Effect of the Velocity Second Slip Boundary Condition on the Peristaltic Flow of Nanofluids in an Asymmetric Channel: Exact Solution

and Applied Analysis 3 where ψ, θ, σ, and p are the stream function, temperature distribution, nanoparticle concentration, and pressure gradient, respectively. In addition, N b , N t , G r , and B r are the Brownian motion parameter, thermophoresis parameter, local temperature Grashof number, and nanoparticle Grashof number, respectively. The above system of PDEs. (6)–(9) has to be solved subject to the following boundary conditions: ψ = F 2 , ∂ψ ∂y = −β 1 ∂2ψ ∂y − β 2 ∂3ψ ∂y − 1 at h 1 = 1 + a cos (x) , ψ = − F 2 , ∂ψ ∂y = β 1 ∂2ψ ∂y + β 2 ∂3ψ ∂y − 1 at h 2 = −d − b cos (x + φ) , θ + γ ∂θ ∂y = 0, at y = h 1 , θ − γ ∂θ ∂y = 1, at y = h 2 , σ + γ 1 ∂σ ∂y = 0, at y = h 1 , σ − γ 1 ∂σ ∂y = 1, at y = h 2 , (10) whereβ 1 ,β 2 , γ, and γ 1 represent the first-order slip parameter, second-order slip parameter, thermal slip parameter, and concentration slip parameter, respectively. It should be noted that the full formulation of the above model can be checked in [22], and there is no need to repeat it again here. However, as mentioned in Section 1 in the present work, we focused on obtaining the general closed form solution in the presence of the second-order slip parameter, as a very important one to predict the investigated physical model. 3. The General Closed Form Solution As mentioned by Ebaid and Aly [28], we have the following exact solutions for the temperature distribution and nanoparticle concentration: θ (x, y) = f 4 e−Nfy + 1 N b f 3


Introduction
Peristalsis is an interest subject and has recently attracted much attention due to its importance in engineering and medical applications. In human body, peristalsis is found in the swallowing food through the esophagus, chyme motion in the gastrointestinal tract, vasomotion of small blood vessels such as venules, capillaries and arterioles, urine transport from kidney to bladder, and intrauterine fluid flow within the uterine cavity. Various experimental and theoretical studies have been conducted to understand peristaltic flow in asymmetric channel or an axisymmetric tube; see [1][2][3][4][5][6][7]. According to De Vries et al. [8], they observed that myometrial contractions are peristaltic-type motion and therefore these contractions of the uterine wall may occur in both symmetric and asymmetric directions. Due to the importance of this field of research, a great effort was devoted to study this type of flow for Newtonian and non-Newtonian fluids in an asymmetric channel [9][10][11][12][13][14][15][16][17].
Recently, a few papers have been published in the field of nanofluid flows under the peristaltic action [18][19][20][21][22][23]. Nanofluids usually lead to the enhancement of the thermal conductivity of the base fluid [24]. The concept of nanofluids, analogous to that of nanoparticles, may be initiated by Choi et al. [25]. The fact that nanofluids have higher thermal conductivity than the other heat transfer fluids because of their nanostructure has attracted many engineers and theoretical scientists to investigate their behavior. This higher thermal conductivity may be very useful in the treatment of tumors by injecting the blood vessel nearest to the tumor with magnetic nanoparticles along with placing a magnet close to the tumor, where these particles act like heat sources, in the presence of the applied magnetic field of alternating nature. Hence, the undesirable tissues (cancer's tissues) are destroyed when the temperature reaches 42-45 ∘ C [6]. Besides, the drug may be placed on the magnetic nanoparticles and is injected near the tumor. The drug is then absorbed by the tumor through a high gradient magnetic field, which is concentrated near the tumor center [26]. In fact, the drug absorption due to high concentration of magnetic particles increases and magnetic force prevents uniform drug distribution in circulatory system. This approach reduces the side effect and allows using high dose of anticancer drug [27].
Very recently, Ebaid and Aly [28] investigated the system of partial differential equations describing the peristaltic flow of a nanofluid with slip effect of the velocity, temperature, and concentration has been analytically solved. The obtained exact solutions have been applied to study effects of the slip parameter, thermophoresis, Brownian motion parameters, and many other parameters on the pressure rise, velocity profiles, temperature distribution, nanoparticle concentration, and pressure gradient. These exact solutions were proved to reduce to the results in the literature at special cases when 2 = 0. Moreover, on comparing the present results with those in [22], remarkable differences were noticed for behavior of the included previous physical phenomena.
The main feature of the previous studies on the nanofluid flow under peristaltic action [18][19][20][21][22][23] is the ignorance of the second slip effect, while this effect has been recently discussed by many authors; see for example [29][30][31][32][33]. These papers showed clearly that the second-order slip flow model is necessary to predict the flow characteristics accurately. Therefore, as an extension of these studies and for the first time, in this paper we aim to investigate effect of the second slip on the peristaltic flow of nanofluids in an asymmetric channel. Hence, we focus here on the problem discussed by Akbar et al. [22] and Ebaid and Aly [28] with slight differences in the boundary conditions.

