Impact of couple stress and variable viscosity on heat transfer and flow between two parallel plates in conducting field

1.   Introduction

In recent years, the major problem encountered by many researchers has been in the sector of magnetohydrodynamic (MHD) flows and heat transfer because of their significant applications in MHD pumps, MHD generators, aerodynamic heating, the petroleum industry, plasma studies, etc. ^[1,2,3]. The heat transfer in non-Newtonian fluids is necessary for the dismemberment of molten plastics, substantial oils and greases, synthetic fibers, and foodstuffs because non-Newtonian fluids have multiple applications in many areas of industry, biology, and engineering ^[4,5]. Stokes ^[6,7] proposed a couple stress fluid as one non-Newtonian fluid.

Falade et al. ^[8] and Farooq et al. ^[9] examined the heat transfer problem of couple stress fluid flow with variable viscosity between parallel plates. Srinivasacharya et al. ^[10] analyzed the Hall and Ion-slip effects with a variable heat source and MHD on couple stress fluid flow between two circular cylinders using the homotopy analysis method (HAM) and illustrated the impact of different appearing parameters on velocity and temperature fields. Numerous research studies ^{[11,12,13,14,15,16]} have expanded on the existing problems of couple stress fluid flow by including MHD with various effects and geometries. The impact of heat transfer and MHD on third-grade fluid flow between two parallel plates has been numerically investigated by ^[17,18,19] and examined in terms of the characteristics of velocity and temperature fields. Makinde et al. ^[20] showed how MHD and radiative heat transport work together to affect an optically thin fluid with a porous medium. Ahmed ^[21] looked into the steady MHD flow of an electrically conducting viscous fluid with heat and mass transfer in a porous medium and thermal diffusion. Umavathi et al. ^[22] examined the MHD Poiseuille-Couette flow of two non-mixable fluids, one of which conducts electricity and the other of which doesn't. They also looked at how new factors affect the speed and temperature distributions. Ahmed et al. ^[23] analysed the MHD flow of an electrically conducting viscous fluid with heat transfer across infinite annular vertical cylinders subjected to a time-dependent pressure gradient and thermal radiation. Ogunmola et al. ^[24] and Shah et al. ^[25] investigated the nature of velocity and temperature distributions by changing relevant factors and taking into account third-grade fluid flow and variable viscosity. Hassan et al. ^[26] studied the influence of varying viscosity and MHD on a heat-producing porous couple stress fluid with a convective cooling wall. Makinde et al. ^[27] investigated the MHD heat and mass transport effects on nanofluids using Poiseuille-Couette flow with the consideration of Hall effects and varying viscosity. Umavathi et al. ^[28] presented the variation in velocity and temperature domains by changing the physical characteristics of a couple stress fluid sandwiched between two viscous fluid layers. Imran et al. ^[29] investigated the applications of non-integer Caputo time fractional derivatives to natural convection flow subject to arbitrary velocity and Newtonian heating. Shah et. al. ^[30] presented the General solution for MHD-free convection flow over a vertical plate with ramped wall temperature and chemical reaction.

The innovative characteristics and aims of this work are noted by the fact that transportation heat couple stress model involves MHD, viscous dissipation, temperature-dependent viscosity, and inclination of the channel. This mathematical model is appropriate for the advantages of biological and lubrication systems. The behaviour of the fluid velocity, energy, skin friction and Nusselt number at the surfaces, total volume flow rate, and heat flow rate across the channel for various influential factors in the presence and absence of couple stress is discussed using graphs and tables.

2.   Basic equations and description of the flow model

2.1.   Basic equations

To discard the induced magnetic field, assuming the magnetic Reynolds number is very small.

The current density is given by

The Lorentz force is defined as

The dimensional version of the continuity, momentum, and energy equations is as follows:

where $\vec q$ is the vector velocity, $\varphi$ is the viscous dissipation of energy, C_p is the specific heat, $\vec J$ is the current density and B₀ is the applied magnetic field.

