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A Two-Layer Mathematical Model of Blood Flow in Porous Constricted Blood Vessels

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Abstract

In this paper, we discussed a mathematical model for two-layered non-Newtonian blood flow through porous constricted blood vessels. The core region of blood flow contains the suspension of erythrocytes as non-Newtonian Casson fluid and the peripheral region contains the plasma flow as Newtonian fluid. The wall of porous constricted blood vessel configured as thin transition Brinkman layer over layered by Darcy region. The boundary of fluid layer is defined as stress jump condition of Ocha-Tapiya and Beavers–Joseph. In this paper, we obtained an analytic expression for velocity, flow rate, wall shear stress. The effect of permeability, plasma layer thickness, yield stress and shape of the constriction on velocity in core & peripheral region, wall shear stress and flow rate is discussed graphically. This is found throughout the discussion that permeability and plasma layer thickness have accountable effect on various flow parameters which gives an important observation for diseased blood vessels.

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Acknowledgements

Bhupesh Dutt Sharma was supported by the MHRD Fellowship for Research Scholars administered by MNNIT Allahabad.

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Authors and Affiliations

  1. Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Allahabad, 211004, India

    Bhupesh Dutt Sharma & Pramod Kumar Yadav

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  1. Bhupesh Dutt Sharma
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Correspondence to Bhupesh Dutt Sharma.

