Oldroyd 6-constant Electro-magneto-hydrodynamic fluid flow through parallel micro-plates with heat transfer using Darcy-Brinkman-Forchheimer model: A parametric investigation

1.   Introduction

Several research studies have been accomplished recently on the heat transfer through microporous media and micro-channels with applications in thermal control in radiators, heat pipes, and microelectronics. Micro-channel systems can provide efficient heat removal at substantially smaller dimensions for cooling systems in aerospace engineering ^[1]. The use of micro-radiators for thermal control enables the reduction of the maximum temperature and temperature gradients on the equipment when subjected to large heat fluxes ^[1]. Heat transfer mechanisms are becoming increasingly important in the modeling of several other applications including magnetic fluid power production ^[2], geophysical flows, and nuclear reactor cooling ^[3]. Heat transfer fluid flow through a porous media is also associated with several engineering applications in nuclear waste disposal systems, thermal energy transfer, and thermal management ^[4]. The subjects of heat transfer and flow in porous media also play an important role in the energy sector, including electro-kinetic energy conversion devices, hydrogen storage systems, shale reservoirs, and membrane-based water desalination with reverse osmosis.

Given its scientific and engineering significance, many researchers investigated heat transfer in various geometric domains and for various fluid models. Moradi et al. ^[5] investigated heat transfer over a porous medium using a double tube filled with water-based MWCNT nanofluids. Natural convection and heat transfer with a fluid via a coarse-grained permeable medium were investigated by Ataei-Dadavi et al. ^[6]. Pandya et al. ^[7] reviewed the heat transfer effectiveness for plate heat exchangers with nanofluids. Miri, Joibary and Siavashi ^[8] explored the role of Reynolds asymmetry and the use of porous media in a counter-flow dual-pipe heat exchanger to improve the heat transfer. Zhang et al. ^[9] used the Darcy–Brinkman–Forchheimer model for porous media to investigate the bioconvection flow over a permeable Riga plate. Also, Selimefendigil and Oztop ^[10] investigated the impact of nanofluids and dual porous layers in order to evaluate the heat transfer from a plate using impinging flow single-jets and multi-jets.

Heat transfer with magnetic fields is technologically important because of its numerous applications in the metallurgical industry (fluid metal flows), micro-pump flows, biological flows, and medicinal uses. In particular the electro-magneto-hydrodynamics micro-pump has attracted the attention of several researchers, because of its possible uses in microfluidic systems ^[11,12]. Magnetic fields generate the Lorentz force by the interactions of an externally induced electrical current in channel filled with electrically conductive fluids ^[13,14,15]. The effects of, viscous dissipation, joule heating, and thermal properties on EMHD fluid flow over a microchannel with constant heat flux were discussed by Chakraborty et al. ^[16]. Sarkar et al. ^[17] investigated the behavior of EMHD flow towards a microchannel when it was subjected to slip and electro-kinetic phenomena. Rashid et al. ^[18] studied the effects of EMHD and permeability on the fluid flow across a corrugated microchannel with varying viscosity. Reza et al. ^[19,20] investigated a third-grade fluid flow in a microchannel with EMHD and porous effects. Obalalu et al. ^[21] explored the effect of phase change heat transfer with a non-Newtonian EMHD nanofluid with activation energy and chemical reactions.

The main objective of this research study is to investigate the Oldroyd 6-constant, Electro-magneto-hydrodynamic fluid flow, across parallel and porous micro-plates with heat transfer, using the Darcy-Brinkman-Forchheimer model. This investigation is motivated by their fundamental importance in industrial engineering applications as well as their value for experimental purposes. Flows in microfluidic channels are also relevant in a variety of sectors ^[22,23], including biomedical and biochemical processes, heat exchange due to surface forces and physical particle separation. The fluid flow in the stated field is efficiently controlled by using electric fields, pressure gradients, magnetic fields, or appropriate combinations of these driving forces. The EMHD micro-pump is one of the several pumping concepts that has use in micro-fluidics systems. Because the operating concept of the EMHD micro-pump is based on the Lorentz force (created by the interaction of an externally imposed electric current along the channel with ferrofluid with a transverse magnetic field orthogonal to the current) ^[24].

In addition, the Oldroyd 6-constant fluid (non-Newtonian) has a number of industrial uses. Several fluid models, including Oldroyd's, have been presented in recent years. This model incorporates the elastic and memory effects that dilute solutions demonstrate. Most of the Oldroyd-B fluid models available in the literature ^[25,26] are confined to 3-constant models, and, hence, cannot cover the full characteristics of viscoelastic flows. This is why we chose the Oldroyd 6-constant fluid model to investigate its properties. After employing the assumptions to the mathematical modeling, the governing differential equations of the model are nonlinear and coupled. Because of this, numerical simulations are carried out using the shooting the method with the Mathematica software. When compared with other similar approaches, the shooting method delivers results quickly with simple computations. Results are presented for the velocity profile, the Nusselt number, and the temperature distributions for all the important to the flow parameters.

