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

Abstract

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.

1. 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 [1–11]. 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].

Further, present application is in the embryo transfer within the uterine cavity, where physiologists observed that the intra-uterine fluid flow due to myometrial contractions is peristaltic-type motion. In addition, De Vries et al. [12] found that the myometrial contractions may occur in both symmetric and asymmetric directions. Therefore, a great effort was devoted to study the peristaltic flow of Newtonian and nonNewtonian fluids in an asymmetric channel [13–23], and recently of Jeffrey and Johnson-Segalman fluids; see, for example, [24–28].

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 [29–34]. 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.

2. 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 𝑐. In the Cartesian coordinates system (𝑥,𝑦), the upper wall ℎ₁ and lower wall ℎ₂ are given by, see Figure 1, ℎ_𝑖=𝐴𝑑_𝑖+𝑎_𝑖cos2𝜋𝜆(𝑥−𝑐𝑡)+𝐵𝜑, (1) where 𝐴=1,−1 and 𝐵=0,1 when 𝑖=1,2, respectively. Further, 𝑎₁ and 𝑎₂ are the amplitude of the waves, 𝜆 is the wave length, 𝑑₁+𝑑₂ is the width of the channel, the phase difference 𝜑 varies in the range 0≤𝜑≤𝜋, where 𝜑=0, and 𝜋 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]: 𝑎²₁+𝑎²₂+2𝑎₁𝑎₂cos𝜑≤𝑑₁+𝑑₂², (2) with the following nondimensional phenomena [33]: 𝑎𝑎=¹𝑑₁𝑎,𝑏=²𝑑₁𝑑,𝑑=²𝑑₁. (3)

Figure 1 

Schematic diagram of a two-dimensional asymmetric channel.

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: 𝜓_{𝑦𝑦𝑦𝑦}+𝐺_𝑟𝜃_𝑦+𝛽_𝑟𝜎_𝑦𝜃=0,_𝑦𝑦+𝑁_𝑏𝜃_𝑦𝜎_𝑦+𝑁_𝑡𝜃_𝑦²𝜎=0,_𝑦𝑦+𝑁_𝑡𝑁_𝑏𝜃_𝑦𝑦=0,𝑑𝑝=𝜓𝑑𝑥^𝑦𝑦+𝐺_𝑟𝜃+𝛽_𝑟𝜎_𝑦, (4)(5)(6)(7) where 𝜓, 𝜃, 𝜎, and 𝑝 are the stream function, temperature distribution, nanoparticles 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 nanoparticles Grashof number, respectively. The system (4)–(6) has to be solved subject to the following boundary conditions on 𝜓, 𝜃, and 𝜎: 𝐹𝜓=2,𝜓_𝑦=−𝛽𝜓_𝑦𝑦−1atℎ₁(𝑥),𝐹=1+𝑎cos𝜓=−2,𝜓_𝑦=𝛽𝜓_𝑦𝑦−1atℎ₂=−𝑑−𝑏cos𝑥+𝜑,𝜃+𝛾𝜃_𝑦=0,at𝑦=ℎ₁,𝜃−𝛾𝜃_𝑦=1,at𝑦=ℎ₂,𝜎+𝛾₁𝜎_𝑦=0,at𝑦=ℎ₁,𝜎−𝛾₁𝜎_𝑦=1,at𝑦=ℎ₂. (8)(9)(10)

3. 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 𝜕²𝜃𝜕𝑦²+𝑁_𝑏𝑓₁(𝑥)𝜕𝜃𝜕𝑦=0. (11)