2.2.   Description of the flow model

It is considered the steady flow of a couple stress fluid passing through two infinitely long parallel plates separated by 2h, which are inclined from the surface at an angle of α. Plates are stable, and fluid flow is hydrodynamically and thermodynamically fully developed. The fluid is moving due to the axial pressure gradient and gravitational force. Θ₀ and Θ₁ are the temperatures at the bottom and top plates, respectively. A uniform magnetic field of intensity B₀ is applied, which is in a perpendicular direction to the fluid flow. The viscosity is temperature dependent. We have chosen a cartesian coordinate system such that the x-axis is along the central line of the channel and the y-axis is in a direction perpendicular to it, as seen in Figure 1.

The channel is long enough so that all the physical quantities are functions of y alone, except the pressure. The velocity and temperature are as follows:

Under the above assumptions, the Eqs (4) and (5) are transformed into

The boundary conditions of the flow model are

The temperature-dependent viscosity by Reynold's model ^{[16,31,32,33]} is given by

Introducing the non-dimensional parameters ^[4,16] are as follows:

where ${G_1} = - \frac{{\partial p}}{{\partial x}}$ is the pressure gradient parameter and ${G_2} = \frac{{{h^2}}}{{{U_0}{\mu _0}}}\rho g$ is the gravitational force parameter.

Equations (7) and (8) in the non-dimensional form (remove asterisks)

and the boundary conditions (9) reduced into

The dimensionless form of Eq (10) is

Use the Taylor's series, Eq (15) reduced into

Substitute Eq (16) in Eqs (12) and (13), then the equations are transformed in the form:

2.3.   Solution of the problem

Using the regular perturbation technique by taking the viscosity variation parameter m as a perturbation parameter where 0 < m < < 1, to solve the above non-linear coupled equations, and the velocity and temperature expressions are given by:

Substitute Eq (19) in Eqs (14), (17) and (18) and separating each order of approximation provides:

Zeroth order equations:

First order equations:

Solve the zeroth and first order equations with their corresponding boundary conditions.

The solution for velocity and temperature distributions are obtained as:

where Ω_i and γ_j, i = 1, 2……22 and j = 1, 2……47 are constants.

2.4.   Volume flow rate, total heat flow rate, coefficient of Skin friction and Nusselt number

The volume flow rate across the channel

By Eq (26),

Total heat flow rate across the channel is given by

At the lower plate, the skin friction coefficient C_f and Nusselt number Nu are given by

At the upper plate, C_f and Nu are given by

The negative sign is present because the top plate is positioned in the opposite direction of the fluid flow.

3.   Results and discussion

This section includes the variation of velocity and temperature profiles and a discussion of different values of the pressure gradient parameter G₁, viscosity variation parameter m, the Brinkman number Br, the Hartmann number M, the angle of inclination $\alpha$, and the couple stress parameter B. We also discussed the variation of volume flux, skin friction coefficient C_f, and the Nusselt number Nu on both plates through surface graphs.

Figure 2 shows that with the increase of m, the fluid viscosity decreases, and the viscous heating enhances, due to which the velocity increases.

Figure 3 depicts that the higher the pressure gradient, the faster the stream flows, and hence velocity increases.

According to Figure 4, a rise in B slows the motion of the fluid particles, providing them with enough force to pull off the frictional hindrance inboard the fluid and increase velocity in this manner.

Figure 5 shows that on increasing the values of Br, viscous dissipation dominates, due to which the velocity increases because the Brinkman number is the proportion of viscous heating to molecular conduction.

Figure 6 portrays that by raising the values of M, the Lorentz force gets stronger, which creates a resistive force in the fluid flow and causes the speed to slow down.

Figure 7 illustrates that as the angle of the inclined channel gets bigger, so does the velocity because fluid moves faster in a vertical channel than in a horizontal channel due to gravity.