2.   Mathematical and physical formulation of the Oldroyd 6-constant fluid

Consider an electrically conducting Oldroyd 6-constant ^[27] incompressible fluid that moves through parallel micro-plates under the influence of electro-magnetic forces. The fluid is moving through a porous medium. To investigate the behavior of the Oldroyd 6-constant fluid through the porous medium, the Darcy-Brinkman-Forchheimer model is used. A system of rectangular coordinates is selected such as $\tilde x -$ and $\tilde z -$ are assigned along the tangential direction on the charged plates, while the $\tilde y -$ axis is assigned along the perpendicular direction, as shown in Figure 1.

Due to the presence of electrical field $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E}$ along the $\tilde z -$ axis in the transverse direction, and the homogenously applied magnetic field $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B}$ along the $\tilde y -$ axis, a magnetic Lorentz force is generated which acts along the $\tilde x -$ axis. The magnetic Reynolds number is assumed to be very small and, therefore, all other effects of magnetism are neglected except for the Lorentz force. The length of the micro-channel is denoted by $L$ along the $\tilde x -$ axis, the width is denoted by $W$, and the height of the micro-channel is denoted by 2H (typically its range is $100 - 200$μm). The length of the microchannel is much larger than the height and the width, that is $L > > 2H$ and $L > > W$. According to these assumptions, the two-dimensional micro-channel Oldroyd 6-constant fluid flow is reduced to a one-dimensional steady-state flow for which the velocity is independent of $\tilde z$. The continuity equation for this flow is:

Using Darcy-Brinkman-Forchheimer model and body forces, the momentum equation becomes:

where $\xi$ is the stress tensor of the Oldroyd 6-constant fluid, $g$ is the gravity, ${B_f}$ is the body force, $p$ is the pressure, $\beta$ the thermal expansion coefficient, $K$ is the permeability of the porosity, ${c_F}$ is the Forchheimer coefficient, $\rho$ is the density, $\tilde t$ is the time, and ${\tilde T_m}{\text{ and }}{\tilde T_s}$ are the average temperature and the surface temperature. The body force is the induced Lorentz force, defined as follows ^[28]:

Where $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {J}$ is the vector that represents the local ion current density and is given by the expression:

where $\sigma$ represents the electrical conductivity, $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B}$ the magnetic field, and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E}$ the electric field.

The stress tensor of the Oldroyd 6-constant fluid is expressed by the following equation:

Where ${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {R} _1}$ the first Rivlin-Erickson tensor, ${\lambda _1}, {\lambda _2}, {\lambda _3}, {\lambda _4}, {\lambda _5}$ are constants of the material, and $\mu$ is the viscosity of the fluid. The total derivative, ${D \mathord{\left/ {\vphantom {D {D\tilde t}}} \right. } {D\tilde t}}$ is defined as follows:

Where $\frac{{\text{d}}}{{{\text{d}}\tilde t}}$ denotes the material derivative. The detailed derivation of Eqs (5) and (6) are presented in the Appendix.

The energy equation for this fluid, which is subjected to viscous dissipation and Joule heating effects is:

where the symbol "$:$" indicates the tensorial product, $\tilde T$ represents the temperature, ${S_h}$ denotes the specific heat, and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {Q}$ is the heat flux vector.

For the chosen geometry shown in Figure 1, the velocity vector is defined by the following equation:

Substituting Eq (5) in Eq (2), we obtain:

The pressure function only depends on the $\tilde y$ coordinates. Therefore, we obtain the following expression for the momentum equation:

where ${\beta _1} = {\lambda _2}{\lambda _1} - \left({{\lambda _4} - {\lambda _2}} \right)\left({{\lambda _5} + {\lambda _3}} \right), {\beta _2} = {\lambda _3}{\lambda _1} - \left({{\lambda _3} - {\lambda _1}} \right)\left({{\lambda _5} + {\lambda _3}} \right)$ are derived from Eq (5).

The boundary conditions of the flow are:

The governing equations of this model may be cast in dimensionless form using the following dimensional parameters:

Where, $\upsilon$ represents the kinematic viscosity of the fluid, $Ha$ is the Hartmann number, $T$ is the dimensionless temperature, ${E_1}$ is a dimensionless parameter related to the electric strength, $Da$ is the Darcy parameter and represents the porosity, Gr is the thermal Grashof number, $F$ represents the Forchheimer number, which accounts for the non-Darcian effects of the porous medium and its range is from 0 to $\infty$.

