Exact Analytical Solution of the Peristaltic Nanofluids Flow in an Asymmetric Channel with Flexible Walls and Slip Condition: Application to the Cancer Treatment

In the cancer treatment, magnetic nanoparticles are injected into the blood vessel nearest to the cancer's tissues. The dynamic of these nanoparticles occurs under the action of the peristaltic waves generated on the flexible walls of the blood vessel. Studying such nanofluid flow under this action is therefore useful in treating tissues of the cancer. In this paper, the mathematical model describing the slip peristaltic flow of nanofluid was analytically investigated. Exact expressions were deduced for the temperature distribution and nano-particle concentration. In addition, the effects of the slip, thermophoresis, and Brownian motion parameters on the temperature and nano-particle concentration profiles were discussed and further compared with other approximate results in the literatures. In particular, these results have been obtained at the same values of the physical examined parameters that was considered in Akbar et al., “Peristaltic flow of a nanofluid with slip effects,” 2012. The results reveal that remarkable differences are detected between the exact current results and those approximately obtained in the literatures for behaviour of the temperature profile and nano-particles concentration. Accordingly, the current analysis and results are considered as optimal and therefore may be taken as a base for any future comparisons.


Introduction
In the recent times, peristalsis has attracted much attention due to its important engineering and medical applications, like chyme movement in the intestine, movement of eggs in the fallopian tube, transport of the spermatozoa in cervical canal, transport of bile in the bile duct, transport of cilia, circulation of blood in small blood vessels and in the intrauterine fluid flow within the uterine cavity. Since the first investigation of Latham [1], several theoretical and experimental studies have been conducted to understand peristaltic action [1][2][3][4][5][6][7][8][9][10][11]. In particular, to describe peristaltic flow in a symmetric channel or axisymmetric tubes containing Newtonian or nonNewtonian fluids, many models have been investigated by Zien and Ostrach [2], Lee and Fung [3], Srivastava et al. [4], El Shehawey and Mekheimer [6], Ramachandra and Usha [7], and Mekheimer and Abd elmaboud [10].
Although a huge number of studies for the peristaltic flow of classical fluids are available, only few papers are available for the peristaltic flow of nanofluids [29][30][31][32][33][34]. In this regard, Akbar et al. [33] may be the first authors to study the peristaltic nanofluids, in which the peristaltic flow and heat transfer of a nanofluid in an asymmetric channel have been analyzed. They have used the homotopy perturbation method to obtain the analytical approximate solutions for the temperature distribution and the nanoparticles concentration.
The nanofluids are a new class of fluids designed by dispersing nanometer-sized materials (nanoparticles, nanofibers, nanotubes, nanowires, nanorods, nanosheet, or droplets) in base fluids. Choi [35] reported that an innovative technique to improve heat transfer is by using nanoscale particles in the base fluid. Further, Choi et al. [36] showed that the addition of a small amount (less than 1% by volume) of nanoparticles to conventional heat transfer liquids increased the thermal conductivity of the fluid up to approximately two times.
In the tumors treatment, one of the effective methods is to inject the blood vessel nearest to the tumor with magnetic nanoparticles along with placing a magnet close to the tumor. These particles act like heat sources, in the presence of the applied magnetic field of alternating nature. Mekheimer and Abd elmaboud [10] found that the cancer's tissues are destroyed if the temperature reaches 42-45 ∘ C. On the other hand, in this application the drug may be placed on the magnetic nanoparticles and is injected near the tumor. Then, the drug is absorbed by the tumor through a high gradient magnetic field, which is concentrated near the tumor center [37]. Regarding, Habibi et al. [38] mentioned that the drug absorption due to high concentration of the 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. It should be noted that Majumder et al. [39] indicated that nanofluidic flow usually exhibits partial slip against the solid surface, which can be characterized by the so-called slip length, around 3.4-68 mm for different liquids.
The aim of this paper is to declare the exact effects of the slip, thermophoresis, and Brownian motion parameters on the temperature and nanoparticle concentration profiles of nanofluid flow in an asymmetric channel. It is well known that the exact solution of any physical model is optimal when available and would lead to the correct physical interpretations of the involved phenomena. Therefore, an approach is presented to achieve this goal for the resulting system of linear and nonlinear partial differential equations derived by Akbar et al. [33]. Then, these exact solutions are invested in obtaining the correct behaviour of the physical quantities.

