Entropy Generation via Ohmic Heating and Hall Current in Peristaltically-Flowing Carreau Fluid

The core objective of the present study is to examine entropy generation minimization via Hall current and Ohmic heating. Carreau fluid considerations interpret the unavailability of systems’ thermal energy (for mechanical work). The magneto hydrodynamic flow is in the channel, which is not symmetric. We have solved analytically the resulting nonlinear mathematical model. Moreover, physical exploration of important parameters on total entropy generation, temperature, and Bejan number is plotted and discussed. We observed that the generation of entropy takes place throughout the confined flow field y = W1 and y = W2 because of the viscous dissipation effect. In addition, reducing the operating temperature minimizes the entropy.


Introduction
Currently, scientists have a major concern about finding a way to control the wastage of heat energy. In thermodynamics, entropy defines thermal irreversibility, often referred to as the destruction of useful energy. Production of entropy is associated with all real-life process. Entropy generation analysis is important in exploring the sources and location of irreversibilities, which are responsible for the destruction of useful energy. The losses in heat energy are mainly due to friction, compression and expansion, heat transfer, magnetic field, and chemical reactions. Minimizing the loss of heat and improving the efficiency of the thermal system are possible only through minimization of entropy generation. Therefore, it is extremely important to study entropy in all real process. Different techniques are being used to decrease the entropy generation, such as the reduction in size of chip components in a computer, cooling fans preventing overheating, porous media, and the heat exchanger.
The laws of thermodynamics define the transformation of energy. The quantity of energy in the heat transfer process is an important factor and is governed by the first law. Hayat et al. [1] studied the impact of the Cattaneo-Christov heat flux model in the flow of variable thermally-conductive fluid. Khan et al. [2] explained the homogeneous-heterogeneous reactions in Casson fluid flow. Most of the engineering problems concern with the quality of energy and the degree of degradation of energy. The second law of thermodynamics defines the decrease in the quality of energy, such as the reduction in the quality of energy measured as entropy. In order to minimize the entropy generation within the fluid flow problem, it is important to learn the distribution of entropy generation. Bejan [3] laid the foundation of entropy generation and analyzed its minimization. Afridi et al. [4] developed the analysis of heat and mass transfer in entropy generation. In another study, Afridi et al. [5] analyzed entropy in hydromagnetic boundary flow. Rashidi et al. [6] studied the entropy generation on peristaltic MHD where W 1 and W 2 represent the lower and upper walls, b 1 , b 2 the amplitudes, φ the phase difference, s the wave speed, λ the wavelength, and t the time. The geometry of the flow problem is given in Figure 1. The Carreau fluid of constant density, moving in a channel, which is asymmetric in nature, is considered here. The rectangular coordinates are ( , ) with U as the axial velocity component. Channel walls are maintained at temperatures 0 T and 1 T . The velocity field is mathematically defined by ( , , 0). The walls of the geometry are given as: where and represent the lower and upper walls, , the amplitudes, the phase difference, the wave speed, the wavelength, and ̅ the time. The geometry of the flow problem is given in Figure 1.
Here, J shows the current density B for the magnetic field, and σ represents the electric conductivity of the fluid. The constitutive laws of mass, momentum, and energy via the Joule heating, Hall current, and viscous dissipation are: where , S pI τ = − + ρ shows density, κ is the thermal conductivity, p C the specific heat, and τ the extra stress tensor.

Fluid Model
The stress-strain relationship of the Carreau fluid model is: The Lorentz body force and Joule heat affecting the flow are determined by: Here, J shows the current density B for the magnetic field, and σ represents the electric conductivity of the fluid. The constitutive laws of mass, momentum, and energy via the Joule heating, Hall current, and viscous dissipation are: dρ where S = −pI + τ, ρ shows density, κ is the thermal conductivity, C p the specific heat, and τ the extra stress tensor.

Fluid Model
The stress-strain relationship of the Carreau fluid model is: and here, η ∞ , η 0 are infinite and initial shear rate viscosities and . γ = 1 2 traceA 1 2 . The components of extra stress tensors τ ij are:

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As p p(y), therefore Equation (17) yields: In the above equations, B r , W e , M f , m e , ψ, and θ are the notations for the Brinkman number, Weissenberg number, Hartman number, Hall parameter, stream function, and temperature, respectively. The associated non-dimensional boundary conditions are: where F = ∂ψ ∂y dy is related to the fixed frame by F = Q − 1 − d.

