MHD Convection Fluid and Heat Transfer in an Inclined Micro-Porous-Channel

1 Introduction

The mechanics of fluid flow in micro-channels has emerged as an active area of research in view of the enormous applications in various branch of industry such as biomedical industry, computer chip manufacturing and separation processes in chemical industry operations among others. Literatures on convective flow in vertical micro-channel are numerous. Extensive investigation have been conducted recently on micro-geometry flow under different physical situations (Chen and Weng [1], Jha et al. [2], Khadrawi et al. [3], Hadded et al. [4], Buonomo and Manca [5], Weng and Chen [6], Jha et al. [7], Avci and Aydin [8, 9], Jha and Aina [10]).

However, the magnetohydrodynamic (MHD) phenomenon has received considerable attention of engineers and scientists over the last two decades owing to its relevance in many MHD applications like MHD generator, MHD accelerators, electric transformers and cooling of metallic plate in cooling bath. Magnetohydrodynamic (MHD) pumps are already in use in chemical energy technology for pumping electrically conducting fluids at some of the atomic energy centre. Several studies have been reported on MHD convective flow under different physical situations. Record of such investigations can be found in the works of Cramer and Pai [11], Chawla [12], Das et al. [13], Sheikholesslami and Gorgi-Bandpy [14], Sheikholesslami et al. [15], Sheikholesslamiet al. [16], Chauhan and Rastogi [17], Ibrahim and Makinde [18], Farhad et al. [19, 20]. Hasan Nihal Zaidi and Naseem Ahmad [21] have discussed MHD convection flow of two immiscible fluids in an inclined channel with heat generation or absorption. Wang et al. [22] experimentally investigated the effect of inclination angle on the convective boiling performance of a micro-channel heat sink. They found that the heat transfer coefficient for 45^∘ upward considerably exceeds other configurations. The combined influence of externally applied transverse magnetic field and suction/injection on steady natural convection flow of conducting fluid in a vertical micro-channel was carried out by Jha et al. [23]. In another work, Jha et al. [24] examined the effect of wall surface curvature on transient MHD free convective flow in vertical micro-concentric-annuli. Jha et al. [25] studied exact solution of steady fully developed natural convection flow of viscous, incompressible, and electrically conducting fluid in a vertical annular micro-channel. Recently, Jha and Aina [26] presented the MHD natural convection flow in a vertical micro-porous-annulus (MPA) in the presence of radial magnetic field. Also, the MHD natural convection flow in vertical micro-concentric-annuli (MCA) in the presence of radial magnetic field has been analyzed by Jha et al. [27]. Farhad et al. [28] investigated the effect of magnetic field on blood flow of Casson fluid in axisymmetric cylindrical tube. Farhad et al. [29] presented a study on the application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters’- B fluid model. Hidden phenomena of magnetohydrodynamic time dependent flow in porous medium with heat transfer was carried out by Farhad et al. [30]. In a related work, Sheikh et al. [31] investigated on modern approach of Caputo–Fabrizio time-fractional derivative to MHD free convection flow of generalized second-grade fluid in a porous medium. Other literatures on the above subject can be found in [32, 33, 34, 35, 36, 37, 38, 39].

Some recent works related to the present investigation are found in the literature [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. Nanofluid treatment in a porous enclosure considering thermal non – equilibrium model under the impact of magnetic field has been analyzed by Sheikholeslami and Shehzad [40]. Sheikholeslami and Seyednezhad [41] numerically studied Fe₃0₄-Ethylene glycol nanofluid electrohydrodynamics flow and natural convection heat transfer in a porous medium. Sheikholeslami and Moradi [42] studied the influence of thermal radiation on nanofluid behaviour in existence of Couloumb forces via CVFEM. Sheikholeslami [43] numerically investigate for C_u0-H₂0 nanofluid flow in a porous channel with magnetic field using mesoscopic method. Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure was carried out by Sheikholeslami [44]. Sheikholeslami and Rokni [45] present numerical simulation for impact of Coulomb force on nanofluid heat transfer in a porous enclosure in presence of thermal radiation. In another related article, Sheikholeslami and Shehzad [46] presented numerical analysis of Fe₃0₄-H₂0 nanofluid flow in permeable media under the effect of external magnetic source. Influence of magnetic field on nanofluid free convection in an open porous cavity by means of Lattice Boltzmann method was carried out by Sheikholeslami [47]. The effect of a constant magnetic field on the nanofluid natural convection in a porous media with a Sinusoidal hot cylinder was presented by Sheikholeslami [48]. Recently, Sheikholeslami [49] investigate the impact of Lorentz forces on magnetic nanofluid free convection in a porous media. Sheikholeslami [50] reported a simulation of nanofluid natural convection in a porous enclosure in existence of magnetic field. Sheikholeslami and Moradi [51] examined the influence of thermal radiation on heat transfer of nanfluid in permeable media in presence of variable magnetic field. The effect of Lorentz forces on nanofluid hydrothermal treatment in a porous curved cavity using CVFEM was described by Sheikholeslami [52].