The Mathematical Model
Consider peristaltic transport of an incompressible Newtonian nanofluid in an asymmetric channel with flexible walls. The channel asymmetry is generated by propagation of waves on the channel walls traveling with different amplitudes and phases but with the same constant speed . In the Cartesian coordinates system ( , ) of the fixed frame, the upper and lower walls ℎ 1 and ℎ 2 , respectively, are given by (see Figure 1) where 1 and 1 are amplitude of the waves, is the wave length, and 1 + 2 is the width of the channel. The phase difference varies in the range 0 ≤ ≤ , where = 0 and = correspond to symmetric channel with waves out of the phase and in the phase, respectively. Further, 1 , 1 , 1 , 2 , and have to satisfy the following condition [34]: with the following nondimensional phenomena [22]: Furthermore, in the moving frame of references ( , ), we have which have been used with the following nondimensional phenomena [22]: to obtain the present physical model. On taking into account (i) heat transfer along with nanoparticle phenomena, (ii) long wavelength, and low Reynolds number approximation, recently, Akbar et al. [22] found that the flow is governed by the following system of partial differential equations in nondimensional form: where , , , and are the stream function, temperature distribution, nanoparticle concentration, and pressure gradient, respectively. In addition, , , , and are the Brownian motion parameter, thermophoresis parameter, local temperature Grashof number, and nanoparticle Grashof number, respectively. The above system of PDEs. (6)-(9) has to be solved subject to the following boundary conditions: where 1 , 2 , , and 1 represent the first-order slip parameter, second-order slip parameter, thermal slip parameter, and concentration slip parameter, respectively. It should be noted that the full formulation of the above model can be checked in [22], and there is no need to repeat it again here. However, as mentioned in Section 1 in the present work, we focused on obtaining the general closed form solution in the presence of the second-order slip parameter, as a very important one to predict the investigated physical model.

The General Closed Form Solution
As mentioned by Ebaid and Aly [28], we have the following exact solutions for the temperature distribution and nanoparticle concentration: 1 can be obtained from the following implicit algebraic equation: It can be seen from (14) that it is of complex structure to be solved exactly in its general form. However, it has an exact solution for 1 when = 1 and given by Regarding the stream function, it can be solved to give where Applying the boundary conditions on the -equation, we obtain the following system: Abstract and Applied Analysis On solving the system (18a)-(18d), we get To get the pressure gradient / , we obtain from (7) that where The pressure rise Δ in terms of the flow rate is given as follows: Therefore, the exact expression for the pressure rise at the center of the channel is given as

Results and Discussion
This section is devoted for investing the exact solutions which have been obtained in the previous section for the temperature distribution, nanoparticle concentration, velocity, pressure gradient, and pressure rise. These solutions are expected to provide us with the correct physical effect of the second slip for the five investigated physical phenomena. It is observed from (11) that there is no effect for the second slip parameter on the solutions of the temperature distribution and nanoparticle concentration. Although (11) have been used by Ebaid and Aly [28] to discuss the effect of many parameters on the temperature distribution and nanoparticle concentration for comparing with the results in [22], many other parameters were ignored. These important parameters are investigated in the next subsections.  other investigated parameters. From these figures, it can be concluded that there are no remarkable influences for the variation of on in the first part of the channel, while an increase in occurs in the upper part with increasing . The variation of the amplitude on the temperature distribution is presented in Figures 3(a) and 3(b). These figures show that very small decrease in occurs as increases. This means that while the amplitude has remarkable effect on , the influence of the amplitude on may be ignored. In Figures  4(a) and 4(b), the variation of the phase difference on is introduced. Although the increase in between the waves of the channel increases the temperature distribution, this increase is very small. Instead of studying the variation of at separate values of Brownian motion parameter , we present in Figure 5 on . In addition, Figure 5(b) shows other descriptions for the variation of the thermophoresis parameter on the temperature distribution. It can be concluded from the last two figures that always decreases with the increase in and .