According to Figures 8 and 9, raising the values of m and G₁ increases the viscous dissipation due to velocity and gradient, which enhances the fluid temperature.

Figure 10 shows that when B increases, so does the temperature in the flow channel due to the viscous heating of the fluid particles.

Figure 11 displays that if Br increases, then heat production through viscous dissipation enhances, resulting in a temperature increase.

Figure 12 exhibits the variation in M. An increase in Hartmann's number strengthens the Lorentz force, which acts against the fluid flow and so lowers the fluid temperature.

Figure 13 represents that as raising the value of α, causing the channel to become vertical, due to which the frictional force in fluid flow increases and so does the temperature.

Figure 14 shows that when the Brinkman number and viscosity parameter rise, so does heat production from viscous dissipation. This makes the volume flow rate across the channel go up.

Figure 15(a) portrays the coefficient of skin friction C_f at the lower plate. It shows that when both Br and $m$ grow at the same time, and when one of the parameters rises while the other remains unchanged, then the C_f increases. Figure 15(b) illustrates C_f at the upper plate. It shows that C_f grows with the increase of Br, whereas C_f initially increases for 0≤m≤0.3 and starts to decrease gradually for further increases in m.

Figure 16(a) and Figure 16(b) depict that when both $Br$ and $m$ increase in parallel time and if one of the parameters grows while the other stays constant, then the heat transfer irreversibility due to convection is more dominant and the temperature gradient rises, and hence the Nusselt number Nu at both the plates increases.

Table 1 represents the findings for the total rate of heat flow by varying the viscosity parameter, couple stress parameter, and the Hartmann number. The total heat flow rate increases as m and B increase, but higher Hartmann numbers lower the heat flow rate.

4.   Validation

4.1.   M=α=0

The current investigation in the absence of a magnetic field with the horizontal channel (α = 0) and the transformation ${y^{'}} = \frac{{y + h}}{2}$ validates the previously investigated work by Falade et al. ^[8].

The results are identical, which are represented in Figures 17 and 18 and Table 2 for the velocity profile by varying the Brinkman number, viscosity parameter, and couple stress parameter.

4.2.   M=B=α=0

The current study in the absence of a magnetic field and couple stress with the horizontal channel (α = 0) and the transformation ${y^{'}} = \frac{{y + h}}{2}$ validates the previously investigated work by O D Makinde ^[32].

For the velocity profile with variable viscosity parameter and Brinkman number, the findings shown in Figures 19 and 20 are equal.

5.   Conclusions

The impact of MHD and temperature-dependent viscosity on couple stress fluid flow has been studied between two heated inclined parallel plates. The effects of relevant parameters on the fields of velocity, temperature, volume flow rate, total heat flow rate, coefficient of skin friction, and Nusselt number are concluded as follows:

● Velocity and temperature enhance with an increase in the pressure gradient, viscosity variation, couple stress parameters, Brinkman number, and angle of the inclined channel, whereas both reduce as the Hartmann number grows.

● The rate of volume flow across the channel increases as the viscosity variation parameter and Brinkman number increase.

● As the viscosity variation and couple stress parameters rise, the total rate of heat flow across the channel enhances, but it declines as the Hartmann number goes up.

● At the lower plate, skin friction increases as the viscosity variation parameter and Brinkman number increase. At the upper plate, it grows as the Brinkman number increases, whereas with the variation of the viscosity parameter, it initially increases and then gradually declines.

● The Nusselt number at both plates increases as the viscosity variation parameter and Brinkman number increase.

Acknowledgments

Geetika would like to express her heartfelt gratitude to CSIR-UGC for their financial support with grant no. 09/1313(0001)/2020-EMR-I. This study was supported by the National Research Foundation of Korea (NRF) funded by the Korean government (MSIT) [NRF-2021R1I1A3047845, NRF-2022R1A4A3023960] and BK21 Four program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education of Korea (Center for Creative Leaders in Maritime Convergence).

Conflict of interest

The authors declare no conflict of interest.