Analysis of Heating Effects and Different Wave Forms on Peristaltic Flow of Carreau Fluid in Rectangular Duct

The existing analysis deals with heat transfer occurrence on peristaltic transport of a Carreau fluid in a rectangular duct. Flow is scrutinized in a wave frame of reference moving with velocity c away from a fixed frame. A peristaltic wave propagating on the horizontal side walls of a rectangular duct is discussed under lubrication approximation. In order to carry out the analytical solution of velocity, temperature, and pressure gradient, the homotopy perturbation method is employed. Graphical results are displayed to see the impact of various emerging parameters of the Carreau fluid and power law index. Trapping effects of peristaltic transport is also discussed and observed that number of trapping bolus decreases with an increase in aspect ratio β.


Introduction
The applications of peristaltic flows in medical and engineering sciences have attracted the attention of a number of researchers. Applications of peristalsis occur in swallowing food through the esophagus, urine transport from the kidney to the bladder through the ureter, transport of the spermatozoa in the efferent ducts of the male reproductive tract, movement of the ovum in the fallopian tube, movement of the chyme in the gastrointestinal tract, the transport of lymph in the lymphatic vessels, and vasomotion in the small blood vessels such as the arterioles, veins, and capillaries. The peristaltic phenomenon was first discussed by Latham [1]. Based on his experimental theory, numerous researchers have inspected the phenomenon of peristaltic transport under many conjectures [2][3][4][5][6][7][8][9][10].
Another fascinating area in connection with peristaltic motion is the heat transfer which has industrial applications like sanitary fluid transport, blood pumps in the heart-lungs machine and transport of corrosive fluids where the contact of fluid with the machinery parts are prohibited. Only a limited attention has been focused to the study of peristaltic flows with heat transfer [11][12][13][14][15].
An immense amount of literature is presented on twodimensional peristaltic flow problems. The study of peristaltic phenomenon in a rectangular channel was first examined by [16]. Based on the theory of [16], several researchers have studied the phenomenon of peristaltic transport in a rectangular duct under various approximations [17][18][19][20][21][22][23]. In the papers cited above, the phenomena of heat transfer are not taken into account. Keeping in mind the present information, the heat transfer phenomena on the peristaltic flow of non-Newtonian fluid have not been discussed in a threedimensional channel. So, the aim of the present problem is to discuss the effects of heat transfer on peristaltic flow of a non-Newtonian fluid in a rectangular duct with different wave forms. The governing equations for the three-dimensional rectangular channel are first modeled for Carreau fluid and then simplified under the long wavelength and low Reynolds number approximation. Homotopy perturbation technique is carried out to calculate the analytical solution of the highly nonlinear partial differential equations. The expressions for velocity, pressure rise, pressure gradient, and temperature have been computed and discussed through graphs. Threedimensional graphical representations for the velocity field are also discussed through graphs. In the end, different wave forms are also used.

Mathematical Formulation of the Problem
Let us consider the peristaltic flow of an incompressible Carreau fluid in a duct of rectangular cross section having the channel width 2d and height 2a. We are considering the Cartesian coordinates system in such a way that the X-axis is taken along the axial direction, Y-axis is taken along the lateral direction, and Z-axis is along the vertical direction of a rectangular duct.
The peristaltic waves on the walls are represented as where a and b are the amplitudes of the waves, λ is the wave length, c is the velocity of propagation, t is the time, and X is the direction of wave propagation. The walls parallel to the XZ plane remain undisturbed and are not subjected to any peristaltic wave motion. We assume that the lateral velocity is zero as there is no change in the lateral direction of the duct cross section. Let ðU, 0, WÞ be the velocity for a rectangular duct. The governing equations for the flow problem are in which ρ is the density, P is the pressure, t is the time, andS , s is the stress tensor for the Carreau fluid; C ′ is the specific heat; and T is the temperature. The stress tensor for the Carreau fluid is defined by [17].

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Here, q is an embedding parameter which has the range 0 ≤ q ≤ 1, under the condition that for q = 0, we get the initial solution, and for q = 1, we seek the final solution. Here, £ is the linear operator which is taken here as £ = ∂ 2 /∂z 2 . We define the initial guess as     Advances in Mathematical Physics

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substituting Equation (30) into Equations (26) and (27), and then comparing the like powers of q, one obtains the following problems with the corresponding boundary conditions.

Zeroth Order System
.
3.4. Second-Order System.  Advances in Mathematical Physics The resulting series solutions after three iterations are determined using Equation (30) as (when q ⟶ 1) and are evaluated as Pr Ec 21 2 + dp dx The volumetric flow rate is given by

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The instantaneous flux is defined as The average volume flow rate over one period ðT = λ/cÞ of the peristaltic wave is defined as The pressure gradient is obtained from Equations (35) and (37) as

Integration of Equation (38) over one wavelength yields
It is noticed here that the limit β ⟶ 0 (keeping it fixed and d ⟶ ∞), the rectangular duct reduces to a twodimensional channel. It is also noticed that when β = 1 the rectangular duct becomes a square duct.