Analysis of Entropy Generation
Fluid irreversibilities in the current problem are due to heat diffusion, viscous dissipation, and the magnetic field, respectively. Based on these, the dimensional entropy generation is defined as: The characteristic entropy is defined as S 0 = d 1 2 κ . The total entropy generation rate, denoted by N ts , is the relation between the actual entropy to the characteristic entropy. In dimensional form, entropy generation via stream function presentation: Here, the temperature difference parameter is labelled by ξ = T 0 T 1 −T 0 , and N ts = N H + N F + N M . The Bejan number identified by B e is the proportion of heat irreducibility to the total entropy generation. Basically, the B e number comprehends the mechanism of the production of entropy.
B e = 1 2 defines equal irreversibility, due to heat and other contributing factors. For dominating heat irreversibilities, B e = 1, while B e = 0 implies that the contributing factors of fluid friction and magnetic field are noteworthy. The Bejan number ranges between zero and one.

Solution Methodology
Our problem is non-linear and coupled in nature. The computation of the exact solution is not possible; therefore, perturbation techniques are employed to solve the resulting governing equations. We apply regular perturbation of the fluid parameter, the Weissenberg number W e 2 as: Substituting these into Equations (19) and (20), we construct the zeroth order and first order systems with reference to the fluid parameter.

Zeroth Order System and Boundary
Conditions

Discussion and Results
The impact of different physical parameters, i.e., B r , W e , M f , m e , Bi 1 , Bi 2 , and φ, on total entropy generation, entropy production due to the friction and heat diffusion, temperature, Bejan number, heat transfer rate, pressure gradient, streams lines, and velocity profile are discussed in this section.  Figure 2a,c portrays that N ts was gradually enhancing for increasing values of the Hartman and Bejan numbers. With the application of the magnetic field, the temperature increased. Joule heating produced more heat, so entropy production increased. At the lower wall, entropy generation was maximum as compared to the other wall (due to temperature gradient). It can also be noticed that at the lower wall, fluid friction irreducibility was dominant, whereas at the upper wall, heat transfer reduced the entropy generation. Figure 2b shows the effect of the Hall parameter m e , which reduced the entropy generation. In Figure 2c, as we increased the Brinkman number (the conduction of energy that was produced by viscous dissipation), entropy generation increased. Figure 2d depicts that with an increment in temperature difference, the entropy generation gradually decreased. Figure 3a indicates the Bejan number for the variation of the Hartman number. It reveals that heat irreversibility at the bulk fluid region was dominant, while at the edges, magnetic and viscous irreversibility were dominating. Figure 3b presents that Bejan number decreased with the Hall parameter m e at y = 0. Figure 3c,d show that with the increase in the Brickman number and temperature difference parameter, the Bejan number decreased.
of the Hartman and Bejan numbers. With the application of the magnetic field, the temperature increased. Joule heating produced more heat, so entropy production increased. At the lower wall, entropy generation was maximum as compared to the other wall (due to temperature gradient). It can also be noticed that at the lower wall, fluid friction irreducibility was dominant, whereas at the upper wall, heat transfer reduced the entropy generation. Figure 2b shows the effect of the Hall parameter e m , which reduced the entropy generation. In Figure 2c, as we increased the Brinkman number (the conduction of energy that was produced by viscous dissipation), entropy generation increased. Figure 2d depicts that with an increment in temperature difference, the entropy generation gradually decreased. Figure 3a indicates the Bejan number for the variation of the Hartman number. It reveals that heat irreversibility at the bulk fluid region was dominant, while at the edges, magnetic and viscous irreversibility were dominating. Figure 3b presents that Bejan number decreased with the Hall parameter e m at 0. y = Figure 3c and Figure 3d show that with the increase in the Brickman number and temperature difference parameter, the Bejan number decreased.         Figure 4a depicts that the temperature profile gradually rose for increasing values of the Hartman number M f . Basically, magnetic field lines interacted electrically with the fluid and produced Lorentz force. Lorentz force retarded the fluid motion (transforming the kinetic energy of the electrically conducting fluid to heat energy), and fluid temperature rose. It is found from Figure 4b that the temperature is lowered because of the increase in the electrical conductivity of the fluid. Figure 4c elucidates that under the influence of the Brinkman number, the temperature rose. The reason behind this is that for a large value of the Brickman number, the frictional force increased (due to the collision of fluid molecules with each other), and as a result, kinetic energy converted into thermal energy, implying a rise in total fluid temperature. The influence of the Biot number on the temperature is presented in Figure 4d,e. Temperature decreased at the upper wall by the increase of Bi 1 , and it had no visible effect on the lower wall. In contrast, the temperature escalated at the lower wall with the increase of Bi 2 , and a negligible difference was observed on the upper wall. In most of the cases, for small Biot numbers, temperature uniformly distributed inside the fluid, whereas for Biot numbers greater than 0.1, irregularity resulted. Therefore, we tool a special case for a large value of the Biot number. Figure 4f elucidates that temperature increased for increasing values of the power law index.