Appendix A

Appendix A

$$\begin{aligned} v_1&= -8 \sqrt{2} \sqrt{P \theta } (\alpha I_0\left( \gamma R_3\right) (\beta \phi K_0\left( \gamma R_2\right) +\gamma K_1\left( \gamma R_2\right) \sqrt{k})-I_1\left( \gamma R_3\right) (k K_1\left( \gamma R_2\right) \gamma ^2\\&\quad +\, \beta \phi K_0\left( \gamma R_2\right) \sqrt{k} \gamma )- (\beta \phi I_0\left( \gamma R_2\right) -\sqrt{k} \gamma I_1\left( \gamma R_2\right) ) (\alpha K_0\left( \gamma R_3\right) +\gamma K_1\left( \gamma R_3\right) \sqrt{k})) R_1^{3/2} \\&\quad + 3 P (\alpha I_0\left( \gamma R_3\right) (\beta \phi K_0\left( \gamma R_2\right) +\gamma K_1\left( \gamma R_2\right) \sqrt{k})-I_1\left( \gamma R_3\right) (k K_1\left( \gamma R_2\right) \gamma ^2+ \beta \phi K_0\left( \gamma R_2\right) \sqrt{k} \gamma )\\&\quad - (\beta \phi I_0\left( \gamma R_2\right) -\sqrt{k} \gamma I_1\left( \gamma R_2\right) ) (\alpha K_0\left( \gamma R_3\right) +\gamma K_1\left( \gamma R_3\right) \sqrt{k})) (\lambda _1-1) R_1^2-12 \theta (\alpha I_0\left( \gamma R_3\right) \\&\quad (\beta \phi K_0\left( \gamma R_2\right) +\gamma K_1\left( \gamma R_2\right) \sqrt{k})-I_1\left( \gamma R_3\right) (k K_1\left( \gamma R_2\right) \gamma ^2+\beta \phi K_0\left( \gamma R_2\right) \sqrt{k} \gamma )-(\beta \phi I_0\left( \gamma R_2\right) \\&\quad - \sqrt{k} \gamma I_1\left( \gamma R_2\right) )(\alpha K_0\left( \gamma R_3\right) +\gamma K_1 (\gamma R_3) \sqrt{k})) R_1+r (\alpha I_0\left( \gamma R_3\right) (\beta \phi K_0\left( \gamma R_2\right) +\gamma K_1\left( \gamma R_2\right) \sqrt{k}) \\&\quad - I_1\left( \gamma R_3\right) (k K_1\left( \gamma R_2\right) \gamma ^2+\beta \phi K_0\left( \gamma R_2\right) \sqrt{k} \gamma )- (\beta \phi I_0\left( \gamma R_2\right) -\sqrt{k} \gamma I_1\left( \gamma R_2\right) ) (\alpha K_0\left( \gamma R_3\right) \\&\quad + \gamma K_1\left( \gamma R_3\right) \sqrt{k})) (3 P r+12 \theta -8 \sqrt{2} \sqrt{r} \sqrt{P \theta })-3 P (4 \gamma ((\alpha I_0\left( \gamma R_3\right) -\sqrt{k} \gamma I_1\left( \gamma R_3\right) ) K_1\left( \gamma R_2\right) \\&\quad + I_1\left( \gamma R_2\right) (\alpha K_0\left( \gamma R_3\right) +\gamma K_1\left( \gamma R_3\right) \sqrt{k})) k^{3/2}+ (\alpha I_0\left( \gamma R_3\right) (\beta \phi K_0\left( \gamma R_2\right) +\gamma K_1\left( \gamma R_2\right) \sqrt{k})\\&\quad - I_1\left( \gamma R_3\right) (k K_1\left( \gamma R_2\right) \gamma ^2+\beta \phi K_0\left( \gamma R_2\right) \sqrt{k} \gamma )-(\beta \phi I_0\left( \gamma R_2\right) -\sqrt{k} \gamma I_1\left( \gamma R_2\right) ) (\alpha K_0\left( \gamma R_3\right) \\&\quad + \gamma K_1\left( \gamma R_3\right) \sqrt{k})) R_2^2-2 \sqrt{k} \phi (-\alpha I_0\left( \gamma R_3\right) K_0\left( \gamma R_2\right) +\gamma I_1\left( \gamma R_3\right) \sqrt{k} K_0\left( \gamma R_2\right) +I_0\left( \gamma R_2\right) \\&\quad (\alpha K_0\left( \gamma R_3\right) +\gamma K_1\left( \gamma R_3\right) \sqrt{k})) R_2) \lambda _1 \div 12 (\alpha I_0\left( \gamma R_3\right) \left( \beta \phi K_0\left( \gamma R_2\right) +\gamma K_1\left( \gamma R_2\right) \sqrt{k}\right) \\&\quad -I_1\left( \gamma R_3\right) \left( k K_1\left( \gamma R_2\right) \gamma ^2+\beta \phi K_0\left( \gamma R_2\right) \sqrt{k} \gamma \right) -\left( \beta \phi I_0\left( \gamma R_2\right) -\sqrt{k} \gamma I_1\left( \gamma R_2\right) \right) (\alpha K_0\left( \gamma R_3\right) . \\&\quad +\gamma K_1\left( \gamma R_3) \sqrt{k}\right) ) \lambda _1\\ v_2&= \frac{1}{4} P \bigg (-4 \gamma k^{3/2} \left( K_1\left( \gamma R_2\right) \left( \alpha I_0\left( \gamma R_3\right) -\gamma \sqrt{k} I_1\left( \gamma R_3\right) \right) +I_1\left( \gamma R_2\right) \left( \gamma \sqrt{k} K_1\left( \gamma R_3\right) +\alpha K_0\left( \gamma R_3\right) \right) \right) \\&\quad +R_2^2 (-\alpha I_0\left( \gamma R_3\right) \left( \gamma \sqrt{k} K_1\left( \gamma R_2\right) +\beta \phi K_0\left( \gamma R_2\right) \right) +\left( \gamma \sqrt{k} K_1\left( \gamma R_3\right) +\alpha K_0\left( \gamma R_3\right) \right) (\beta \phi I_0\left( \gamma R_2\right) \\&\quad -\gamma \sqrt{k} I_1\left( \gamma R_2\right) )+I_1\left( \gamma R_3\right) \left( \beta \gamma \sqrt{k} \phi K_0\left( \gamma R_2\right) +\gamma ^2 k K_1\left( \gamma R_2\right) \right) )+ 2 \sqrt{k} R_2 \phi (I_0\left( \gamma R_2\right) \\&\quad \left( \gamma \sqrt{k} K_1\left( \gamma R_3\right) +\alpha K_0\left( \gamma R_3\right) \right) +\gamma \sqrt{k} I_1\left( \gamma R_3\right) K_0\left( \gamma R_2\right) -\alpha I_0\left( \gamma R_3\right) K_0\left( \gamma R_2\right) ) \div \alpha I_0\left( \gamma R_3\right) \\&\quad \left( \gamma \sqrt{k} K_1\left( \gamma R_2\right) +\beta \phi K_0\left( \gamma R_2\right) \right) -\left( \gamma \sqrt{k} K_1\left( \gamma R_3\right) +\alpha K_0\left( \gamma R_3\right) \right) \left( \beta \phi I_0\left( \gamma R_2\right) -\gamma \sqrt{k} I_1\left( \gamma R_2\right) \right) \\&-I_1\left( \gamma R_3\right) \left( \beta \gamma \sqrt{k} \phi K_0\left( \gamma R_2\right) +\gamma ^2 k K_1\left( \gamma R_2\right) \right) +r^2).\\ v_3&=( \sqrt{k} P (2 (\alpha \left( -\sqrt{k}\right) I_0\left( \gamma R_3\right) \left( \gamma \sqrt{k} K_1\left( \gamma R_2\right) -\beta \phi K_0(r \gamma )+\beta \phi K_0\left( \gamma R_2\right) \right) -\sqrt{k} (\gamma \sqrt{k} K_1\left( \gamma R_3\right) \\&\quad +\alpha K_0\left( \gamma R_3\right) ) \left( \gamma \sqrt{k} I_1\left( \gamma R_2\right) +\beta \phi I_0(r \gamma )-\beta \phi I_0\left( \gamma R_2\right) \right) +\gamma k I_1\left( \gamma R_3\right) (\gamma \sqrt{k} K_1\left( \gamma R_2\right) -\beta \phi K_0(r \gamma )\\&\quad +\beta \phi K_0\left( \gamma R_2\right) )) +R_2 \phi (I_0(r \gamma ) \left( \gamma \sqrt{k} K_1\left( \gamma R_3\right) +\alpha K_0\left( \gamma R_3\right) \right) +\gamma \sqrt{k} K_0(r \gamma ) I_1\left( \gamma R_3\right) -\alpha K_0(r \gamma )\\&\quad I_0\left( \gamma R_3\right) ))) \div 2 (\alpha I_0\left( \gamma R_3\right) \left( \gamma \sqrt{k} K_1\left( \gamma R_2\right) +\beta \phi K_0\left( \gamma R_2\right) \right) -\left( \gamma \sqrt{k} K_1\left( \gamma R_3\right) +\alpha K_0\left( \gamma R_3\right) \right) \\&\quad \left( \beta \phi I_0\left( \gamma R_2\right) -\gamma \sqrt{k} I_1\left( \gamma R_2\right) \right) -I_1\left( \gamma R_3\right) \left( \beta \gamma \sqrt{k} \phi K_0\left( \gamma R_2\right) +\gamma ^2 k K_1\left( \gamma R_2\right) \right) \bigg ).\\ Q&= - 176 (\alpha I_0\left( \gamma R_3\right) (\beta \phi K_0\left( \gamma R_2\right) +\gamma K_1\left( \gamma R_2\right) \sqrt{k})-I_1\left( \gamma R_3\right) (k K_1\left( \gamma R_2\right) \gamma ^2+\beta \phi K_0\left( \gamma R_2\right) \sqrt{k} \gamma )\\&\quad -(\beta \phi I_0\left( \gamma R_2\right) -\sqrt{k} \gamma I_1\left( \gamma R_2\right) ) (\alpha K_0\left( \gamma R_3\right) +\gamma K_1\left( \gamma R_3\right) \sqrt{k})) \sqrt{2} \sqrt{P \theta } R_1^{7/2}-21 P (\alpha I_0\left( \gamma R_3\right) \\&\quad (\beta \phi K_0\left( \gamma R_2\right) +\gamma K_1\left( \gamma R_2\right) \sqrt{k})-I_1\left( \gamma R_3\right) (k K_1\left( \gamma R_2\right) \gamma ^2+\beta \phi K_0\left( \gamma R_2\right) \sqrt{k} \gamma )-(\beta \phi I_0\left( \gamma R_2\right) \\&\quad -\sqrt{k} \gamma I_1\left( \gamma R_2\right) ) (\alpha K_0\left( \gamma R_3\right) +\gamma K_1\left( \gamma R_3\right) \sqrt{k}))(2 \lambda _1-1) R_1^4+56 \theta (\alpha I_0\left( \gamma R_3\right) (\beta \phi K_0\left( \gamma R_2\right) \\&\quad +\gamma K_1\left( \gamma R_2\right) \sqrt{k})-I_1\left( \gamma R_3\right) (k K_1\left( \gamma R_2\right) \gamma ^2+\beta \phi K_0\left( \gamma R_2\right) \sqrt{k} \gamma )-(\beta \phi I_0\left( \gamma R_2\right) -\sqrt{k} \gamma I_1\left( \gamma R_2\right) )\\&\quad (\alpha K_0\left( \gamma R_3\right) +\gamma K_1\left( \gamma R_3\right) \sqrt{k})) R_1^3+21 P (\alpha I_0\left( \gamma R_3\right) (\beta \phi