Expressions for Different Wave Shapes
The nondimensional expressions for five considered wave forms are given by [11]. The expression for the triangular, square, and trapezoidal waves are derived from the Fourier series.

Numerical Results and Discussion
In this segment, the graphical representation of the proposed problem is discussed. With the help of mathematics software 9 Advances in Mathematical Physics Mathematica, the expression for pressure rise and pressure gradient is calculated. In order to see the behavior of pressure rise with volume flow rate Q for different values of aspect ratio β, power law index n, amplitude ratio ϕ, and Weissenberg number We. It is observed from Figure 1 that in the peristaltic pumping ðΔp > 0, Q > 0Þ and retrograde pumping ðΔp > 0, Q < 0Þ regions, the pumping rate decreases with an increase in aspect ratio β, while in the copumping region ðΔ p < 0, Q > 0Þ, the behavior is quite opposite; here, the pumping rate increases with an increase in aspect ratio β. Figures 2 and 3 show the variation of pressure rise with volume flow rate Q for different values of n and ϕ. It is observed from Figure 2 that in the retrograde pumping ðΔp > 0, Q > 0Þ and peristaltic pumping ðΔp > 0, Q < 0Þ regions, the pumping rate increases with an increase in n and ϕ, while in the copumping region ðΔ p < 0, Q > 0Þ, the pressure rise decreases with an increase in values of n and φ. In order to see the behavior of pressure rise with volume flow rate Q for different values of We, Figure 4 is plotted. It is depicted from Figure 4 that in the peristaltic pumping ðΔp > 0, Q > 0Þ and retrograde pumping ðΔp > 0, Q < 0Þ regions, the pumping rate decreases with an increase in We, while in the copumping region ðΔ p < 0, Q > 0Þ , the behavior is quite opposite; here, the pumping rate increases with an increase in We. The pressure gradient for different values of aspect ratio β, n, amplitude ratio ϕ, and We against space variable x is plotted in Figures 5-8. It is depicted from Figures 5-8 that for x ∈ ½0, 0:2 and x ∈ ½0:8, 1, the pressure gradient is small, i.e., the flow can easily pass without the imposition of a large pressure gradient, while in the region x ∈ ½0:2, 0:8 that the pressure gradient decreases with an increase in aspect ratio β and We and increases with an increase in amplitude ratio ϕ 10 Advances in Mathematical Physics and n, so much pressure gradient is required to maintain the flux to pass. Figures 9-12 show the stream lines for different wave forms. It is observed from Figures 9-12 that the pressure gradient retains the same shape of the waves as we consider. Figures 13-16 show the temperature profile for different values of We, β, Pr, and Ec. It is depicted from Figure 13 that the magnitude value of the temperature profile increases with an increase in We. It is observed from Figure 14 that the magnitude value of the temperature profile decreases with an increase in β Figures 15 and 16 Figures 17 and 18 that the magnitude value of the velocity profile decreases with an increase in β volume flow rate Q. It is depicted from Figure 19 that the magnitude value of the velocity profile increases with an increase in We. Stream lines for different values of aspect ratio β amplitude ratio ϕ, power law index n, and Weissenberg number We is plotted in Figures 20-23. It is observed from Figure 20 that the number of the trapping bolus decreases with an increase in aspect ratio β. It is observed from Figures 21 and 22 that the size of the trapping bolus increases with an increase in power law index n and amplitude ratio ϕ Figure 23 shows the stream lines for different values of the Weissenberg number We. It is depicted from Figure 23 that the number of the trapping bolus increases in the lower half of the channel with an increase in We, while in the upper half of the channel the size

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Advances in Mathematical Physics of the trapping bolus increases. Figure 24 shows the stream lines for different wave forms.

Conclusion
We have discussed the impact of heat transfer on peristaltic flow of a Carreau fluid in a rectangular duct with different wave forms. The governing equations for a threedimensional rectangular channel are the first model for Carreau fluid and then simplified under long wave length and low Reynolds number approximation. The analytical solutions of highly nonlinear equations are calculated. The results are discussed through graphs. It is observed that pumping rate decreases with an increase in aspect ratio β and We in the peristaltic ðΔp > 0, Q > 0Þ and retrograde pumping ðΔp > 0, Q < 0Þ regions, while in the copumping region ðΔp < 0, Q > 0Þ, the behavior is quite opposite. It is also noted that in the retrograde pumping ðΔp > 0, Q < 0Þ and peristaltic pumping ðΔp > 0, Q > 0Þ regions, the pumping rate increases with an increase in n and φ. It is observed that the profile of temperature increases with an increase in We, Pr, and Ec, while temperature profile shows the opposite behavior in the case of β. It is also observed that number of trapping bolus increases in the lower half of the channel with an increase in We, while in the upper half of the channel, size of the trapping bolus increases.

Data Availability
Data used to support the findings of this work are included in the article.

Conflicts of Interest
The authors declare no conflict of interest.