Analysis of Hall current and nonuniform heating effects on magneto-convection between vertically aligned plates under the influence of electric and magnetic fields

1 Introduction

Natural convective flow resulting from the buoyancy forces between two parallel plates has received substantial attention from engineers and scientists due to the broad range of scope in scientific and technological processes. These studies are essential in thermal hydraulics nuclear reactor (Sudo et al. [1]), thermal support dynamics in services (Rodrigues et al. [2]), channel–chimney systems (Manca et al. [3]), cooling electronic components (Florio and Harnoy [4]), and frost formation (Fossa and Tanda [5]). These applications of free convective flow attract researchers to work in this area. The heat transfer and fluid flow between the two vertical parallel plates were analyzed by Kato et al. [6] and Campo et al. [7] in different physical conditions. Nelson and Wood [8] scrutinized the flow characteristics in natural convective laminar flow in the vertical parallel plates. Their numerical results were in good agreement with the experimental results of Aihara [9], who examined the convective heat transfer through vertical parallel plates. A laminar flow of Bingham fluid via vertical parallel plates had been analyzed by Bayazitoglu et al. [10]. Jha and Ajibade [11] discussed the effect of the time-periodic thermal boundaries on the viscous dissipative free convective flow in vertical parallel plates.

The aforementioned studies have been carried without the magnetic fields. But, we know that a magnetic field has an attribute to modify the energy and flow of the system. This trait of the magnetic field has many applications in technical and scientific fields such as liquid metal coating, electromagnetic casting, confinement of plasma and others (Ni [12] and Khan et al. [13]). Therefore, many researchers worked in this context and investigated magnetohydrodynamics (MHD) flow problems [14–21] in different geometries and physical circumstances. A study by Rehman et al. [22] examined the stability of MHD couple stress concerning the effects of shape factor and magnetic field. The study by Kumam et al. [23] explored the effects of the relative magnetic field and slip on the behavior of Casson dusty fluid in a two-phase fluctuating flow over inclined parallel plates. Magneto-convective flow is well known term it stands for magnetohydrodynamics convective flow [24].

Hall effect in magnetohydrodynamics is the development of a transverse electric field in an electrically conducting fluid when it carries an electric current and placed in a magnetic field that is perpendicular to the current. The magnetic field sensing equipment, proximity detectors, and Hall effect tong tester are such applications of the Hall effect that attracted the researchers. The Hall and radiation effects were studied by VeeraKrishna et al. [25] over an unsteady magneto-convective flow in a porous medium through two vertical plates. They obtained that the primary and secondary velocity profiles enhanced with the Hall parameter and reduced with the radiation parameter. Kumar et al. [26] found in their study of MHD flow with alternating conducting vertical walls that the Hall parameter enhanced the primary rate of mass flow while the secondary rate of mass flow decreases.

MHD flow in the existence of an electric field for the flow formation problem has been increasing the interest of the researchers. A qualitative analysis of electric field had a significant bearing in industrial and chemical process applications (Tsuda [27]), such as in tissue engineering (Markx [28]), biology and medicine (Hart and Palisano [29]). Sheikholeslami et al. [30] explored the implementation of an electric field to enhance the rate of heat transfer of heat of ferrofluid in an enclosure with double moving walls. The transmission of heat in Williamson fluid flow in the appearance of applied electric and magnetic fields past a heated surface with varying thickness had been studied by Aliy and Kishan [31].

Furthermore, the idea of nonuniform heating in flow field is an emerging notion, having opportunities in design engineering and nuclear equipments. Such applications of fluid flow past nonuniformly heated surfaces caught the attention of researchers toward this arising stream of fluid dynamics. Roy and Basak [32] keenly inspected the behavior of free convection flow over the nonuniform and uniform heating wall(s). They revealed that for nonuniform heating, the heat transfer rate represents sinusoidal variations with Ra = 1 0 5 . Mythili et al. [33] had shown in their study of free convective Casson fluid flow across a vertical cone that the heat transfer can be controlled with the nonuniform heating. The heat transfer of super-critical water in the nonuniform heating tube was analyzed by Bai et al. [34]. They found the nonuniformity in the distribution of temperature along the circumference, and also observed that the increment in heat transfer arises in a portion of the cross section. Pordanjani et al. [35] scrutinized the effect of nonuniform and uniform boundary conditions over entropy formation and convective heat transfer of the nanofluid in a square cavity. Kumar et al. [36] considered the Williamson fluid convective flow with nonuniform heat generation/absorption and observed the reduction in mass transport rates with enhancement in local Williamson parameter.

