On laminar hydromagnetic mixed convection flow in a vertical channel with symmetric and asymmetric wall heating conditions
Introduction
Mixed convection flow in a vertical channel has been the subject of many previous investigations due its possible application in many industrial and engineering processes. These include cooling of electronic equipment, heating of the Trombe wall system, gas-cooled nuclear reactors and others. Tao [17] analyzed laminar fully developed mixed convection flow in a vertical parallel-plate channel with uniform wall temperatures. Aung and Worku [1], [2] discussed the theory of combined free and forced convection in a vertical channel with flow reversal conditions for both developing and fully developed flows. Aung and Worku [2] assumed that the walls of the channel were having asymmetric temperatures. The case of developing mixed convection flow in ducts with asymmetric wall heat fluxes was analyzed by the same authors [3]. A comprehensive review of the literature dealing with mixed convection in internal flow was reported by Aung [4]. Cheng et al. [9], Hamadah and Wirtz [11] and Ingham et al. [12] also reported on flow reversal situations in mixed convection in a vertical channel for different wall heating conditions. Kou and Lu [13] analyzed mixed convection in a porous medium channel and discussed the conditions for flow reversal situations.
The use of electrically conducting fluids under the influence of magnetic fields in various industries has led to a renewed interest in investigating hydromagnetic flow and heat transfer in different geoemetries. For example, Sparrow and Cess [15] considered the effect of a magnetic field on the free convection heat transfer from a surface. Raptis and Kafoussias [14] analyzed flow and heat transfer through a porous medium bounded by an infinite vertical plate under the action of a magnetic field. Garandet et al. [10] discussed buoyancy driven convection in a rectangular enclosure with a transverse magnetic field. Chamkha [7] analyzed free convection effects on three-dimensional flow over a vertical stretching surface in the presence of a magnetic field.
The study of internal heat generation or absorption in moving fluids is important in view of several physical problems such as those dealing with chemical reactions and those concerned with dissociating fluids (see, for instance, [18], [19]). Other investigations dealing with internal heat generation or absorption can be found in the works of Sparrow and Cess [16] and Chamkha [7].
The aim of the present work is to extend studies available in the literature and especially the work of Barletta [5] on laminar fully developed mixed convection in a vertical parallel-plate channel by including internal heat generation or absorption and magnetic field effects. This will be done for three types of left–right walls thermal conditions. These conditions are the isothermal–isothermal, isoflux–isothermal and the isothermal–isoflux thermal wall conditions.
Section snippets
Governing equations
Consider steady laminar mixed convective flow in a parallel-plate vertical channel in the presence of a magnetic field and a heat source or sink. The vertical plates are separated by a distance L and are maintained at either constant temperature or constant heat flux (see Fig. 1). Let x and y represent the vertical and horizontal distances, respectively, with the origin being at the center of the channel. Let the magnetic field be applied in the horizontal direction normal to the flow
Analytical solutions
Various analytical solutions for special cases of the problem described above are possible only in the absence of both viscous and magnetic dissipation terms. These are reported for the following nine special cases:
Case 1. Hydromagnetic mixed convection flow in a vertical channel with isothermal walls.
For this special case the viscous and magnetic dissipations and the heat generation or absorption effect are neglected (Br=0 and φ=0). Using this, Eq. (8) reduces toSolution of Eq. (11)
Heat transfer aspects
The Nusselt numbers at each of the channel walls are important physical characteristics. These can be defined for the three different thermal boundary conditions considered in the present work as follows:
Reversed flow conditions
Depending on the value of the mixed convection parameter Gr/Re and the wall thermal boundary conditions, a flow reversal condition may occur. It is beneficial to understand when this situation occurs and determine a reversed flow zone associated with each of the problems discussed earlier. The occurrence of a reversed flow condition is ensured when the slopes of the velocity profile at the walls have the same sign. That isThe critical condition for flow reversal at the
Selected graphical results
Fig. 2 displays typical velocity profiles U in a vertical channel with asymmetric (Rt=1.0) isothermal–isothermal wall heating conditions for different values of the mixed convection parameter Gr/Re. For a vanishing value of Gr/Re, the usual symmetric Hagen–Poiseuille velocity profile is obtained. For an upward flow, increases in the value of Gr/Re have the tendency to increase the momentum of the flow close to the hot right wall causing the velocity profile to become asymmetric. As mentioned
Conclusion
This work focused on the laminar fully developed mixed convective flow of an electrically conducting fluid in a vertical channel in the presence of a magnetic field and heat generation or absorption effects. Three different combinations of thermal left–right wall conditions were prescribed. These thermal left–right wall conditions were isothermal–isothermal, isoflux–isothermal, and isothermal–isoflux conditions. Various analytical solutions for the velocity and temperature profiles for
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