Exact Solution of Non-Newtonian Blood Flow with Nanoparticles through Porous Arteries: A Comparative Study

: In this paper, the mathematical model describing the third-grade non-Newtonian blood flow suspended with nanoparticles through porous arteries is exactly solved. The present physical model was solved in the research literature via the optimal homotopy analysis method and the collocation method, where the obtained solution was compared with the numerical fourth-order Runge-Kutta solution. However, the present paper only introduces a new approach to obtain the exact solution of the concerned system and implements such exact solution as a reference to validate the published approximate solutions. Several remarks on the previously published results are observed and discussed in detail through tables and graphs. In view of the present calculations, the obtained results in the literature by Ghasemi et al. [Ghasemi, Hatami, Sarokolaie et al. (2015)] may need revisions. Furthermore, it is found that the obtained approximate results in the relevant literature agree with the current exact ones up to only two or three decimal places, at most. Hence, the present approach along with the obtained results reflexes the effectiveness and efficiency of our analysis when compared with the corresponding study in the literature. Moreover, the present results can be directly invested for similar future problems of the same constructions.


Introduction
Presently, the study of nanofluid flow has gained the attention of many researchers [Metal and Gegel (1979); Karwe and Jaluria (1991); Chan (2009); Choi and Eastman (1995); Saidur, Leong and Mohammad (2011); Mujumdar (2007, 2008)] and is now of practical applications in engineering and medical sciences. Experiments showed that the thermal conductivity of a base-fluid can be enhanced by adding a small fraction of nanoparticles to such base-fluids [Xuan (2003); Chamkha, Rashad, El-Zahar et al. (2019); El-Zahar, Rashad and Seddek (2019)], and accordingly the heat transfer coefficient was improved for Newtonian nanofluids [Bachok, Ishak and Pop (2010); Aly and Ebaid (2016); Hamad (2011); Ebaid and Al Sharif (2015); Ebaid, Al Mutairi and Khaled (2014); Mabood, Khan and Rashidi (2017); Aly and Ebaid (2013)]. The progress in nanofluids including both types of Newtonian and non-Newtonian fluids are listed in the literature [Yadav, Bhargava and Agrawal (2013); Valinataj-Bahnemiri, Ramiar, Manavi et al. (2015); Yadav, Lee, Cho et al. (2016); Yadav (2017aYadav ( , 2017b; Hady, Ibrahim, Abdel-Gaied et al. (2011); Hatami and Ganji (2013) ;Hady, Eid and Ahmed (2014); Eid (2016Eid ( , 2017]. Usually, the mathematical models describing such kind of problems are governed by systems of coupled linear/nonlinear ordinary and partial differential equations. To solve such systems, the researchers often resort to approximate numerical/analytical methods. However, the accuracy of the approximate methods should be validated in order to trust their results. It has been shown in the literature ; ; Khaled, Ebaid and Almutairi (2014); Ebaid and Alatawi (2014); Almazmumy and Ebaid (2017) ;Ebaid, El-Zahar, Aljohani et al. (2019)] that some of the published approximate numerical/analytical solutions were not accurate enough. Moreover, the exact solution of any system is capable of detecting the accuracy of the approximated one by performing comparisons. As an example, Ghasemi et al. [Ghasemi, Hatami, Sarokolaie et al. (2015)] analyzed the system describing the blood flow suspended with nanoparticles through porous arteries using the Optimal Homotopy Analysis Method (OHAM) and the Collocation Method (CM) to obtain approximate analytical and numerical solutions, respectively. Besides, they implemented the fourth-order Runge-Kutta method (4RKM) to validate their approximate analytical and numerical solutions where the exact solution is still unavailable. Therefore, the objective of this paper is to derive the exact solution of the system describing the blood flow suspended with nanoparticles through porous arteries under the influence of a magnetic field in the form: are, respectively, the fluid velocity, the temperature, and the nanoparticles concentration. Also, µ is the viscosity of nanofluid, M is the magnetic parameter, P is the porosity parameter, c is the pressure gradient, Λ is the third-grade parameter, Br is the Brownian diffusion constant, Gr is the Grashof number, Nb is Brownian motion parameter, Nt is the thermophoresis parameter, and α and 1 α are the material modules. The boundary conditions in Eqs. (4)-(6) describe the values of ( ) v r , ( ) r θ , and ( ) r φ at the boundary of the inner and the outer coaxial cylinders. In terms of the temperature, the viscosity of the non-Newtonian nanofluid was considered as Vogel's model and given by: where A and B are constants and the subscript 0 denotes to the ambient condition. The paper is organized as follows. The exact solutions of Eqs.
(2)-(3) with the boundary condition in Eqs. (5)-(6) will be obtained in Section 2. Section 3 is devoted to discuss two special cases. In Section 4, the numerical results of ( ) v r , ( ) r θ , and ( ) r φ will be conducted for the purpose of comparisons with the corresponding results obtained by Ghasemi et al. [Ghasemi, Hatami, Sarokolaie et al. (2015)] literature. The obtained results are summarized and concluded in Section 5.
Eq. (11) can be simplified by assuming that 1 , a Nb Nt Nt and therefore Eq. (11) reduces to which is Bernoulli non-linear differential equation in ψ . In order to solve Eq. (13), the next assumption is useful Accordingly, Eq. (13) reduces to the following linear differential equation in χ : The solution of Eq. (15) can be obtained as where 3 a is a constant. Therefore Hence, the temperature ( ) r θ can be calculated from the last equation in Eq. (12) and given as and thus ( ) where 4 a is an additional constant. Applying the conditions in Eq. (5) on Eq. (19), we then get the following system: Solving this system, the constants 3 a and 4 a are determined in a simplest form as Inserting Eqs. (22) and (23) We have from Eq. (8) that Using Eqs. (9) and (24) in Eq. (25) we obtain the following final form of ( ) Here, the exact solutions of ( ) 3.2 At 1 → λ As 1 λ → , we have from Eqs. (9) and (12) that The relations in Eq. (30) are compacted to give the following restriction on α , Nb , and Nt : In this case, Eq. (13) becomes The solution of Eq. (32) can be obtained as where 5 a is a constant. Therefore Inserting Eqs. (39) and (40) where the relation in Eq. (31) is also implemented to obtain the last equation.