The research mentioned above, and their applications give us an impetus to study the influence of the Hall effect and electric field on fluid flowing through the magnetic field between nonuniformly heated parallel plates, which is still to be investigated to the best of the knowledge of the author. The aim of this study is to bridge that gap by utilizing advanced numerical approaches to comprehensively examine the interplay between Hall currents, electric field, and nonuniform heating. In this study, we observe that advancing the Hartmann number diminishes the primary fluid velocity. This maintained fluid slowing allows precise regulation in devices such as MHD pumps, where managing flow rates is essential. It enhances the accuracy of applications such as optimized heat transfer through targeted delivery of nanoparticles in a cooling mechanism.

Additionally, raising the Hall parameter speeds up fluid flow, which can enhance the efficiency of turbine systems. This faster fluid flow in MHD generators interacts more effectively with magnetic fields, making energy transformation more efficient. Therefore, we have investigated the impact of the Hall parameter on magneto-convection flow via vertical parallel plates when subjected to electric and magnetic fields under the nonuniform heating at the boundary.

2 Mathematical formulation

This mathematical model described the magneto-convective flow of an electrically conducting fluid whose thermophysical characteristics are supposed to be constant. The fluid is confined between two vertical parallel plates. The Cartesian coordinates are taken into account with x -axis along the plate while y -axis normal to the plates. The leading edge of the plate coincides with the z -axis. The plate at y = 0 is heated nonuniformly, while the plate at y = 1 is maintained at constant temperature. A constant magnetic field of strength B 0 and an electric field of strength E 0 are applied in y -direction and opposite z -direction, respectively. The electric and magnetic fields are strong enough to generate the Hall current but unable to produce an induced magnetic field in fluid. The physical description of the flow model is given in Figure 1. The fluid flow and heat transmission were also induced by the viscous dissipation and Julian heating. The basic equations of the fluid flow have been derived from the Continuity equation, Navior–Stokes equations and energy equation. The derived equations of flow are given as follows [31,37]:

The prescribed initial and boundary conditions for the designated flow model are as follows:

where g ′ ( x ′ ) = T 0 ′ + Δ T ( 1 − cos 2 π x L ) . Also, J x = − σ B 0 1 + m 2 ( w − m u + m E 0 B 0 ) , J y = 0 , and J z = σ B 0 1 + m 2 ( u − m w − E 0 B 0 ) .

Figure 1

Geometry of this physical model.

The dimensionless physical parameters are proposed as follows:

By using these dimensionless parameters in Eqs (1)–(5), we obtain the following set of non-dimensional equations:

Also, the non-dimensional initial and boundary conditions are derived as

where g ( x ) = 1 − cos ( 2 π x ) .

3 Numerical methods

The flow regime, ruled by the partial differential equations (PDEs) (6)–(10), has been solved using an implicit Crank–Nicolson scheme [38]. In this numerical technique, first, we have transformed the PDEs 6–10 into finite difference equations using the corresponding finite difference operators with grid discretization, and these equations are given as