K_0\left( \gamma R_2\right) +\gamma K_1\left( \gamma R_2\right) \sqrt{k})-I_1\left( \gamma R_3\right) \\&\quad (k K_1\left( \gamma R_2\right) \gamma ^2+\beta \phi K_0\left( \gamma R_2\right) \sqrt{k} \gamma )-(\beta \phi I_0\left( \gamma R_2\right) -\sqrt{k} \gamma I_1\left( \gamma R_2\right) ) (\alpha K_0\left( \gamma R_3\right) +\gamma K_1\left( \gamma R_3\right) \sqrt{k}))\\&\quad \lambda _1 R_1^2+21 P (2 (\alpha I_0\left( \gamma R_3\right) (\beta \phi K_0\left( \gamma R_2\right) +\gamma K_1\left( \gamma R_2\right) \sqrt{k})-I_1\left( \gamma R_3\right) (k K_1\left( \gamma R_2\right) \gamma ^2+\beta \phi K_0\left( \gamma R_2\right) \\&\quad \sqrt{k} \gamma )-(\beta \phi I_0\left( \gamma R_2\right) -\sqrt{k} \gamma I_1\left( \gamma R_2\right) ) (\alpha K_0\left( \gamma R_3\right) +\gamma K_1\left( \gamma R_3\right) \sqrt{k})) R_2^4-4 \sqrt{k} \phi (-\alpha I_0\left( \gamma R_3\right) \\&\quad K_0\left( \gamma R_2\right) +\gamma I_1\left( \gamma R_3\right) \sqrt{k} K_0\left( \gamma R_2\right) +I_0\left( \gamma R_2\right) (\alpha K_0\left( \gamma R_3\right) +\gamma K_1\left( \gamma R_3\right) \sqrt{k})) R_2^3+(-\alpha I_0\left( \gamma R_3\right) \\&\quad ((8 k+1) \beta \phi K_0\left( \gamma R_2\right) +\gamma (1-8 \phi ) K_1\left( \gamma R_2\right) \sqrt{k})+\gamma I_1\left( \gamma R_3\right) \sqrt{k} ((8 k+1) \beta \phi K_0\left( \gamma R_2\right) +\gamma (1-8 \phi )\\&\quad K_1\left( \gamma R_2\right) \sqrt{k})+((8 k+1) \beta \phi I_0\left( \gamma R_2\right) +\gamma (8 \phi -1) I_1\left( \gamma R_2\right) \sqrt{k}) (\alpha K_0\left( \gamma R_3\right) +\gamma K_1\left( \gamma R_3\right) \sqrt{k})) R_2^2 \\&\quad -8 \sqrt{k} \gamma \phi (2 (I_1\left( \gamma R_2\right) (k \beta \gamma K_1\left( \gamma R_3\right) {+}\alpha \beta K_0\left( \gamma R_3\right) \sqrt{k}){+}\beta (\alpha I_0\left( \gamma R_3\right) -\sqrt{k} \gamma I_1\left( \gamma R_3\right) ) K_1\left( \gamma R_2\right) \sqrt{k}) \\&\quad +\alpha (I_1\left( \gamma R_3\right) K_0\left( \gamma R_3\right) +I_0\left( \gamma R_3\right) K_1\left( \gamma R_3\right) ) R_3) R_2-8 k (-2 (\alpha I_0\left( \gamma R_3\right) (\beta \phi K_0\left( \gamma R_2\right) +\gamma K_1\left( \gamma R_2\right) \\&\quad \sqrt{k})-I_1\left( \gamma R_3\right) (k K_1\left( \gamma R_2\right) \gamma ^2+\beta \phi K_0\left( \gamma R_2\right) \sqrt{k} \gamma )-(\beta \phi I_0\left( \gamma R_2\right) -\sqrt{k} \gamma I_1\left( \gamma R_2\right) ) (\alpha K_0\left( \gamma R_3\right) \\&\quad + \gamma K_1\left( \gamma R_3\right) \sqrt{k})) R_3^2{-}2 \alpha \beta \gamma \phi (I_1\left( \gamma R_3\right) K_0\left( \gamma R_3\right) {+}I_0\left( \gamma R_3\right) K_1\left( \gamma R_3\right) ) R_3{+}(\alpha I_0\left( \gamma R_3\right) (\beta \phi K_0\left( \gamma R_2\right) \\&\quad +\gamma K_1\left( \gamma R_2\right) \sqrt{k}){-}I_1\left( \gamma R_3\right) (k K_1\left( \gamma R_2\right) \gamma ^2+\beta \phi K_0\left( \gamma R_2\right) \sqrt{k} \gamma ){-}(\beta \phi I_0\left( \gamma R_2\right) -\sqrt{k} \gamma I_1\left( \gamma R_2\right) )\\&\quad (\alpha K_0\left( \gamma R_3\right) +\gamma K_1\left( \gamma R_3\right) \sqrt{k})) R_4^2)) \lambda _1 \div 336 (\alpha I_0\left( \gamma R_3\right) (\beta \phi K_0\left( \gamma R_2\right) +\gamma K_1\left( \gamma R_2\right) \sqrt{k})-I_1\left( \gamma R_3\right) \\&\quad (k K_1\left( \gamma R_2\right) \gamma ^2+\beta \phi K_0\left( \gamma R_2\right) \sqrt{k} \gamma )-(\beta \phi I_0\left( \gamma R_2\right) -\sqrt{k} \gamma I_1\left( \gamma R_2\right) ) (\alpha K_0\left( \gamma R_3\right) +\gamma K_1\left( \gamma R_3\right) \sqrt{k})) \lambda _1 \end{aligned}$$

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Sharma, B.D., Yadav, P.K. A Two-Layer Mathematical Model of Blood Flow in Porous Constricted Blood Vessels. Transp Porous Med 120, 239–254 (2017). https://doi.org/10.1007/s11242-017-0918-9

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