Comparisons with published results
In their research paper, the authors Ghasemi et al. [Ghasemi, Hatami, Sarokolaie et al. (2015)] obtained the approximate analytical solutions for the present problem via the OHAM and the CM. They have implemented 4RKM as a reference method to stand on the accuracy of their approximate solutions. However, it has been shown in the previous section that the exact solutions for ( )   [Ghasemi, Hatami, Sarokolaie et al. (2015)] are listed in Tabs. 1-3, respectively, along with the obtained corresponding errors. The results of Tabs. 1-3 reveal that the CM and the OHAM are of similar accuracy when 2 Nt Nb = = , but the situation may be changed at another set of values of the parameter Nt and Nb as discussed below. The numerical results listed in Tabs. 4-6 show that the CM is more accurate than the OHAM when 1 Nt Nb = = and this conclusion can also be observed from Tabs. 7-9 when 0.5 Nt Nb = = . As a final remark, we have noticed that the calculations reported by the authors Ghasemi et al. [Ghasemi, Hatami, Sarokolaie et al. (2015)] (Tabs. 1 and 2) for the values of ( ) we can easily detect that the resulting values are completely different than those published ones, listed in (Tab. 1 and 2). In addition, we noticed that the previously obtained approximate results via the CM and OHAM agree with the current exact ones up to only two or three decimal places, at most. Consequently, the above discussion confirms the advantage of our approach over the previous published study.

Conclusion
The exact solution of the system describing the third grade non-Newtonian blood flow suspended with nanoparticles through porous arteries was provided. Moreover, the exact solutions for two useful special cases were obtained. The developed exact solutions, for the phenomenon at hand, were compared with the published approximate results in the research literature using the OHAM and the CM. The comparisons revealed that, in most cases, the CM was of higher accuracy than the OHAM. Furthermore, the CM and OHAM results obtained by Ghasemi et al. [Ghasemi, Hatami, Sarokolaie et al. (2015)] agree with the current exact solutions up to only two decimal places in most of the investigated cases.
The developed exact approach may be trustily recommended to study similar physical models in future.

Funding Statement:
The authors would like to thank the referees for their valuable comments and suggestions which helped to improve the manuscript. Moreover, the third author acknowledges that this publication was supported by the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University.

Conflicts of Interest:
The authors declare no conflict of interest.