where ( i , j ) is the grid location in x and y directions, respectively. Δ x , Δ y , and Δ t are step size in x , y , and t directions, respectively, while ( n , n + 1 ) represents the n th and ( n + 1 ) th iterations and ξ stands for either u , w , or T . By properly choosing values of the constants f 1 , f 2 , A 1 , A 2 , A 3 , A 4 , A 5 and a , we obtain the finite difference equations of primary velocity, secondary velocity, and temperature. For primary velocity equation equation, f 1 → T , f 2 → w , A 1 = 1 , A 2 = Gr , A 3 = − m ha 2 1 + m 2 , A 4 = 0 , A 5 = Ha 2 1 + m 2 , A 6 = Ha 2 E 1 1 + m 2 , A 7 = 0 , and a = 1 ; for secondary velocity equation, f 1 → u , f 2 → 0 , A 1 = 1 , A 2 = m ha 2 1 + m 2 , A 3 = 0 , A 4 = 0 , A 5 = Ha 2 1 + m 2 , A 6 = − m E 1 Ha 2 1 + m 2 , A 7 = 0 , and a = 1 ; for temperature equation, f 1 → u , f 2 → w , A 1 = 1 Pr , A 2 = A 3 = Ec Ha 2 1 + m 2 , A 4 = E 1 Ec Ha 2 1 + m 2 , A 5 = 0 , A 6 = Ec Ha 2 E 1 2 1 + m 2 , A 7 = Ec , and a = 2 .

After obtaining finite difference equations, we have to convert them into algebraic equations. These algebraic equations are presented in the form of a tridiagonal system and solved by the Thomas algorithm. In this numerical procedure, we have chosen a step size of 0.025 for increments in the x and z directions, while the time increment is set to 0.0025 for our numerical computations. A grid of 41 × 41 points has been thoroughly employed in the computation to ensure accurate results.

For every increasing time step Δ t , we compute the values of u , v , w , and T appearing in equations (11) and (12) in the previous time step and use them to evaluate u , v , w , and T at the next time step. In each time step, we first solve the energy equation and then use the computed values to solve for the primary velocity component ( u ). Subsequently, we compute the secondary velocity component ( w ), and finally, we solve the continuity equation to obtain the v velocity field.

In the unsteady state, the nature of flow regime has been determined at a certain time after performing definite iterations. The velocity and temperature profiles attain the steady state when the convergence criterion given below is satisfied:

where ζ stands for the considered flow fields, and n and ( n + 1 ) are the iterated values of flow field at n and ( n + 1 ) time step and ( i , j ) are the grid locations.

The authenticity of our computation procedure has been checked under limiting conditions by ignoring the effect of Julian heat, viscous dissipation, and electric field at steady state with isothermal boundary condition for semi-infinite vertical plate. For this analysis, the parameter values are set as Ec = 0 , E 1 = 0 , Ha = 1 , Gr = 1 , and Pr = 0.72 . The validation process involves comparing our numerical results with established theoretical or experimental data. By doing this, we can assess the accuracy and reliability of our numerical scheme. The validation of our numerical scheme is accomplished by comparison of current results with the analogous results in Table 1 of Pop and Watanabe [37] and found to be in excellent agreement.

The obtained results for the temperature and velocity fields are used to calculate the local and average Nusselt numbers, as well as the skin friction, at plate 1. These values are determined using the following relationships:

4 Results and discussion

In this study, we have explored the free convection MHD flow of an electrically conducting fluid in the presence of external electric and magnetic fields between two vertical parallel plates under the nonuniform heating. We have numerically solved this flow model to determine the temperature, primary ( u ), and secondary ( w ) velocity profiles for a range of distinct values of the Hartmann number, Eckert number, electric field parameter, Hall current parameter, and time parameter. In contrast, Ha = 2 , Ec = 0.5 , E 1 = 0.1 , Gr = 5 , Pr = 7 , and m = 0.5 have been considered as default values throughout the numerical simulation.

For the considered physical parameters, Figures 2 and 3 present the primary and secondary velocity profiles, respectively, at distinct cross-sections of x ( x = 0.25 , 0.5, and 0.75). A comprehensive analysis of Figures 2 and 3 reveals that both the velocity profiles exhibit their maximum magnitude at x = 0.5 , and their minimum magnitude at the cross-section x = 0.25 . The impact of Ha on the u is demonstrated in Figure 2(a) and it is observed that the profile of u decreases with respect to Ha as an opposing Lorentz force effected between the fluid layers. However, Figure 3(a) reveals that Ha has completely reverse effects on w when compared with u , that is, an increment in Ha results in an increase in the w profile.

Figure 2

Velocity profile at t = 0.5 for distinct values of parameters: (a) Ha, (b) E 1 , (c) Ec, and (d) m .