This equation can be exactly solved to give the temperature distribution, and therefore the nanoparticles concentration, as 𝜃𝑥,𝑦=𝑓₄(𝑥)𝑒^{−𝑁𝑏𝑓1(𝑥)𝑦}+1𝑁_𝑏𝑓₃(𝑥)𝑓₁(𝑥),𝜎𝑁𝑥,𝑦=−^𝑡𝑁_𝑏𝑓₄(𝑥)𝑒^{−𝑁𝑏𝑓1(𝑥)𝑦}+𝑓₁(𝑥)𝑦+𝑓₂(𝑥)−𝑁_𝑡𝑁²_𝑏𝑓₃(𝑥)𝑓₁(𝑥), (12)(13) where 𝑓_𝑖(𝑥), 𝑖=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 𝑓₄=11+𝛾𝑁_𝑏𝑓₁𝑟^𝑓1₂−1−𝛾𝑁_𝑏𝑓₁𝑟^𝑓1₁,𝑓₃=−𝑁_𝑏𝑓₁1−𝛾𝑁_𝑏𝑓₁𝑟^𝑓1₁1+𝛾𝑁_𝑏𝑓₁𝑟^𝑓1₂−1−𝛾𝑁_𝑏𝑓₁𝑟^𝑓1₁. (14)

Further, applying the boundary conditions (10) on (13), and then solving the given system, results 𝑓₂=𝑁_𝑡𝑁²_𝑏𝑓₃𝑓₁−𝛾₁𝑓₁−1𝑁_𝑏𝑁_𝑡𝑓₄𝑟^𝑓1₁−𝛾₁+ℎ₁𝑓₁, (15) where 𝑟₁=𝑒^{−𝑁𝑏ℎ1},𝑟₂=𝑒^{−𝑁𝑏ℎ2}. (16)

The above analysis leads to the following implicit algebraic equation in 𝑓₁(𝑥): 𝑁_𝑡𝑁_𝑏𝛾₁𝑁_𝑏𝑓₁−1𝑟^𝑓1₁+𝛾₁𝑁_𝑏𝑓₁+1𝑟^𝑓1₂𝛾𝑁_𝑏𝑓₁−1𝑟^𝑓1₁+𝛾𝑁_𝑏𝑓₁+1𝑟^𝑓1₂+2𝛾₁+ℎ₁−ℎ₂𝑓₁=−1. (17)

4. Exact Solutions of the Physical Variables

4.1. Exact Expression of the Stream Function 𝜓(𝑥,𝑦)

Now, we search for the exact expression of the stream function 𝜓(𝑥,𝑦). By integrating the 𝜓(𝑥,𝑦) expression in (4) twice, we obtain 𝜓=𝑓₈+𝑓₇1𝑦+2𝑓₆𝑦²+16𝑓₅𝑦³+𝑔𝑦, (18) where Ω₁(𝑥)=𝛽_𝑟𝑁_𝑡𝑁_𝑏−𝐺_𝑟1𝑁_𝑏𝑓₃𝑓₁−𝛽_𝑟𝑓₂,Ω₂(𝑥)=𝛽_𝑟𝑁_𝑡𝑁_𝑏−𝐺_𝑟𝑓₄,𝑔𝑦=16Ω₁𝑦³−1𝛽24^𝑟𝑓₁𝑦⁴−Ω₂𝑁_𝑏𝑓₁³𝑒^{−𝑁𝑏𝑓1𝑦}. (19)

Applying the boundary conditions (8) on the 𝜓-equation given in (18) and (19), we obtain the following system: 𝑓₈+𝑓₇ℎ_𝑖+12𝑓₆ℎ²_𝑖+16𝑓₅ℎ³_𝑖=𝑅_𝑖(𝑥),𝑓₇+𝐴𝛽+ℎ₁𝑓₆+12ℎ²_𝑖+𝐴𝛽ℎ_𝑖𝑓₅=𝑆_𝑖(𝑥), (20) where 𝑅_𝑖(𝑥)=𝐴2𝐹−𝑔ℎ_𝑖,𝑆_𝑖(𝑥)=−1−𝑔^′ℎ_𝑖−𝐴𝛽𝑔^′′ℎ_𝑖, (21) where again 𝐴=1,−1 when 𝑖=1,2, respectively.

