flow is considered in a vertical channel with steady-periodic temperature regime on the boundaries. The effect of internal heating by viscous dissipation is taken into consideration. Separating the velocity and temperature fields into steady and periodic parts, the resulting second order ordinary differential equations are solved to obtain the expressions for velocity, and temperature. The amplitudes and phases of temperature and velocity are also obtained as well as the rate of heat transfer and the skin-friction on the plates. In presence of viscous dissipation, fluids of relatively small Prandtl number has higher temperature than the channel plates and as such, heat is being transferred from the fluid to the plate.     

Flow and heat transfer for a power-law electrically conducting fluid flowing between parallel plates under transverse magnetic field with viscous dissipation

The flow and heat transfer problem with viscous dissipation for electrically conducting non-Newtonian fluids with power-law model in the thermal entrance region of two parallel plates with magnetic field under constant heat flux and constant wall temperature conditions has been studied. The governing equations have been solved numerically using quasilinearization technique and implicit finite-difference scheme. It has been found that the effect of viscous dissipation on heat transfer is quite significant for heating and cooling conditions at the wall.     

Effect of viscous dissipation on natural convection heat and mass transfer from vertical cone in a non-Newtonian fluid saturated non-Darcy porous medium

In this paper we investigate the influence of viscous dissipation and Soret effect on natural convection heat and mass transfer from vertical cone in a non-Darcy porous media saturated with non-Newtonian fluid. The surface of the cone and the ambient medium are maintained at constant but different levels of temperature and concentration. The Ostwald-de Waele power law model is used to characterize the non-Newtonian fluid behavior. The governing equations are non-dimensionalized into non-similar form and then solved numerically by local non-similarity method. The effect of non-Darcy parameter, viscous dissipation parameter, Soret parameter, buoyancy ratio, Lewis number and the power-law index parameter on the temperature and concentration field as well as on the heat and mass transfer coefficients is analyzed.     

Buoyancy-assisted flow of yield stress fluids past a cylinder: Effect of shape and channel confinement

The aiding-buoyancy mixed convection heat transfer in Bingham plastic fluids from an isothermal cylinder of elliptical and circular shape in a vertical adiabatic channel is numerically investigated. For a fixed shape of the elliptical cylinder E = 2 (ratio of major to minor axes), the effect of confinement is studied for three values of blockage ratio, B, defined as the ratio of the channel width to the circumference of the cylinder/π, as 6.5, 2.17 and 1.3. In order to delineate the role of cross-section of the cylinder, results are also presented here for a circular cylinder of the same heat transfer area as the elliptical cylinder. The results presented herein span the range of conditions as: Bingham number, 0 ≤ Bn ≤ 100, Reynolds number, 1 ≤ Re ≤ 40, and Prandtl number, 1 ≤ Pr ≤ 100 over the range of Richardson number Ri = 0 (pure forced convection) to Ri = 10. Extensive results on drag coefficient, local and surface averaged values of the Nusselt number and yield surfaces are presented herein to elucidate the combined effects of buoyancy, blockage ratio and fluid yield stress. The morphology of the yield surfaces shows that the unyielded plug regions formed upstream and downstream of the cylinder grow faster at low Reynolds numbers with the increasing yield stress effects under the weak buoyancy forces, i.e., small values of Grashof or Richardson number. The heat transfer enhancement is observed with the increasing channel-confinement due to the sharpening of the temperature gradients near the surface of the cylinder. The average Nusselt number shows a positive dependence on the Reynolds number, Prandtl number and Richardson number irrespective of the shape of the cylinder or the type of fluid. By employing the modified definitions of the dimensionless parameters (based on the two choices of the overall effective fluid velocity), predictive correlations have been established for estimating the value of the average Nusselt number in a new application.