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Two-phase analysis of blood flow through a stenosed artery with the effects of chemical reaction and radiation

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Abstract

The paper presents a study related to the two-phase analysis of pulsatile blood flow through a narrowed stenosed artery with radiation and the chemical effects. In the model, a vertical artery is considered in which the flow of blood is assumed vertical upward and the direction of an external applied magnetic is in the radial direction of the flow. To understand the behavior of blood flow, graphs of the velocity profile, wall shear stress, flow rate, flow impedance and concentration profile are portrayed with different values of the magnetic and radiation parameters. In order to validate the results, a comparative study is presented between the single-phase and two-phase model of the blood flow, which shows that the two-phase model fits more accurately with the experimental data than the single-phase model, as mean errors are \(0.3\%\) for the two-phase model while it is \(1\%\) for single-phase model. For pulsatile flow, the phase difference between the pressure gradient and the flow rate is displayed with the effects of the magnetic field and different heights of the stenosis.

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Acknowledgements

Authors are sincerely thankful to the Department of Science and Technology, Government of India (SR/FST/MSI-090/2013(C)) for their financial support.

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Authors and Affiliations

  1. Department of Analytics, IFIM Business School, Bangalore, 560100, India

    Bhavya Tripathi

  2. Department of Mathematics, Birla Institute of Technology and Science, Pilani, 333031, India

    Bhupendra Kumar Sharma

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Correspondence to Bhavya Tripathi.

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Appendix A

Appendix A

$$\begin{aligned} {\beta _1}&=-\left( \frac{{K_0}{{N}^2}}{{\alpha _0}}+{i\frac{P_e}{\rho _0}\left( \frac{K_0}{s_0}\right) }\right) , \quad \beta _2 = -\left( {{N}^2}+i{P_e}\right) .\\ {\gamma _1}&=-\left( i{Re}{D_0}{S_c}+\frac{E}{E_0}{D_0}{S_c}\right) , \quad {\gamma _2} = -\left( i{Re}{S_c}+{E}{S_c}\right) \\ \lambda _1&= -\left( M^2+\frac{{\mu _0}{Re}}{\rho _0}i\right) \quad \lambda _2= -\left( M^2+{{Re}i}\right) \\ {U_1}&=\left( \frac{{J_0}({\sqrt{\beta _2}}{R_1})}{{J_0}({\sqrt{\beta _1}}{R_1})}-\frac{{J_0}({\sqrt{\beta _2}}{R})}{{Y_0}({\sqrt{\beta _2}}{R})}{\frac{{Y_0}({\sqrt{\beta _2}}{R_1})}{{J_0}({\sqrt{\beta _1}}{R_1})}}\right) ,\\ {U_2}&=\frac{{Y_0}({\sqrt{\beta _2}}{R_1}))}{{J_0}({\sqrt{\beta _1}}{R_1}){Y_0(\sqrt{\beta _2}R)}}\\ {U_3}&={\sqrt{\beta _1}}{U_1}{J_1}(\sqrt{\beta _1}{R_1})-\sqrt{\beta _2}\left( {{{J_1}({\sqrt{\beta _2}}{R_1})}-{\frac{{J_0}({\sqrt{\beta _2}}{R})}{{Y_0(\sqrt{\beta _2}R)}}{{Y_1}}({\sqrt{\beta _2}}{R_1})}}\right) \\ {U_4}&=\left( \frac{{J_0}({\sqrt{\gamma _2}}{R_1})}{{J_0}({\sqrt{\gamma _1}}{R_1})}-\frac{{J_0}({\sqrt{\gamma _2}}{R})}{{J_0}({\sqrt{\gamma _1}}{R_1})}{\frac{{Y_0}({\sqrt{\gamma _2}}{R_1})}{{Y_0}({\sqrt{\gamma _2}}{R})}}\right) ,\\ {U_5}&=\frac{{Y_0}({\sqrt{\gamma _2}}{R_1}))}{{J_0}({\sqrt{\gamma _1}}{R_1}){Y_0(\sqrt{\gamma _2}R)}}\\ {U_6}&={\sqrt{\gamma _1}}{U_5}{J_1}(\sqrt{\gamma _1}{R_1})-\sqrt{\gamma _2}{\left( {{J_1}({\sqrt{\gamma _2}}{R_1})}-{\frac{{J_0}({\sqrt{\gamma _2}}{R})}{{Y_0(\sqrt{\gamma _2}R)}}.{{Y_1}({\sqrt{\gamma _2}}{R_1})}} \right) }\\ D_1&=-{A_1}(R_1){J_0}(\sqrt{\lambda _1}{R_1})-{B_1}(R_1){Y_0}(\sqrt{\lambda _1}{R_1})+{A_2}(R_1){J_0}(\sqrt{\lambda _2}{R_1})\\&\quad +{B_2}(R_1){Y_0}(\sqrt{\lambda _2}{R_1})\\ D_2&= -{A_2}(R){J_0}(\sqrt{\lambda _2}{R})-{B_2}(R){Y_0}(\sqrt{\lambda _2}{R})\\ D_3&= -\frac{\partial {A_1(R_1)}}{\partial {r}} {J_0}(\sqrt{\lambda _1}{R_1})+{A_1}(R_1){\sqrt{\lambda _1}}{J_1}(\sqrt{\lambda _1}{R_1}) -\frac{\partial {B_1(R_1)}}{\partial {r}} {Y_0}(\sqrt{\lambda _1}{R_1})\\&\quad +{B_1}(R_1){\sqrt{\lambda _1}}{Y_1}(\sqrt{\lambda _1}{R_1})+\frac{\partial {A_2(R_1)}}{\partial {r}} {J_0}(\sqrt{\lambda _2}{R_1})+\frac{\partial {B_2(R_1)}}{\partial {r}} {Y_0}(\sqrt{\lambda _2}{R_1})\\&\quad -{A_2}(R_1){\sqrt{\lambda _2}}{J_1}(\sqrt{\lambda _2}{R_1})-{B_2}(R_1){\sqrt{\lambda _2}}{Y_1}(\sqrt{\lambda _2}{R_1}) \end{aligned}$$

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Tripathi, B., Sharma, B.K. Two-phase analysis of blood flow through a stenosed artery with the effects of chemical reaction and radiation. Ricerche mat 73, 151–177 (2024). https://doi.org/10.1007/s11587-021-00571-7

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