Mathematical Analysis on an Asymmetrical Wavy Motion of Blood under the Influence Entropy Generation with Convective Boundary Conditions

In this article, we discuss the entropy generation on the asymmetric peristaltic propulsion of non-Newtonian fluid with convective boundary conditions. The Williamson fluid model is considered for the analysis of flow properties. The current fluid model has the ability to reveal Newtonian and non-Newtonian behavior. The present model is formulated via momentum, entropy, and energy equations, under the approximation of small Reynolds number and long wavelength of the peristaltic wave. A regular perturbation scheme is employed to obtain the series solutions up to third-order approximation. All the leading parameters are discussed with the help of graphs for entropy and temperature profiles. The irreversibility process is also discussed with the help of Bejan number. Streamlines are plotted to examine the trapping phenomena. Results obtained provide an excellent benchmark for further study on the entropy production with mass transfer and peristaltic pumping mechanism.


Introduction
In our daily life, living organisms require energy to do physical work and keep the body temperature under the influence of heat exchange to the environment, as well as to generate, replace, and propagate molecules to the relevant constituents. Such type of energy comes from the oxidation process of organic substances i.e., amino acids, fats, and carbohydrates fed to the organisms. As compared to the other heat engines (i.e., in which the chemical energy gets transformed to the thermal energy, and then is transformed to mechanical work), living organisms can transform the nutrient's chemical energy into work. It happens due to the oxidation of nutrients located internally in the organisms i.e., metabolism, pass through different steps, which helps to hold some energy from ATP (adenosine triphosphate). The ATP utilized entirely by all beings for the direct transformation of mechanical energy and also actively supports other biological reactions [1]. In recent years, various authors [2][3][4][5][6] have examined the heat production of mammals via calorimetry, and presented that for of the peristaltic blood of nonlinear Williamson fluid. An assumption of long peristaltic wavelength is taken into account and Reynolds number is considered to be very small (Re ≈ 0). A regular perturbation method is used to obtain series solutions. The novelty of all the leading parameters is discussed and illustrated. The trapping mechanism is also examined to determine the nonlinear asymmetric peristaltic motion.

Governing Equations
In this section we analyze the incompressible peristaltic propulsion of Williamson fluid in a two-dimensional channel with a width d 1 + d 2 . The flow is initialized by a sinusoidal wave propagating with a constant speed c along the layout of channel (see Figure 1). The addition here is the extra equations of energy and entropy generation. It is assumed that the temperature at the upper wall of the channel is T 1 and lower wall has the temperature T 0 such that T 0 < T 1 . It depicts the physical reasoning that heat will transfer from lower to upper wall. The wall surfaces are suggested as: and where a 1 and b 1 are the wave amplitudes, λ the wave length, t the time, c the velocity of the propagation, and X is the direction of wave propagation. The phase difference φ has the range 0 ≤ φ ≤ π i.e., waves out of phase φ = 0 associated to the symmetric channel, and φ = π associated to the waves are in phase. Moreover, a i , b i , d i , d j , and φ satisfy the condition: The equations of momentum in component forms are described as: The stress tensor for the Williamson fluid model reads as: where µ ∞ , µ 0 the infinite and zero shear rate viscosity, Γ the time constant, and . γ reads as: where S t is the second invariant strain tensor. For the present flow problem, we considered µ ∞ = 0 (the infinite shear rate viscosity is very small as compared to zero shear rate viscosity) and Γ . γ < i i.e. i = 1. Then, Equation (6) takes the following form: The energy equation to represent the heat exchange in the channel is as stated below. The law of conservation of energy in the dimensional mathematical pattern is given by: In the above equation, S h is the specific heat coefficient, K the thermal conductivity, and ρ the density of the governing fluid.
Introducing wave frame coordinates transformations with propagation velocity c away from the fixed frame read as: {x Defining the dimensionless quantities as: where θ is the dimensionless temperature profile. By invoking the above transformations in Equations (4)-(6), we arrive at (after ignoring the bars): Re Reδ and where, In the above equation, We the Weissenberg number, E c is the Eckert number, P r the Prandlt number, Re the Reynolds number, and B r the Brinkman number. Under the assumptions of long wavelength and low Reynolds numbers (δ ≈ 1, Re ≈ 0), Equations (12)-(14) take the form: This equation implies that p = p(y) so ∂p/∂x can be written as dp/dx. At We = 0, the above equation turns into viscous fluid flow. The associated no slip and convective boundary conditions selected for the problem read as: where B i is the Biot number.

Entropy Generation Analysis
According to the theory of thermodynamics, the physical process can be divided in to two types: Irreversible and reversible process. The characterization of such kind of procedures is associated with the change of entropy. Particularly, we say that the process is reversible if there is no change in the entropy, whereas, if the change occurs i.e., entropy is not zero, it shows that the process is irreversible. Therefore, the production of entropy is the measure of the irreversibility of a process. All the processes that arise in nature are irreversible and this reveals a significant obstacle in the study of that process.
The entropy generation in the dimensional form can be defined as: Here we define some new dimensionless quantities in addition to those used above: Using Equation (22) in Equation (21), we get the dimensionless form of entropy generation: In the above expression, ∆ shows the entropy production characteristics and temperature difference parameter. Equation (23) is divided into two parts. The first is due to the finite temperature difference whereas the second part defines the fluid frictional irreversibility.
The Bejan number is describe as the entropy production ratio because of heat transfer irreversibility to the total entropy production: Bejan number lies between 0 to 1. Be < 1 represents that the total entropy production dominates the total entropy production due to heat transfer. Be = 1 represents when the total entropy production is equal to entropy production due to heat transfer irreversibility.