Figure 3

Temperature profile at t = 0.5 for distinct values of parameters: (a) Ha, (b) E 1 , (c) Ec, (d) m ≤ 1 , and (e) m ≥ 1 .

Figures 2(b) and 3(b) demonstrated the u and w profiles for the electric field parameter ( E 1 ). From these figures, we depicted that both the velocity profiles increase as E 1 increased. It is also observed from Figures 2(b) and 3(b) that in the vicinity of the plate at y = 0 , which is heated nonuniformly, the u and w profiles have maximum magnitude at a fixed cross-section x = 0.5 , then reduce for x = 0.75 , and reach their minimum magnitude at x = 0.25 . However, on moving away from this plate, the maximum magnitudes of the u and w profiles are obtained at the cross-section x = 0.75 .

The impacts of viscous dissipation on the u and w profiles are presented in Figures 2(c) and 3(c), and we found that an increase in Ec enhances the viscous dissipative heat, which further causes a rise in the u and w profiles. Proceeding to analyze the Hall effect on u and w profiles, Figures 2(d), 3(d), and 3(e) come into focus. Figure 2(d) depicted that the profile of u enhanced with an increasing Hall parameter ( m ) . Additionally, the secondary velocity profile also increased with an increase in m up to m = 1 (Figure 3(d)). This can be attributed to the Hall current generating an additional velocity component in the secondary direction, which is orthogonal to both the magnetic field and the primary flow. As a result, the secondary velocity experiences an enhancement under these conditions. But beyond m = 1 , a decrement in the secondary velocity profile was noted (Figure 3(e)). This occurs because, at higher values of the Hall current parameter, the Hall effect induces stronger secondary forces, driving the fluid in a direction perpendicular to the primary velocity. As the magnitude of these forces increases, they may cause resistance within the thin boundary layer near the plate, leading to a reduction in the secondary velocity.

The temperature profile against y is explained in Figure 4 for various flow parameters, including Ha, E 1 , and Ec, at distinct cross-sections: x = 0.25 , 0.5, and 0.75. The assorted study of Figures 4(a)–(c) reveals that nonuniform heating maximizes the thermal energy at the middle position of the plate, i.e., x = 0.5 , while the minimum temperature is observed at cross-section x = 0.25 . This pattern indicates a continuous temperature increase for x < 0.5 and a decrease for x > 0.5 , demonstrating the dynamic nature of temperature distribution due to nonuniform heating. The respective effects of Ha, E 1 , and Ec on the temperature profile are individually detailed in Figures 4(a)–(c). Further, it becomes evident from Figure 4(a) that in the neighborhood of a nonuniformly heated plate, the temperature profile decreases as the value of Ha increases at x = 0.5 and 0.75. However, a completely opposite nature of the Hartmann number is observed for the fixed cross-section x = 0.25 , and at x = 0.5 beyond y = 0.18 , where the temperature increased with higher values of Ha. The temperature profile for the electric field parameter is depicted in Figure 4(b). As previously established, increasing values of E 1 enhances the flow velocities, which, in turn, generates frictional heat within the flow regime. This additional heat contributes to the rise in the temperature profile.

Figure 4

Temperature profile ( T ) at t = 0.5 for distinct values of parameters: (a) Ha, (b) E 1 , and (c) Ec.

Figure 4(c) demonstrates that the Eckert number thoroughly enhances the temperature of fluid flow at every cross-section x = 0.25 , 0.5, and 0.75 due to the fractional heat produced in the system, which shows an increase in the temperature profile.