On the other hand, suction or injection of fluid through the bounding surfaces, as, for example, in mass transfer cooling, it can significantly change the flow field and, as a consequence, affect the rate of heat transfer from the bounding surfaces. In general, suction tends to increase the skin-friction and heat transfer coefficients whereas injection acts in the opposite manners. Injection or withdrawal of fluid through porous heated or cooled surface is of general interest in practical problems involving film cooling, control of boundary layers, etc. This can lead to enhanced heating (or cooling) of the system and can help to delay the transition from laminar flow [53].

To date, a lot of work has been conducted on micro-geometry flow under different physical situations in the presence of magnetic field. Nevertheless, to the best of the authors’ knowledge, there is no investigation on the influence of transverse magnetic field as well as suction/injection on MHD natural convection flow in an inclined micro-porous-channel. The main aim of the present work is to present a theoretical study for the MHD convection fluid and heat transfer in an inclined micro-porous-channel.

2 Mathematical Analysis

The flow considered is a fully developed steady natural convection flow of viscous, incompressible, electrically conducting fluid through an inclined micro-porous-channel under the effect of transverse magnetic field. The system under consideration is sketched in Figure 1, where the chosen coordinate’s (X, Z) axes, making an angle α with the horizontal are drawn. The fluid is bounded by porous channel walls separated by b. A magnetic field of uniform strength (0, B₀, 0) is assumed to be applied in the direction perpendicular to the direction of flow. It is assumed that the magnetic Reynolds number is very small, which corresponds to negligibly induced magnetic field compared to the externally applied one. Fluid is being injected into the flow region through the cold porous plate and in order to conserve the mass of the fluid in the channel, fluid is being sucked out of the channel at the same rate through the hot porous plate. In addition, the plates are heated asymmetrically with one plate maintained at a temperature T₁ while the other plate at a temperature T₂ where T₁ > T₂. Due to this temperature gradient between the porous plates, natural convection flow occurs in the channel. Following Hasan and Naseem [21] and Jha et al. [23], the governing equations for the transport processes in dimensionless form in the presence of velocity slip and temperature jump are given under the Boussinesq’s approximation as follows [21, 23]:

Fig. 1

Physical model

The dimensionless quantities used in the above equations are:

The physical quantities used in the above equations are defined in the nomenclature.

The boundary conditions which describe velocity slip and temperature jump conditions at the fluid – wall interface are:

where:

Here y_s is the ratio of specific heats, Pr is the Prandtl number, f_v and f_t are the tangential momentum and thermal accommodation coefficients, respectively, and range from near 0 to 1, λ is the molecular mean free path, Kn is the Knudsen number, ln is the fluid – wall interaction parameter, and ξ is the wall- ambient temperature difference ratio. Referring to the values of f_v and f_t given in Eckert and Drake [54] and Goniak and Duffa [55], the value of β_v is near unity, and the value ofβ_t ranges from near 1 to more than 100 for actual wall surface conditions and is near 1.667 for many engineering applications, corresponding to f_v = 1, f_t = 1, y_s = 1.4 and Pr = 0.71(β_v = 1, β_t = 1.667).