4.1.1. Obtaining the Values of 𝑓_𝑗,𝑗=5,…,8

On solving the last linear system in (20) with (21), we obtain 𝑓₅=6−2𝑅₁+2𝑅₂+ℎ₁−ℎ₂𝑆₁+𝑆₂ℎ₁−ℎ₂²6𝛽+ℎ₁−ℎ₂,𝑓₆=2−ℎ³₁𝑆₁+2𝑆₂+3ℎ²₁𝑅₁−𝑅₂+−2𝛽+ℎ₂𝑆₂+3ℎ₁2𝛽𝑅₁−2𝛽𝑅₂+ℎ₂−2𝛽+ℎ₂𝑆₁+2𝛽𝑆₂−ℎ₂3−2𝛽+ℎ₂𝑅₁+6𝛽−3ℎ₂𝑅₂+ℎ₂2−3𝛽+ℎ₂𝑆₁+ℎ₂𝑆₂×ℎ₁−ℎ₂²2𝛽+ℎ₁−ℎ₂6𝛽+ℎ₁−ℎ₂⁻¹,𝑓₇=ℎ⁴₁𝑆₂+ℎ³₁2−𝛽+ℎ₂𝑆₁+4𝛽𝑆₂+2ℎ₁32𝛽²−4𝛽ℎ₂+ℎ²₂𝑅₁−𝑅₂+ℎ²₂−3𝛽+ℎ₂𝑆₂+ℎ₂6𝛽−2𝛽+ℎ₂𝑅₁+6𝛽2𝛽−ℎ₂𝑅₂+ℎ²₂−4𝛽+ℎ₂𝑆₁+2𝛽𝑆₂+ℎ²₁6𝛽−ℎ₂𝑅₁−32𝛽−ℎ₂𝑅₂+ℎ₂−2𝛽+ℎ₂𝑆₁+ℎ₂𝑆₂×ℎ₁−ℎ₂²2𝛽+ℎ₁−ℎ₂6𝛽+ℎ₁−ℎ₂⁻¹,𝑓₈=ℎ²₂12𝛽²−8𝛽ℎ₂+ℎ²₂𝑅₁+ℎ³₁2𝛽−ℎ₂4𝑅₂+ℎ₂𝑆₁−2𝑆₂+ℎ⁴₁𝑅₂−ℎ₂𝑆₂+ℎ²₁12𝛽²𝑅₂−6𝛽ℎ₂𝑅₁+3𝑅₂+ℎ³₂2𝑆₁−𝑆₂+3ℎ²₂𝑅₁+𝑅₂−2𝛽𝑆₁+2𝛽𝑆₂−ℎ₁ℎ₂26𝛽²−9𝛽ℎ₂+2ℎ²₂𝑅₁+6𝛽2𝛽−ℎ₂𝑅₂+ℎ²₂−4𝛽+ℎ₂𝑆₁+2𝛽𝑆₂×ℎ₁−ℎ₂²2𝛽+ℎ₁−ℎ₂6𝛽+ℎ₁−ℎ₂⁻¹. (22)(23)

4.2. Exact Expression of the Pressure Gradient 𝑑𝑝/𝑑𝑥

To get the pressure gradient 𝑑𝑝/𝑑𝑥, we obtain from (7) and the above analysis that 𝑑𝑝𝑑𝑥=Ω₃(𝑥)−𝛽_𝑟𝑓₁𝑦+1+𝑁_𝑏𝑓₁Ω₂(𝑥)𝑒^{−𝑁𝑏𝑓1𝑦}, (24) where Ω₃(𝑥)=Ω₁(𝑥)+𝑓₅(𝑥)+𝛽_𝑟𝑓₁(𝑥), (25) and further all other functions are already well defined in the present section.

4.3. Numerical Values of 𝑓₁(𝑥)

In Section 3, the general closed form solutions for the temperature distribution 𝜃 and nanoparticles concentration 𝜎 are obtained and expressed in terms of 𝑓₁, 𝑓₂, 𝑓₃, and 𝑓₄. As 𝑓₂, 𝑓₃, and 𝑓₄ depend on the evaluating of 𝑓₁, it is noticed from (17) that 𝑓₁ is governed by a nonlinear algebraic equation. Once this equation is solved for 𝑓₁, the analytical expressions for 𝜃 and 𝜎 are established.

It should be noted that obtaining the value of 𝑓₁ 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 𝑓₁ at some given cases are presented in Tables 1 and 2. These obtained values for 𝑓₁ play an important role to get several plots for variation of the temperature distribution and nanoparticles concentration, which are introduced in the next section.