Figure 5 shows how temperature, primary velocity, and secondary velocity profiles change over time, with a fixed cross-section at x = 0.5 . Both the velocity and temperature profiles show an upward trend with the increase in time. This shows that the flow and thermal fields become more intense over time. Furthermore, at increasing time values, the profiles are approaching a steady state. In particular, the temperature and velocity profiles (primary and secondary) exhibit reduced variance over time, attaining a constant pattern as time passes. This steady-state tendency indicates that transient impacts are decreasing with time. The variations in the local skin friction profiles ( τ u and τ w ) at plate y = 0 with flow parameters Ha, E 1 , and Ec for the distinct values of time are depicted in Figures 6 and 7, respectively. Collective study of Figures 6 and 7 shows the enhancement in local skin-friction profiles ( τ u and τ w ) for increasing value of time parameter. It has been obtained from Figures 6(a) and 7(a) that the increased values of Hartmann number reduced the τ u ; however, it enhanced the τ w profile. Figures 6(b) and 7(b) elaborate the influence of Ec on τ u and τ w . It is evident that both skin friction profiles increase with augmented values of Ec. The impact of the Hall parameter on τ u and τ w is presented in Figures 6(c) and 7(c). It is apparent from Figure 6(b) that the local skin friction τ u enhanced with m when m ≤ 1 , but a completely reverse trend is noted for m > 1 .

Figure 5

(a) Temperature profile, (b) primary velocity profile and (c) secondary velocity profile for distinct values of time parameter at cross-section x = 0.5 .

Figure 6

Variation in local skin friction ( τ u ) for distinct values of time along with flow parameters (a) Ha, (b) Ec, and (c) m .

Figure 7

Variation in local skin friction ( τ w ) for distinct values of time along with flow parameters (a) Ha, (b) Ec, and (c) m .

The local Nusselt number profile for different flow parameters Ha, Ec, and m at the different extents of time is investigated in Figure 8. It has been shown in Figures 8(a)–(c) that the local Nusselt number profile decreased when the value of the time parameter increased. In addition to this, Figures 8(a)–(c) depict that the rate of heat transfer increases with Ha, while a decrease is obtained for increasing Ec and m . Table 2 provides data representation for the variations in the average skin friction profiles ( ( τ u ) a v and ( τ w ) a v ) and the average Nusselt number profile ( Nu ) a v at t = 0.5 and x = 0 , respectively. For the average primary skin friction profile ( τ u ) a v , we observe that it decreases as the Hartmann number and electric field parameter increase. Conversely, ( τ u ) a v exhibits an increase when Ec and m are increased. Turning to the average secondary skin friction profile ( ( τ w ) a v ), we note an increase in its value as the Hartmann number, Eckert number, and electric field parameter are enhanced. In contrast, ( τ w ) a v decreases with changes in m . Finally, considering the average Nusselt number profile, a decline in its values is noted with an increase in the Hall parameter, Eckert number, and electric field parameter. However, ( Nu ) a v shows an increase as the Hartmann number is increased.

Figure 8

Variation in local Nusselt number for distinct values of time along with flow parameters (a) Ha, (b) Ec, and (c) m .

5 Conclusion

The free convective MHD fluid flow has been studied in this work under the influence of applied electric and magnetic fields in the presence of nonuniform heating via vertical parallel plates. The fluid is considered laminar and incompressible, and a constant magnetic and electric fields of strengths B 0 and E 0 have been applied. The flow equations had been solved numerically by employing an implicit method of Crank–Nicolson type. The numerical results are shown through graphs and the obtained results compared with the existing study. Based on the results and discussion, we conclude the following important points from the study:

1. The maximum magnitude of all considered profiles has been found at the midst section between two parallel plates.

2. The impact of Ha on the primary velocity is to reduce it, whereas the effect of the rest of the parameters is to increase it.

3. The temperature profile increased with an increase in Ec and E 1 while it diminishes with the increase in Ha.

4. The secondary velocity profile decreases with an enhancement in Hall current parameter for m > 1 while increases for m ≤ 1 , Ha, E 1 , and Ec.

5. τ u decreases with increase in Ha, while it increases with increase in t , E 1 , and Ec.

6. An increase in m for m ≤ 1 increases τ w , however for m > 1 , it diminishes with increase in the Hall parameter.

7. The rate of heat transmission reduces as t , Ec, and m increase.

8. ( τ u ) a v decreases with increasing Hartmann numbers and electric field parameters, while increases with elevated Ec and m .

9. ( τ w ) a v increases with increasing Hartmann numbers, Eckert numbers, and electric field parameters but decreases with changes in the Hall current parameter.

10. ( N u ) a v declines with increasing Hall current parameters, Eckert numbers, and electric field parameters but increases with elevated Hartmann number.