By using the method of undetermined coefficients, solutions to Eqns. (1) and (2), subject to the boundary conditions, (4) and (5) have the following exact solutions:

Two important parameters for buoyancy – induced micro-flow and micro-heat transfer are the volume flow rate m and heat transfer rate q, respectively. The dimensionless volume flow rate is:

Substituting equation (6) into equation (8), one obtain

And the rate of heat transfer which is expressed as the Nusselt number is

Therefore:

Using expression (6), we obtain the skin - friction (τ) on the micro-porous-channel walls as follows:

All the constants are defined in Appendix

3 Results and Discussion

In this section, the results obtained for fully developed MHD natural convection flow of electrically conducting fluid in an inclined micro-porous-channel in the presence of suction/injection as well as transverse magnetic field are discussed. The major parameter such as: angle of inclination, rarefaction, fluid wall interaction, suction/injection and Hartmann number on the flow formation are depicted graphically. The present parametric study has been performed in the continuum and slip flow regimes (Kn ≤ 0.1). Also, for air and various surfaces, the values of β_ν and β_t range from near 1 to 1.667 and from near 1.64 to more than 10, respectively. So, this study has been performed over reasonable ranges of 0 ≤ β_vKn ≤ 0.1, 0 ≤ M ≤ 2, –2 ≤ S ≤ 2 and 0 ≤ ln ≤ 10 [23] with fixed values β_vKn = 0.05, M = 2, S = 0.5, ϕ = 45^∘ and ln = 1.667 all arbitrarily chosen to investigate their effect on the micro-porous-channel fluid flow. The product β_vKn represents a measure of the departure from the continuum regime while dimensionless parameter ln represents a property of the fluid – wall interaction.

The expression for the temperature in equation (7), the effect of suction/injection (S), rarefaction parameter (β_vKn), and fluid-wall interaction parameter (ln) on the temperature profile and rate of heat transfer which is expressed as the Nusselt number are exactly the same as those given by Jha et al. [23]. Therefore, we focus on the effects of angle of inclination, rarefaction, fluid wall interaction, suction/injection and Hartmann number on fluid velocity, volume flow rate and skin friction on the micro-porous channel walls using the aid of line graphs. In the present study, three cases of interest are investigated for different values of the wall ambient temperature difference ratio ξ : ξ = 1 corresponding to the physical case when both micro-porous-channel walls are heated symmetrically, ξ = 0 corresponds to the physical case when one of the walls is heated and other not heated and ξ = –1 implies the physical situation when one of the micro-porous-channel walls is heated and the other cooled.

Using the numerical values of the skin friction on the micro-porous-channel walls, results obtained in the present work was validated with the work of Jha et al. [23] for Gr = 1 and ϕ = 90^∘ and presented in Table 1. From the table, it is obvious that an excellent agreement is established.

Figure 2 displays the effect of the inclination angle on the velocity profile under different wall heating conditions. From the figure, it is observed that for symmetric (ξ = 1) and asymmetric heating (ξ = 0) fluid velocity increases with the increase in the inclination angle. This could be attributed to the enhancement in the convective buoyancy force due to increase in the inclination angle. However, when asymmetric heating (ξ = –1) is considered, increase in the inclination angle results to increase in fluid velocity near the hot wall whereas it decreases at the cold wall.

Fig. 2

Effect on angle of inclination on velocity profile

Figure 3 illustrates the effect of suction/injection parameter (S) on fluid velocity in an inclined micro-porous channel. The figure reveals the significance of suction or injection on buoyancy induced flow formation in the micro-channel. From the figure, it is obvious that injection (S > 0) thickens the thermal boundary layer, resulting to weakening the convective current and consequently decreasing the fluid velocity whereas suction (S < 0) weakens the thermal boundary layer yielding an increase in fluid velocity.

Fig. 3

Effect of suction/injection on velocity profile

The influence of the Hartmann number (M) on fluid velocity in an inclined micro-porous channel is displayed in Fig. 4 for both symmetric (ξ = 1) and asymmetric (ξ = 0, –1) heating conditions. From the figure, it is clear that increase in Hartmann number leads to decrease in fluid velocity for all wall heating conditions. This is due to the physical fact that increase in Hartmann number induces a mechanical damping force which acts in the directions opposite to the fluid motion thereby suppressing the fluid velocity within the micro-channel as observed in the Fig. 4.