Analysis of Velocity
Axial velocity serves to provide salient feature of flow behavior. Figure 5a-c portrays the impact velocity profile in a channel with convective boundaries. We observed that the velocity formed a parabolic trajectory for physical parameters, and maximum velocity occurred at 0 y = . Figure 5a portrays that the axial velocity decreased for the increasing value of the Hartman number. Since the Hartman number directly relates the magnetic force and this force is resistive in nature, therefore the

Analysis of Velocity
Axial velocity serves to provide salient feature of flow behavior. Figure 5a-c portrays the impact velocity profile in a channel with convective boundaries. We observed that the velocity formed a parabolic trajectory for physical parameters, and maximum velocity occurred at y = 0. Figure 5a portrays that the axial velocity decreased for the increasing value of the Hartman number. Since the Hartman number directly relates the magnetic force and this force is resistive in nature, therefore the velocity decreased. Figure 5b demonstrates the influence of the Hall parameter m e . Here, the velocity accelerated at the center of the channel while it reduced at the edges, the reasons behind this being that m e caused an upsurge of the electrical conductivity of the fluid; hence, the velocity increased. Figure 5c shows the comparison of viscous and Carreau fluids. It gained maximum velocity for a Newtonian fluid, while it reduced for the non-Newtonian Carreau fluid. Furthermore, it restored the symmetry about the center line.

Analysis of the Pressure Gradient and the Rate of Heat Transfer
The influence of  Figure 6a reveals that for the increasing value of the Hartman number, the pressure gradient decreased at the narrow part, whereas it increased at the wider region. Figure 6b depicts the influence of the Hall parameter. ⁄ decreased at the wider region, and a negligible difference was observed at the narrow part. Figure 6c portrays that with the increase of the Weissenberg number, the pressure gradient ⁄ increased at the narrow and wider region. Figure 7a, d presents that for higher values of the Biot number and Hall parameter, the heat transfer rate reduced; whereas the heat transfer rate increased for the Brinkman number and Hartman number (Figure 7c, b).

Analysis of the Pressure Gradient and the Rate of Heat Transfer
The influence of M f , m e , W e , and B r is analyzed through Figures 6a-c and 7a-d to peruse the pressure gradient and rate of heat transfer. Figure 6a reveals that for the increasing value of the Hartman number, the pressure gradient decreased at the narrow part, whereas it increased at the wider region. Figure 6b depicts the influence of the Hall parameter. dp/dx decreased at the wider region, and a negligible difference was observed at the narrow part. Figure 6c portrays that with the increase of the Weissenberg number, the pressure gradient dp/dx increased at the narrow and wider region. Figure 7a,d presents that for higher values of the Biot number and Hall parameter, the heat transfer rate reduced; whereas the heat transfer rate increased for the Brinkman number and Hartman number (Figure 7b,c).
peruse the pressure gradient and rate of heat transfer. Figure 6a reveals that for the increasing value of the Hartman number, the pressure gradient decreased at the narrow part, whereas it increased at the wider region. Figure 6b depicts the influence of the Hall parameter. ⁄ decreased at the wider region, and a negligible difference was observed at the narrow part. Figure 6c portrays that with the increase of the Weissenberg number, the pressure gradient ⁄ increased at the narrow and wider region. Figure 7a, d presents that for higher values of the Biot number and Hall parameter, the heat transfer rate reduced; whereas the heat transfer rate increased for the Brinkman number and Hartman

