Hydromagnetic Natural Convection from a Horizontal Porous Annulus with Heat Generation or Absorption

This paper deals with a numerical study of free convection in a horizontal cylindrical annulus filled with a fluidsaturated porous medium in the presence of a transverse magnetic field and the heat generation or absorption effect. It is assumed that the inner and outer walls of the cylindrical annulus are maintained at constant temperatures Ti and To , respectively, as Ti > To . In addition to the heat equation, the model consists of the equation of motion under the Darcy law and Boussinesq approximation. The system of equations is solved numerically by the alternatingdirection implicit finite difference method. This investigation concerns the effects of the magnetic field inclination angle, Hartmann number, and the heat generation or absorption coefficient on heat transfer and the flow pattern. The results demonstrate that the heat transfer rate and flow regime depend mainly on the characteristics mentioned. The obtained data are presented graphically in terms of the streamlines and isotherms.

porous matrix. The polar coordinates system r′ -n is used. Under these assumptions along with the Boussinesq and Darcy approximations, the governing equations of heat and electric transfer can be written as follows: Here the gradient and Laplace operator in the used polar coordinates are defi ned in the following manner: where K is the permeability, k is the unit vector in the vertical direction /| |), = (k g g Q 0 is the constant generated or absorbed heat from the source per unit volume, and φ is the electric potential. The heat capacity of a saturated porous medium is equal to and the equivalent thermal conductivity is calculated as the weighted average of the solid and fl uid conductivities, i.e., m f s As was shown in [16,17], Eqs. (4) and (5) reduce to 2 0. ′ ∇ φ = Unique solution of this equation is φ = 0 due to an electrically insulating boundary around the cavity. It follows that an electric fi eld vanishes everywhere (see [18]).
It is more suitable to deal with dimensionless equations. Thus, using the following scales: r 1 for the radial coordinate, for the pressure, we obtain the partial differential equations in the dimensionless form as Ra where ψ is the stream function such that 1 u r The problem is assumed to be symmetric about the vertical line passing through the center of the system, therefore, only the half of the fl ow domain will be considered. Thus, the boundary conditions can be written as where the radius ratio R = r o /r i represents an another dimensionless key parameter of the problem. Heat transfer across the whole annulus is presented in terms of the average Nusselt number along the hotter (inner) cylinder evaluated as follows: where Nui is the local Nusselt number along the hotter cylinder calculated as the ratio between convective and conductive heat transfer: Numerical Methods. The dimensionless governing equations are discretized by the use of the fi nite difference method coupled with the alternating-direction implicit scheme. To solve these discretized equations, the Thomas tridiagonal matrix algorithm is used in conjunction with iterations. As the study concerns steady-state regimes, the iterative procedure is stopped when the residuals become below 10 -8 .
In order to assess the accuracy of the numerical code developed, the convection problem was solved in the absence of a magnetic fi eld and the heat generation or absorption effects, since solutions for such a problem are available. The results obtained with the use of the present code are in good agreement with the experimental results of [19] and the numerical data reported earlier [19][20][21][22][23][24] (see Table 1). Moreover, Fig. 2 is indicative of good agreement between the obtained streamlines and temperature contour plots and the results of [25][26][27] for R = 2 and Ra = 200. In this comparison, we have used a motionless fl uid and the temperature distribution in the case of pure conduction at the initial conditions of the problem considered.
For obtaining the best compromise between the accuracy and minimized calculation time, a grid sensitivity analysis was performed. The present code was assessed for grid independence by evaluating the average Nusselt number along the hotter cylinder. Therefore, various numerical experiments were tested for R = 2, Ra = 100, Ha = 0.5, Q = 0.01, and θ = 0, as illustrated in Table 2 (where the reference value r Nu corresponds to the 301 × 301 grid). It was found from the comparison performed that the 131 × 131 grid yields the solutions that are reasonably grid-independent.  2. Streamlines (at the left) and isotherms (at the right) for R = 2 and Ra = 200: [25] (a), [26] (b), [27] (c), and the present study (d).
Particular efforts have been focused on the impact of these key parameters on the fl uid fl ow and convective heat transfer. Figure 3a illustrates the effect of the inclination angle on the average Nusselt number for Ra = 50. It is seen that the average Nusselt number decreases when the Hartman number increases. It is interesting to note that the maximum effect of a magnetic fi eld is observed when the inclination angle is equal to zero. Moreover, convective heat transfer is more pronounced for the values of the inclination angle close to π/2. Figure 3b and c illustrates the infl uence of the Hartmann number on the average Nusselt number for Ra = 100 and different values of the heat generation or absorption coeffi cient. It is seen that, regardless of the value of Q, the average Nusselt number decreases when Ha increases. It should also be noted that the effect of the Hartman number on convective heat transfer is less signifi cant when Ha is greater than 15. Besides, it may be concluded that the convective heat transfer rate is a decreasing function of the heat generation or absorption coeffi cient Q. This conclusion is confi rmed by Fig. 4 which shows the average Nusselt number as function of Q for Ra = 100 and different values of the Hartman number.
It is well known that several types of convective fl ow regimes may develop depending on the initial conditions of the problem and the values of the key parameters, such as the radius ratio and Rayleigh number [21,25,28,29]. Thus unicellular, bicellular or multicellular fl ow structures may appear for relatively high values of the Rayleigh number at R = 2. The transition from a unicellular fl ow regime to a bicellular one is connected with the onset of thermoconvective instabilities, as was already demonstrated by earlier investigations [19,23,30]. Figure 5a and b shows the streamlines and isotherms for Ra = 80, Q = 0, and two values of the Hartmann number (Ha = 0 and 5). It is clearly seen that the fl ow structure transfers from a bicellular fl ow regime for Ha = 0 to a unicellular structure at Ha = 5. It should be noted that the number of convective counterrotating cells inside the annulus top decreases as the Hartman number increases. In other words, thermoconvective instability is less pronounced when a magnetic fi eld is relatively strong. However, the performed tests demonstrate that the fl ow structure changes when the heat generation or absorption coeffi cient exceeds a certain given threshold. Thus, the thermoconvective instabilities may develop for relatively high values of Q. Therefore, as shown in Fig. 5c and d, a small increase in Q leads to the development of a new convective cell in the top part of the annulus.
Conclusions. In this paper, we have studied the impact of the magnetic force and heat generation or absorption on convective heat transfer and the fl ow pattern inside horizontal concentric cylinders fi lled with a saturated porous medium. Upon introducing the proper similarity variable, the resulting set of differential equations under the Boussinesq and Darcy approximations was represented in the dimensionless form.
The most important conclusions of the investigation can be summarized as follows: 1. In general, the convective heat transfer rate is highly impacted by the magnetic force inclination angle, Hartmann number, Rayleigh number, and the heat generation or absorption coeffi cient. 2. The impact of the magnetic force becomes more effective when the inclination angle is close to zero. 3. Different fl ow regimes may appear for various combinations of the Rayleigh number, Hartmann number, and the heat generation or absorption coeffi cient.  4. The magnetic force has a signifi cant effect on thermoconvective instabilities. A small increase in the Hartmann number infl uences the fl ow structure. 5. The thermoconvective instabilities are more pronounced when the heat generation or absorption coeffi cient increases.