Series Solution
Since Equation (17) is non linear, its exact solution may not be possible, therefore, we employ the regular perturbation method to find the solution. For perturbation solution, we expand u, F and dp/dx as: We n dp n dx , Substituting above expression in Equation (17) and their boundary conditions in Equation (20) and comparing the coefficients of powers of We we get the zeroth and first order systems which can be manipulated easily by a mathematical computing tool Mathematica and are conclusively stated as: where the constant are defined as: The solution for velocity u obtained by above perturbation method can be used in Equation (19). The final solution for θ can be obtained by integrating Equation (19) along with their associated boundary conditions (See Equation (20)) and can be written as: θ = θ 1 + θ 2 y + θ 3 y 2 + θ 4 y 3 + θ 5 y 4 + θ 6 y 5 + θ 7 y 6 + θ 8 y 7 + θ 9 y 8 , where constants of integration θ 1 and θ 2 can be evaluated by using boundary conditions defined in Equation (20) and the expression obtained are very large and therefore are not presented here.

Discussion
In this section, we present our results by varying the quantities under the variation of several factors. Figures of temperature profile θ, entropy generation coefficient N, and streamlines are illustrated below. Figures 2-5 reflect the behavior of θ for some useful parameters. Entropy generation graphs are given in Figures 6-11. The streamlines conducting the flow samples are depicted in Figures 12 and 13. Figure 2 shows the impact of parameters a and b on temperature profile θ. It can be observed from this plot that temperature is getting increased for both parameters from the lower wall to the upper wall. Figure 3 shows the mechanism of the Biot number and Brinkman number. Biot number is an important mechanism to determine the heat transfer. It can be visualized from this figure that an enhancement in Biot number tends to boost the temperature profile while the contrary behavior has been observed with the Brinkman number. Brinkman number is the product of Eckert and Prandtl numbers B r = P r E c , or it is the ratio of the heat generated by viscous dissipation and propagation of heat by molecular conduction, such as, the ratio of the viscous heat production to extrinsic heating. Therefore, the enhancement of Brinkman's number tends to increase the temperature profile. It can be seen in Figure 4 that the volumetric flow rate significantly enhances the temperature profile. It can also be noticed that the temperature profile has a lower magnitude for smaller values of d whereas the behavior is converse for higher values. It can be viewed from Figure 5 that the Weissenberg number causes a remarkable resistance for higher values. By enhancing the Weissenberg number, the elastic forces are more dominant, which diminishes the temperature profile. However, the phase difference φ also produces a significant resistance in the temperature profile. Figures 6-9 are presented for entropy profiles against the leading parameters. It can be viewed from Figure 6 that an increment in a and b tends to boost the entropy profile whereas the entropy profile is increasing along the whole channel. Figure 7 shows that by increasing the Brinkman number, the entropy profile rises, and it decreases by increasing the Weissenberg number. However, the entropy remains positive and growing along the entire channel. It is seen from Figure 8 that the Biot number enhances the entropy profile. It can be seen that at the lower wall, the entropy profile is maximum and minimum at the upper wall, whereas it is uniform in the middle of the channel. The entropy profile for various values of ∆ is presented in Figure 9. It is noticed in this figure that the entropy profile is uniform, and no change occurs in the middle of the channel i.e., y ∈ (0, 0.5). Although it shows a decreasing pattern, but it rises along the upper wall of the channel and remains positive. Figures 10 and 11 are plotted for the Bejan number profile against the governing parameters. It is observed from Figure 10 that the Bejan number profile diminishes for higher values of the Brinkman number and shows a converse behavior for the Weissenberg number. In Figure 11, we can see that the phase difference shows versatile behavior for higher values on the Bejan number profile. When Bejan number rises, then the phase difference's effects are negligible for the domain y ∈ (0, 1.3), while when the Bejan number is small, it decreases in a similar area.
The most interesting and useful phenomena of peristaltic motion are trapping, which is plotted in Figures 12 and 13 via streamlines. It was found that by enhancing the phase difference parameter, the effects are negligible on the trapping bolus despite the fact that an unusual movement in the magnitude of the bolus is noticed. Furthermore, we can see in Figure 13 that an increment in the Weissenberg number profile tends to diminish the width of the trapping bolus. The number of boluses disappeared more quickly in the lower region as compared with the upper one.

Conclusions
In this study, we analyzed the entropy generation on the asymmetric peristaltic propulsion of non-Newtonian fluid with convective boundary conditions. The Williamson fluid model was considered to examine the entropy profile. The mathematical modeling was performed under the approximation of small Reynolds number and long wavelength of the peristaltic wave. A regular perturbation method was employed to get the series solutions up to third-order approximation. The significant results of the governing flow problem are summarized below: (i) It was noticed that the temperature profile revealed an increasing behavior by increasing the amplitude in the upper and lower region; (ii) The Biot number and Brinkman number significantly enhanced the temperature profile, whereas the behavior is converse for the phase difference parameter and Weissenberg number; (iii) Entropy profile represented an increment profile for higher values of Brinkmann number and Biot number, and a decrement behavior for the Weissenberg number; (iv) The Weissenberg number boosedt the Bejan number profile, whereas it decreased due to the Biot number and Brinkman number; (v) Trapping mechanism showed that the phase difference parameter affected the magnitude of the trapped bolus, while the Weissenberg number not only affected the magnitude of the trapped bolus and the number of trapped boluses reduced in the lower region; (vi) The non-Newtonian results in the present study could be reduced to Newtonian fluid flow by taking We = 0.
The present results provide an excellent benchmark for further study on the entropy production with mass transfer and peristaltic pumping mechanism. The mass transfer phenomena with magnetic and porosity effects that were not covered in this paper is a topic for future research.