Table 1 

The numerical values of 𝑓₁ at different values of 𝑁_𝑡 for 𝑎=0.1, 𝑑=1, 𝑏=0.5, 𝑥=1, 𝜙=0.2 when 𝛾=𝛾₁=0.1 for 𝑁_𝑏=0.8 and 5.

Table 2 

The numerical values of 𝑓₁ at different values of 𝛾 and 𝛾₁ for 𝑎=0.1, 𝑑=1, 𝑏=0.5, 𝑥=1, 𝜙=0.2 when 𝑁_𝑏=2 for 𝑁_𝑡=0.8 and 5.

5. 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 𝑁_𝑡 on the temperature profile 𝜃 for different values of the thermophoresis parameter 𝑁_𝑡 is plotted in Figure 2 at two different values for Brownian motion parameter 𝑁_𝑏. It is observed from this figure that the temperature profile increases when thermophoresis parameter 𝑁_𝑡 increases for the small or high value of Brownian motion parameter 𝑁_𝑏. 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.

(a) 𝑁𝑏=0.8

(b) 𝑁𝑏=5

Figure 2 

Variation of the temperature profile 𝜃 at different values of 𝑁𝑡 for 𝑎=0.1, 𝑑=1, 𝑏=0.5, 𝑥=1, 𝜙=0.2 when 𝛾=𝛾1=0.1 at (a) 𝑁𝑏=0.8 and (b) 𝑁𝑏=5.

Figure 3 shows the effect of the slip parameter 𝛾 on the temperature profile 𝜃 at two different values of the thermophoresis parameter 𝑁_𝑡. The results reveal that the temperature profile decreases in a specific domain with increasing 𝛾 for any small or high value of 𝑁_𝑡. After that domain, the behaviour of 𝜃 is different, where it increases with increasing 𝛾. However, the domain in which the temperature profile decreases with increasing 𝛾 becomes wider when 𝑁_𝑡 takes high values; see Figure 3(b).

(a)  𝑁𝑡=0.8

(b) 𝑁𝑡=5

Figure 3 

Variation of the temperature profile 𝜃 at different values of 𝛾 for 𝑎=0.1, 𝑑=1, 𝑏=0.5, 𝑥=1, 𝜙=0.2, 𝛾1=0.5, 𝑁𝑏=2 at (a) 𝑁𝑡=0.8 and (b) 𝑁𝑡=5.

The nanoparticles concentration 𝜎 is depicted in Figures 4 and 5. In Figure 4(a) when the small value of Brownian motion parameter 𝑁_𝑏 is presented, it is observed that the nanoparticles concentration 𝜎 decreases in a certain domain with increasing the slip parameter 𝛾₁. A converse of this behaviour occurs after that domain. At a higher value of Brownian motion parameter 𝑁_𝑏, it is noticed that the nanoparticles concentration 𝜎 decreases in the whole domain with increasing the slip parameter 𝛾₁. 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 𝑁_𝑡 on the nanoparticles concentration 𝜎 is depicted in Figure 5 at two different values for 𝑁_𝑏. The behaviour of 𝜎 always decreases with increasing 𝑁_𝑡. 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.

(a)  𝑁𝑏=0.8

(b) 𝑁𝑏=5

Figure 4 

Variation profile of nanoparticles concentration 𝜎 at different values of 𝛾1 for 𝑎=0.1, 𝑑=1, 𝑏=0.5, 𝑥=1, 𝜙=0.2, 𝛾=0.5, 𝑁𝑏=2 at (a) 𝑁𝑡=0.8 and (b) 𝑁𝑡=5.

(a)  𝑁𝑏=0.8

(b) 𝑁𝑏=5

Figure 5 

Variation profile of nanoparticles concentration 𝜎 at different values of 𝑁𝑏 for 𝑎=0.1, 𝑑=1, 𝑏=0.5, 𝑥=1, 𝜙=0.2 when 𝛾=𝛾1=0.1 at (a) 𝑁𝑏=0.8 and (b) 𝑁𝑏=5.

6. 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.