The Mathematical Investigated Model
In the current work, we consider the peristaltic transport of an incompressible Newtonian nanofluid in an asymmetric channel with flexible walls, generating 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 ( , ), the upper wall ℎ 1 and lower wall ℎ 2 are given by, see Figure 1, where = 1, −1 and = 0, 1 when = 1, 2, respectively. Further, 1 and 2 are the amplitude of the waves, is the wave length, 1 + 2 is the width of the channel, the phase difference varies in the range 0 ≤ ≤ , where = 0, and corresponds to symmetric channel with waves out of the phase and in the phase, respectively. It should be noted that the following condition has to be achieved [40]: with the following nondimensional phenomena [33]: On considering heat transfer along with nanoparticles phenomena under the assumptions of long wavelength and low Reynolds number approximation, Akbar et al. [33] found that the flow is governed by the following system of partial differential equations: where , , , and are the stream function, temperature distribution, nanoparticles concentration, and pressure gradient, respectively. In addition, , , , and are the Brownian motion parameter, thermophoresis parameter, local temperature Grashof number, and nanoparticles Grashof number, respectively. The system (4)-(6) has to be Computational and Mathematical Methods in Medicine 3 solved subject to the following boundary conditions on , , and :

Closed Form Solution of the Model
In the present section, an effective procedure is introduced to obtain the analytical solutions for the resulted system of linear and nonlinear differential equations. On integrating (6) twice and then inserting the resulted equation into (5), we obtain This equation can be exactly solved to give the temperature distribution, and therefore the nanoparticles concentration, as where ( ), = 1, 2, 3, 4 are unknown functions to be determined. On applying the boundary conditions (9) on (12), and then solving the resulted equations, we get Further, applying the boundary conditions (10) on (13), and then solving the given system, results The above analysis leads to the following implicit algebraic equation in 1 ( ): ( , ). Now, we search for the exact expression of the stream function ( , ). By integrating the ( , ) expression in (4) twice, we obtain

Exact Expression of the Pressure Gradient / .
To get the pressure gradient / , we obtain from (7) and the above analysis that where and further all other functions are already well defined in the present section.

Numerical Values of 1 ( ).
In Section 3, the general closed form solutions for the temperature distribution and nanoparticles concentration are obtained and expressed in terms of 1 , 2 , 3 , and 4 . As 2 , 3 , and 4 depend on the evaluating of 1 , it is noticed from (17) that 1 is governed by a nonlinear algebraic equation. Once this equation is solved for 1 , the analytical expressions for and are established. It should be noted that obtaining the value of 1 analytically from (17) in terms of the other parameters set is a very difficult task, and it may be impossible. However, with the help of MATHEMATICA 6 software, the numerical solutions are still available. Values for 1 at some given cases are presented in Tables 1 and 2. These obtained values for 1 play an important role to get several plots for variation of the temperature distribution and nanoparticles concentration, which are introduced in the next section.

Results and Discussion
Besides discussing the effects of various physical parameters on the temperature distribution and nanoparticles concentration, comparing with the approximate solutions obtained by Akbar et al. [33] is also presented.
Effect of on the temperature profile for different values of the thermophoresis parameter is plotted in Figure 2 Computational and Mathematical Methods in Medicine 5  at two different values for Brownian motion parameter . It is observed from this figure that the temperature profile increases when thermophoresis parameter increases for the small or high value of Brownian motion parameter . It should be mentioned here that the present results are derived through exact solutions not as in [33] by an approximate way via the homotopy perturbation method. For the purpose of comparison, remarkable differences can be easily detected between our exact results presented in Figure 2(a) and those obtained by Akbar et al. [33] at the same values of the physical parameters. Regarding this we may point out that the approximate solutions obtained in [33] were not effective enough to give the correct physical curves. Figure 3 shows the effect of the slip parameter on the temperature profile at two different values of the thermophoresis parameter . The results reveal that the temperature profile decreases in a specific domain with increasing for any small or high value of . After that domain, the behaviour of is different, where it increases with increasing . However, the domain in which the temperature profile decreases with increasing becomes wider when takes high values; see Figure 3(b). The nanoparticles concentration is depicted in Figures  4 and 5. In Figure 4(a) when the small value of Brownian motion parameter is presented, it is observed that the nanoparticles concentration decreases in a certain domain with increasing the slip parameter 1 . A converse of this behaviour occurs after that domain. At a higher value of Brownian motion parameter , it is noticed that the nanoparticles concentration decreases in the whole domain with increasing the slip parameter 1 . On comparing the results depicted in Figure 4(a) and those obtained by Akbar et al. [33] at the same values of the physical parameters, slight differences are observed. Therefore, the current exact solutions, which can be verified by direct substitution into the governing differential equations and the boundary conditions, are reported in this paper for the first time. In addition, the effect of the thermophoresis parameter on the nanoparticles concentration is depicted in Figure 5 at two different values for . The behaviour of always decreases with increasing . However, slight differences are also observed between the current results in Figure 5(b) and those obtained in Figure 4(b) by Akbar et al. [33]. Accordingly, the present results can be viewed as optimal and more accurate.

Conclusion
In this paper, exact effects of the slip conditions and peristaltic action on the nanofluid flow in an asymmetric channel were discussed for the variations of the temperature profile and nanoparticles concentration. The flow was described by a system of linear and nonlinear partial differential equations with complex boundary conditions generated on the flexible walls of the channel. The exact solutions have been successfully obtained and reported for the first time. In addition, the obtained exact numerical results for effects of the slip condition, thermophoresis, and Brownian motion parameters on the temperature and nanoparticles concentration profiles show slight differences on comparing with the approximate solutions obtained via the homotopy perturbation method. The current analysis may throw some light on the nanofluid dynamic aspects used in the biomedical applications to treat the cancer's tissues, with the help of magnetic nanoparticles under the peristalsis on the blood vessels.