Accordingly, Eq (12) becomes in dimensionless form:

The boundary conditions in dimensionless form are simplified as follows:

The EMHD fluid's energy equation is as follows:

where ${S_h}$ the specific heat capacity, and ${T_c}$ represents the thermal conductivity of the fluid. In Eq (17), the second term indicates the volumetric heat generation caused by the viscous dissipation, and the third term indicates the impact of Joule heating.

The temperature profile for the fully developed flow depends on the $\tilde y$ direction only, and satisfies the condition:

Using the boundary conditions and the above equation, we obtain the following equations:

where $k$ is constant.

The energy equation with its pertinent boundary conditions may be cast in the following form:

where ${Q_s}$ represents the constant heat flux at the wall. For the geometric configuration examined here, the energy equation for an infinitesimal section with length ${\text{d}}\tilde x$ becomes:

Hence, the mean temperature may be written as follows:

where,

The coefficients of the previous equations that are used in the calculations are:

The energy equation, Eq (20), and its boundary conditions become in dimensionless form:

where $Bm$ is the Brinkman number –the ratio of heat produced because of viscous dissipation to the conduction heat transfer; ${\xi _1}\, {\text{ and }}{\kern 1pt} {\xi _2}$ are defined in Eq (14);${\xi _3}$ represents the ratio of the Joule heating to the heat conduction; and ${\xi _4}$ represents the influence of heat generation because of the influence of the electric and magnetic fields to the heat conduction. These dimensionless parameters are expressed as follows:

The Nusselt number, which denotes the effects of convective heat transfer, is written as follows:

where ${Q_s} = {h_c}\left({{{\tilde T}_s} - {{\tilde T}_m}} \right),$ and ${h_c}$ is the coefficient of convective heat transfer and, ${D_H}\left({ = H} \right)$ denotes the hydrodynamics diameter of the microchannel. Using Eqs (21) and (29), the final form of the Nusselt number at the upper wall is expressed as:

3.   Solutions of the governing equations

Because the final forms of Eqs (15) and (26) are nonlinear, closed-form solutions cannot be obtained. Thus, a numerical scheme is proposed to obtain the solution of the equations. The shooting method is the most effective scheme to solve nonlinear differential equations. To use this numerical scheme, we first reduce Eqs (15) and (26) to first-order differential equation systems and then solve them using the numerical algorithm developed in Mathematica. This technique is advantageous to other similar methods, because it produces better results in a shorter period ^[29,30,31]. As a result, Eqs (15) and (26) are reduced to the following form:

The boundary conditions are:

4.   Graphical and numerical analysis

This section focuses on the graphical presentation of the numerical results as affected by all the variation of the important parameters for flow and heat transfer. In particular, we investigate the behavior of the velocity profile, the Nusselt number, and the temperature profile. The following parametric values are chosen in the calculations: the semi-height of the geometric domain is in the range $H{\text{ = }}100 - 200 \;$μm; the range of the imposed magnetic field is $0.018 - 0.44{\text{T; }}$ the values of the Hartman number are in the range $0.8 \times {10^{ - 6}}{\text{ to 3}}$; the value of the imposed electric field strength is $1 \times 60{\text{V/m}}$; the dimensionless parameter ${E_1}$ is in the range $0.4 \times {10^{ - 8}}{\text{ to }}2.1 \times {10^5};$ The Brinkman number is $Bm = 10.5;$ the Oldroyd 6-constant fluid parameters are ${\xi _1} = 1, {\xi _2} = 0.5;$ the Darcy-Brinkman-Forchheimer parameter is $F = 1;$ and the thermal Grashof number is $Gr = 0.2.$ It must be noted that the results for a simply viscous Newtonian fluid can easily recovered in this model by setting ${\xi _1} = {\xi _2} = 0$.

Table 1 shows the computational results of the Nusselt number against all the important parameters of this problem.

Figure 1 is a schematic representation of the micro-pumps, showing the dimensions and the direction of the heat transfer. Figure 2 shows the results of the Nusselt number in graphical/column form. It is clearly observed in this Figure that the Nusselt number is suppressed due to the strong impact of Hartmann number $Ha$, Darcy parameter Da, and the fluid parameter ${\xi _1}.$ However, the Nusselt number increases with the Brinkman number $Bm,$ the thermal Grashof number $Gr,$ and the fluid parameter ${\xi _2}$.