Fig. 4

Effect on Hartmann number on velocity profile

The effect of rarefaction parameter (β_vKn) on the micro-porous-channel is observed in Figure 5 under different wall heating conditions. It is observed from the figure that, increasing the wall-ambient temperature difference ratio as well as rarefaction parameter leads to enhancement in the velocity slip. This result yields an observable increase in the fluid velocity. In addition, as the wall-ambient temperature difference ratio (ξ) increases, the effect of rarefaction parameter (β_vKn) on the inclined micro-porous channel slip velocity becomes significant.

Fig. 5

Effect of rarefaction parameter on velocity

Figure 6 displays the effect of fluid wall interaction parameter (ln ) on fluid velocity. From the figure, it is observed that velocity slip increases on the micro-porous-channel surfaces with the increase of fluid wall interaction parameter (ln). Also, it is obvious that for symmetric heating (ξ = 1), increase in ln has no influence on fluid velocity within the micro-porous channel since the temperature of the walls are not changing. However, when asymmetric heating is considered (ξ = 0, –1), the effect of temperature jump induced by increase in ln becomes significant as fluid velocity is observed to increase about the cold wall and decrease about the hot wall.

Fig. 6

Effect of fluid wall interaction parameter on velocity profile

Figure 7 describes the combined effect of the inclination angle (ϕ), rarefaction parameter (β_vKn) and wall ambient temperature difference ratio (ξ) on the micro-porous-channel volume flow rate. It is clear from the figure that increase in the inclination angle yields an observable increase in the volume flow for both symmetric (ξ = 1) and asymmetric heating (ξ = 0, –1) cases. However, this increase in volume flow could be further enhanced by the increase in rarefaction parameter as for the symmetric heating case (ξ = 1). Furthermore, it is also clear that for asymmetric heating (ξ = 0), increase in rarefaction parameter has no significant effect on the volume flow.

Fig. 7

Volume flow rate versus β_vKn for different of ϕ

Figure 8 and 9 illustrates the influence of the inclination angle (ϕ), rarefaction parameter (β_vKn) and wall ambient temperature difference ratio (ξ) on the skin friction (τ) at the micro-porous-channel walls(Y = 0) and (Y = 1). From Fig. 8, it is observed that the effect of the rarefaction parameter on the skin friction (τ₀) becomes more pronounced as β_vKn grows for both symmetric (ξ = 1) and asymmetric heating (ξ = 0, –1) cases. Increasing the inclination angle on the other hand yields an observable increase in τ₀. A similar phenomenon is observed in Fig. 9.

Fig. 8

Skin fraction (τ₀) versus β_vKn for different of ϕ

Fig. 9

Skin fraction (τ₁) versus β_vKn for different of ϕ

4 Conclusions

The present paper investigates on the fully developed laminar flow of a viscous incompressible and electrically conducting fluid in an inclined micro-porous-channel subject to a transversely applied magnetic field. The governing equations were solved analytically and expressions for the temperature and velocity profiles where obtained under relevant boundary conditions. Effect of various flow parameters on fluid velocity and volume flow have been highlighted using line graphs and there after discussed. The main findings of the present work are as follows:

1. Increasing the inclination angle enhances the convective current in the micro-porous-channel resulting to increase in fluid velocity as observed in Figure 2.

2. It could also be seen from Figure 3 that injection (S > 0) thickens the thermal boundary layer, thereby weakening the convective current and consequently decreasing fluid velocity whereas suction (S < 0) weakens the thermal boundary layer yielding an increase in fluid velocity.

3. Increase in the inclination angle yields an observable increase in the volume flow for both symmetric (ξ = 1) and asymmetric heating (ξ = 0, –1) cases as observed in Figure 7.

4. Increasing the inclination angle yields an increase in the skin friction (τ₀, τ₁) at both boundaries, whereas the effect of the rarefaction parameter on skin friction (τ₀, τ₁) becomes pronounced as β_vKn becomes large.