The Nanoparticle Concentration .
The nanoparticles concentration is depicted in Figures 6(a) and 6(b). It is observed from these figures that reaches its highest value at the lower wall of the channel and then it decreases in a certain domain across the channel. A converse of this behavior occurs after that domain. It is also detected that at a higher value of Brownian motion parameter the domain in which decreases becomes wider. In addition, the wide of such domain increases with increasing the amplitude of the upper wave. The variation of the lower wave amplitude on is presented in Figures 7(a) and 7(b). For all values of , it is noticed that decreases until it reaches a certain point. After that point the curves become identical whatever the value of becomes. This refers to that there is no effect for on after this point.
In Figures 8(a) and 8(b), the variation of the phase difference on is displayed. Although the increase in the phase difference between the waves of the channel increases the nanoparticles concentration, this occurs in certain domain of the channel. However, after that domain the curves are identical whatever the value of the phase difference. Moreover, instead of studying the variation of at separate values of Brownian motion parameter , the variation of against a continuous range for the values of is displayed in Figure 9(a). In addition, Figure 9(b) shows other descriptions for the variation of the thermophoresis parameter on the nanoparticle concentration. It can be concluded from Figure 9(a) that always decreases when is in the range from 1 to 10 at fixed values for the other parameters. However, the situation is different for the variation of against at certain values of , where it is noticed that when 1 < < 4 then decreases within the channel. In addition, at values of higher than 4, we observe that decreases in certain domain of the channel and changes this behavior in the rest of channel.

The Pressure
Rise Δ . The exact expression for the pressure rise is given by (24) which is valid for any set of the physical parameters. On using this equation, we observe that the pressure rise is always a decreasing function in terms of the flow rate. This notice is clarified in Figures 10(a)-10(d).
It is also observed from Figure 10(a) that the pressure rise increases with increasing the negativity of the second slip parameter 2 in the whole range of the flow rate . The effect of the thermophoresis parameter on the pressure rise is depicted in Figure 10(b). It can be seen from this figure that the pressure rise decreases with the increase in the values of in the whole region of . Figure 10(c) shows that the pressure rise decreases with decreasing the amplitude when lies in the region −2.5 < < 1, while a converse behavior occurs for > 1. Figure 10(d) reveals that the pressure rise increases with increasing the local nanoparticle Grashof number . This behavior differs from the results obtained by Akbar et al. [22] in the absence of the second slip effect. Therefore, the present results reveal that the existence of this parameter has a significant effect on the pressure rise and leads to different behavior when compared with its ignorance.  Figure 11(a) that the velocity increases near the lower wall with increasing 2 , while a converse behavior occurs near the upper wall; that is, the velocity decreases with increasing 2 . Unlike the study made by Akbar et al. [22], the effect of Brownian motion parameter leads to that there are three different regions in which alternating behavior for the velocity occurs. As shown in Figure 11(b), in a central region of the channel, that is, −0.6 < < 0.8, the velocity increases with the increase in , while in the other two regions of the channel, that is, ℎ 2 < < −0.6 and 0.8 < < ℎ 1 , a different situation is detected. At the same three regions just mentioned, it can be seen from Figure 11(d) that effect of the local temperature Grashof number on the velocity profiles happens in a converse manner regarding Brownian motion parameter . However, it is clear from Figure 11 that an increase in the flow rate leads to an increase in the velocity and this is similar to the variation of the velocity profile against flow rate as in Akbar et al. [22]. , and on the pressure gradient are completely different comparing with those obtained in [22] for the effect of the first slip parameter only. This also may refer to the importance of including the second slip parameter 2 in such problems of peristaltic flow. However, the effect of the amplitude on the pressure gradient remains the same even in the presence of 2 , where Figure 12(d) shows that an increase in the amplitude leads to an increase in the pressure gradient. Besides, one of the main features of these graphs is that the maximum pressure gradient occurs at the same point when = 0.5. In the absence of 2 , this value was shifted slightly to the left; that is, it occurred at = 0.45 as in [22]. The final notice here is that the present numerical discussion is based on the obtained exact analytical solutions for the current physical model, unlike the approximate series solutions obtained in [22] whose disadvantages have been proved by Ebaid and Aly [28].

Conclusion
The effect of the second slip condition on the peristaltic transport of a nanofluid in an asymmetric channel was investigated for the first time in the present paper. The system describing the problem was exactly solved via a simple, but very effective, analytical procedure. Based on the obtained exact solutions, some important results were introduced through graphs to indicate effect of the second slip on the velocity profiles, pressure rise, and pressure gradient. Moreover, the variation of many physical parameters such as amplitudes of the lower and upper waves, phase difference on the temperature distribution, and nanoparticle concentration were also discussed for a new set of the presented parameters rather than those in the literature. It was found that the temperature distribution always decreases with the increase in and . In addition, the amplitude has remarkable effect on , while the influence of the amplitude on may be ignored. For all values of , nanoparticles concentration decreases until it reaches a certain point; however, there is no effect for after this point. In a certain domain of the channel, the phase difference between the waves of the channel increases ; however, the curves are identical whatever value of the phase difference after that. Finally, it should be mentioned that the second slip condition had a significant effect on the various phenomena involved in the current model. Accordingly, the present suggested technique may be useful for obtaining exact solutions for many other similar problems. In addition, such type of the second slip should be included for modelling the peristaltic flow.