Trapping Phenomenon
Streams lines were plotted to depict the flow pattern. The trapping phenomenon for fluid parameters M f (Hartman number), m e (Hall parameter) and W e (Weissenberg number) was described through plotting the streams lines. Figures 8-10 show that the bolus size decreased for increasing values of the Hartman number and Weissenberg number. Figure 10a-c depict the opposite trend that is for higher values of hall parameter bolus size not only increases but number of closed stream lines also increases in count.

Analysis of Entropy Generation due to Heat Diffusion and Viscous Dissipation
The thermal entropy generation rate and viscous entropy generation are very important in entropy generation. The influence of the Hartman number ( ) and Weissenberg number ( ) was studied to configure the entropy generation due to the heat diffusion and viscous dissipation effect. Figure 11a elucidates that for the increasing value of the Hartman number, the heat diffusion rate increased. This physically happens due to the strong magnetic field, which boosts the temperature. Therefore, the diffusion rate increased. Figure 11b illustrates that for higher values of , the heat diffusion rate decreased. Variation in the entropy generation rate due to viscous dissipation for different values of the Hartman number is observed through Figure 12a,b. The increase in the thermal entropy generation rate at the walls was observed due to resistive forces, while heat production dropped off due to low viscosity at the center of the channel.

Analysis of Entropy Generation Due to Heat Diffusion and Viscous Dissipation
The thermal entropy generation rate and viscous entropy generation are very important in entropy generation. The influence of the Hartman number (M f ) and Weissenberg number (W e ) was studied to configure the entropy generation due to the heat diffusion and viscous dissipation effect. Figure 11a elucidates that for the increasing value of the Hartman number, the heat diffusion rate increased. This physically happens due to the strong magnetic field, which boosts the temperature. Therefore, the diffusion rate increased. Figure 11b illustrates that for higher values of W e , the heat diffusion rate decreased. Variation in the entropy generation rate due to viscous dissipation for different values of the Hartman number is observed through Figure 12a,b. The increase in the thermal entropy generation rate at the walls was observed due to resistive forces, while heat production dropped off due to low viscosity at the center of the channel.

Analysis of Entropy Generation due to Heat Diffusion and Viscous Dissipation
The thermal entropy generation rate and viscous entropy generation are very important in entropy generation. The influence of the Hartman number ( ) and Weissenberg number ( ) was studied to configure the entropy generation due to the heat diffusion and viscous dissipation effect. Figure 11a elucidates that for the increasing value of the Hartman number, the heat diffusion rate increased. This physically happens due to the strong magnetic field, which boosts the temperature. Therefore, the diffusion rate increased. Figure 11b illustrates that for higher values of , the heat diffusion rate decreased. Variation in the entropy generation rate due to viscous dissipation for different values of the Hartman number is observed through Figure 12a,b. The increase in the thermal entropy generation rate at the walls was observed due to resistive forces, while heat production dropped off due to low viscosity at the center of the channel.

Conclusions
We analyzed the entropy generation via the Ohmic heating and Hall current in peristalticallyflowing Carreau fluid. The conclusions are stated below.

•
Entropy generation is not zero at the centerline y = 0.

•
Heat irreversibility, at the bulk fluid region, is dominant, while at the edges, magnetic and viscous irreversibility dominates.

•
The entropy generation profile is parabolic.

Conclusions
We analyzed the entropy generation via the Ohmic heating and Hall current in peristaltically-flowing Carreau fluid. The conclusions are stated below.

•
Entropy generation is not zero at the centerline y = 0.

•
Heat irreversibility, at the bulk fluid region, is dominant, while at the edges, magnetic and viscous irreversibility dominates.

•
The entropy generation profile is parabolic. • Entropy production boosts for increasing values of the Hartman number and Brinkman number.

•
Increasing the value of ξ, which is the temperature difference parameter, reduces both the entropy generation and Bejan number.