Figure 3 is plotted for the velocity profile to show the behavior of the fluid parameters ${\xi _1}{\text{ and }}{\xi _2}.$ One can see in this Figure that increasing the values of ${\xi _1}$ suppress the magnitude of the velocity. On the other hand, the fluid parameter ${\xi _2}$ acts in reverse and increases the velocity. The solid line corresponds to the case of a viscous fluid.

The impact of the buoyancy forces on the working fluid is represented by the thermal Grashof number. Figure 4 shows the variation of the thermal Grashof number, $Gr$, and is effects on the fluid velocity. The thermal Grashof number augments the velocity profile along the entire channel, as shown in this graph.

The effects of the Darcy-Brinkman-Forchheimer parameter, F, is shown in Figure 5. It is apparent that increasing F significantly retards the flow. A similar behavior was observed in the viscous fluid flows through a Riga plate ^[32].

Figure 6 shows the influence of the Hartmann number, Ha, on the fluid velocity. The existence of an external magnetic field introduces resistance to the fluid flow, as it is apparent in the Figure. The resistance is caused by the Lorentz force, which is a consequence of the combined application of the electric and magnetic fields. The case $Ha = 0$ corresponds to the absence of the magnetic field and the velocity profile attains its maximum. As expected, the velocity field is also retarded with the Darcy parameter, Da, whose influence is plotted in Figure 7. The effects of the parameters Ha and Da are very similar on the entire velocity profile. Figure 8 shows the variation of electric field E₁ strength on the velocity profile. It is observed in the Figure that increasing values of the electric field enhances the velocity profile, since electric field produces larger aiding forces which boost the velocity profile.

Figures 9–16 depict the effect of the several parameters on the temperature profile. As shown in Figure 9, the Brinkman number Bm, which is the product of the Prandtl number and the Eckert number, enhances the temperature profile. The higher values of the Brinkman number impede the heat diffusion caused by the viscous dissipation, causing the temperature profile to grow.

Figure 10 shows the effect of the fluid parameters ${\xi _1}{\text{ and }}{\xi _2}$. It is observed that the variation of the two parameters affects very little the temperature profile. In contrast, the variation of the Grashof number has a very significant effect on the temperature profile as shown in Figure 11.

The variation of the Darcy-Brinkman-Forchheimer parameter, F, on the temperature profile is shown in Figure 12, where it may be seen that increasing F has a small but detrimental effect on the temperature profile. Figure 13 shows that the magnetic field parameter, Ha, diminishes the temperature profile. This effect also causes significant entropy production and exergy destruction ^[33]. On the contrary, the temperature profile was dramatically decreased at the larger values of the parameter ${\xi _3}$, as it may be observed in Figure 14. The effect of the Darcy number, Da, is similar since it lowers the temperature profile, as it is shown in Figure 15. It may be concluded from the last two cases, that increasing the parameters ${\xi _3}$ and Da reduces the thermal dissipation and exergy destruction ^[33]. The last Figure 16 shows the variation of the electric field strength on the temperature profile. It can be seen that larger values ${E}_{1}$ cause significant increment of the temperature profile. It must be noted that the case ${E}_{1} = 0$ corresponds to the absence of electric field and its influence on the flow regime.

5.   Conclusions

The Darcy-Brinkman-Forchheimer model was used to investigate the behavior of an Oldroyd 6-constant fluid in an electro-magneto-hydrodynamic fluid flow through parallel micro-plates with heat transfer. The effects of the viscous dissipation and Joule heating are considered in the model. The governing equations for this type of fluid are nonlinear and coupled, which implies that exact solutions are not possible to obtain. Hence, we used the shooting method to obtain numerical solutions. The governing equations are first developed and then cast in dimensionless form. With the results of the numerical solution and the effects of all the pertinent dimensionless variables are discussed using graphs and tables. The following are the key conclusions of the study:

i.        The Nusselt number increases with the Brinkmann and Grashof numbers, but decreases with the Hartmann number.

ii.        The Grashof number significantly increases the fluid velocity, but the Darcy parameter and the magnetic field, which creates the Lorenz force impede the fluid motion.

iii.        The Oldroyd 6-constant fluid parameters affect both the velocity and temperature profiles, but their effects on the temperature profile are very small.

The Darcy-Brinkman-Forchheimer parameter slightly decreases the temperature profile. The parameter associated with the magnetic field, Ha, acts to increase the temperature profile.

Conflict of interest

The authors declare no conflict of interest.

Appendix

The scalar forms (time-dependent) of Eq (5) are found as:

The time-independent equations are as follows: