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Transport of a reactive solute in a pulsatile non-Newtonian liquid flowing through an annular pipe

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Abstract

The impact of heterogeneous (kinetic reversible phase exchange and irreversible absorption) chemical reactions along with a homogeneous first-order reaction is considered for the dispersion of a solute in a solvent flowing through an annular pipe under a periodic pressure gradient. A Casson model is used to describe the non-Newtonian viscosity of the liquid. The Aris–Barton method of moments is employed to study the behavior of the dispersion coefficient. The axial distribution of the mean concentration is determined using the Hermite polynomial representation of central moments. This study focuses on the transport phenomena in terms of the dispersion coefficient due to multiple kinds of reaction, yield stress, radius ratio, etc., which could be useful for analysis of flow of physiological blood-like liquids.

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Acknowledgements

The authors are grateful to the editor and reviewers for constructive comments and suggestions that helped to improve this article. S.D. is grateful to the National Institute of Technology, Agartala, India for financial support to pursue this work.

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Author notes

  1. B. S. Mazumder

    Present address: Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, India

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology, Agartala, 799046, India

    Sudip Debnath, Apu Kumar Saha & Ashis Kumar Roy

  2. Fluvial Mechanics Laboratory, Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata, 700108, India

    B. S. Mazumder

  3. Department of Aerospace Engineering and Applied Mechanics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, 711103, India

    B. S. Mazumder

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  1. Sudip Debnath
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Appendix

Appendix

The first correction of the shear stress (\(\tau _1\)) is derived as follows:

Since

$$\begin{aligned} \frac{\partial {u_0}}{\partial {t}}=\frac{\partial {u_0}}{\partial {p}}\frac{\mathrm {d}p}{\mathrm {d}t}+\frac{\partial {u_0}}{\partial {\lambda }}\frac{\mathrm {d} \lambda }{\mathrm {d}t}, \end{aligned}$$

Eq. (35) can be written as

$$\begin{aligned} \tau _1= & {} \frac{1}{r}\int _{r}^{\lambda }\left( \frac{\partial {u_0}}{\partial {p}}\frac{\mathrm {d}p}{\mathrm {d}t}+\frac{\partial {u_0}}{\partial {\lambda }}\frac{\mathrm {d} \lambda }{\mathrm {d}t}\right) r\, \mathrm {d}r+\frac{D_1(t)}{r}\\= & {} \frac{1}{r}\left[ \frac{\mathrm {d}p}{\mathrm {d}t}\int _{r}^{\lambda }\frac{\partial {u_0}}{\partial {p}}r \mathrm {d}r+\frac{\mathrm {d}\lambda }{\mathrm {d}t}\int _{r}^{\lambda } \frac{\partial {u_0}}{\partial {\lambda }} r \mathrm {d}r\right] +\frac{D_1(t)}{r}. \end{aligned}$$

Explicitly,

$$\begin{aligned} \tau _1= & {} \frac{1}{r}\left[ \frac{\mathrm {d}p}{\mathrm {d}t}\int _{r}^{\lambda _1} \frac{\partial {u_0^+}}{\partial {p}} r \mathrm {d}r+\frac{\mathrm {d} \lambda }{\mathrm {d}t} \int _{r}^{\lambda _1}\frac{\partial {u_0^+}}{\partial {\lambda }}r \mathrm {d}r\right] \nonumber \\&\quad +\left( \frac{ \lambda ^2-\lambda _1^2}{2r}\right) \left[ \frac{\mathrm {d}p}{\mathrm {d}t} \frac{\partial {u_{0\mathrm{p}}^-}}{\partial {p}}+\frac{\mathrm {d} \lambda }{\mathrm {d}t}\frac{\partial {u_{0\mathrm{p}}^-}}{\partial {\lambda }} \right] +\frac{D_1(t)}{r}\quad \text{ if } \; r_\mathrm {i}\le r\le \lambda _1, \end{aligned}$$
(A.1a)
$$\begin{aligned} \tau _1= & {} \left( \frac{\lambda ^2-r^2}{2r}\right) \left[ \frac{\mathrm {d}p}{\mathrm {d}t} \frac{\partial {u_{0\mathrm{p}}^-}}{\partial {p}}+\frac{\mathrm {d} \lambda }{\mathrm {d}t}\frac{\partial {u_{0\mathrm{p}}^-}}{\partial {\lambda }} \right] +\frac{D_1(t)}{r} \quad \text{ if } \; \lambda _1\le r\le \lambda _2, \end{aligned}$$
(A.1b)
$$\begin{aligned} \tau _1= & {} \left( \frac{\lambda ^2-\lambda _2^2}{2r}\right) \left[ \frac{\mathrm {d}p}{\mathrm {d}t} \frac{\partial {u_{0\mathrm{p}}^-}}{\partial {p}}+\frac{\mathrm {d} \lambda }{\mathrm {d}t}\frac{\partial {u_{0\mathrm{p}}^-}}{\partial {\lambda }} \right] +\frac{D_1(t)}{r} \nonumber \\&\quad -\frac{1}{r}\left[ \frac{\mathrm {d}p}{\mathrm {d}t}\int _{\lambda _2}^{r}\frac{\partial {u_0^{++}}}{\partial {p}} r \mathrm {d}r+\frac{\mathrm {d}\lambda }{\mathrm {d}t}\int _{\lambda _2}^{r}\frac{\partial {u_0^{++}}}{\partial {\lambda }} r \mathrm {d}r\right] \quad \text{ if } \; \lambda _2\le r\le r_\mathrm {o}. \end{aligned}$$
(A.1c)

Now, using the values of

$$\begin{aligned} \frac{\partial {u_0^+}}{\partial {p}} ,\frac{\partial {u_0^+}}{\partial {\lambda }}, \frac{\partial {u_{0\mathrm{p}}^-}}{\partial {p}}, \frac{\partial {u_{0\mathrm{p}}^-}}{\partial {\lambda }},\frac{\partial {u_0^{++}}}{\partial {p}} ,\frac{\partial {u_0^{++}}}{\partial {\lambda }}, \end{aligned}$$

Equations (A.1a)–(A.2c) can be rewritten as

$$\begin{aligned} \tau _1= & {} \frac{1}{4r}\frac{\mathrm {d}p}{\mathrm {d}t}\left[ 2\lambda ^2\left\{ 2\left( \lambda _1^2 \log (\lambda _1)-r^2 \log (r)\right) -\left( \lambda _1^2 -r^2\right) \left( 1+2 \log (r_\mathrm {i})\right) \right\} +\left( \lambda _1^2-r^2\right) \left( 2 r_\mathrm {i}^2-\lambda _1^2-r^2\right) \right. \nonumber \\&\quad +2\left( \lambda ^2-\lambda _1^2\right) \left\{ 2\lambda ^2 \log \left( \frac{\lambda _1}{r_\mathrm {i}}\right) -\left( \lambda _1^2- r_\mathrm {i}^2\right) \right\} -2(2\varLambda )^\frac{1}{2}\left\{ 2\int _{r}^{\lambda _1}\left( \int _{r_\mathrm {i}}^{r} \left( \frac{\lambda ^2-r^2}{r}\right) ^{\frac{1}{2}} \mathrm {d}r\right) r \,\mathrm {d}r \right. \nonumber \\&\quad \left. \left. +\left( \lambda ^2-\lambda _1^2\right) \int _{r_\mathrm {i}}^{\lambda _1}\left( \frac{\lambda ^2- r^2}{r}\right) ^\frac{1}{2} \mathrm {d}r\right\} \right] + \frac{p(t)\lambda }{r} \frac{\mathrm {d}\lambda }{\mathrm {d}t}\left[ 2\left( \lambda _1^2 \log (\lambda _1)-r^2 \log (r)\right) \right. \nonumber \\&\quad -\left( \lambda _1^2-r^2\right) \left( 1+2 \log (r_\mathrm {i})\right) +2\left( \lambda ^2-\lambda _1^2\right) \log \left( \frac{\lambda _1}{r_\mathrm {i}}\right) \nonumber \\&\left. \quad -(2\varLambda )^\frac{1}{2}\left\{ 2 \int _{r}^{\lambda _1}\left( \int _{r_\mathrm {i}}^{r} \frac{1}{ \sqrt{r(\lambda ^2-r^2)}} \mathrm {d}r \right) r\, \mathrm {d}r +\left( \lambda ^2-\lambda _1^2\right) \int _{r_\mathrm {i}}^{\lambda _1} \frac{1}{\sqrt{r(\lambda ^2-r^2)}} \mathrm {d}r\right\} \right] \nonumber \\&\quad +\frac{D_1(t)}{r} \quad \text{ if } \; r_\mathrm {i}\le r\le \lambda _1,\end{aligned}$$
(A.2a)
$$\begin{aligned} \tau _1= & {} \left( \frac{\lambda ^2- r^2}{2 r}\right) \frac{\mathrm {d}p}{\mathrm {d}t}\left[ 2\lambda ^2 \log \left( \frac{\lambda _1}{r_\mathrm {i}}\right) -\left( \lambda _1^2-r_\mathrm {i}^2\right) \right. \left. -(2\varLambda )^\frac{1}{2} \int _{r_\mathrm {i}}^{\lambda _1}\left( \frac{\lambda ^2-r^2}{r}\right) ^\frac{1}{2} \mathrm {d}r\right] \nonumber \\&\quad +\left( \frac{\lambda ^2- r^2}{ r}\right) \frac{\mathrm {d}\lambda }{\mathrm {d}t} p(t) \lambda \left[ 2 \log \left( \frac{\lambda _1}{r_\mathrm {i}}\right) \right. \left. -(2\varLambda )^\frac{1}{2}\int _{r_\mathrm {i}}^{\lambda _1} \frac{1}{\sqrt{r(\lambda ^2-r^2)}}\mathrm {d}r\right] +\frac{D_1(t)}{r}\quad \text{ if } \; \lambda _1\le r\le \lambda _2, \end{aligned}$$
(A.2b)
$$\begin{aligned} \tau _1= & {} \frac{1}{4r}\frac{\mathrm {d}p}{\mathrm {d}t}\left[ 2\lambda ^2\left\{ \left( r^2-\lambda _2^2\right) \left( 1+2 \log (r_\mathrm {o})\right) -2\left( r^2 \log (r)\right. \right. \right. \left. \left. \left. -\lambda _2^2 \log (\lambda _2)\right) \right\} +\left( r^2-\lambda _2^2\right) \left( r^2+\lambda _2^2-2 r_\mathrm {o}^2\right) \right. \nonumber \\&\quad +\left. 2\left( \lambda _2^2-\lambda ^2\right) \left\{ \lambda _1^2-r_\mathrm {i}^2-2\lambda ^2 \log \left( \frac{\lambda _1}{r_\mathrm {i}}\right) \right\} \right. \left. +\, 2(2\varLambda )^\frac{1}{2}\left\{ \int _{\lambda _2}^{r} 2\left( \int _{r}^{r_\mathrm {o}}\left( \frac{r^2-\lambda ^2}{r}\right) ^\frac{1}{2} \mathrm {d}r\right) r\, \mathrm {d}r \right. \right. \nonumber \\&\quad +\left( \lambda _2^2-\lambda ^2\right) \left. \left. \int _{r_\mathrm {i}}^{\lambda _1}\left( \frac{\lambda ^2-r^2}{r}\right) ^\frac{1}{2} \mathrm {d}r\right\} \right] +\frac{p(t)}{r} \lambda \frac{\mathrm {d}\lambda }{\mathrm {d}t}\left[ \left( r^2-\lambda _2^2\right) \left( 1+2 \log (r_\mathrm {o})\right) \right. \left. \right. \nonumber \\&\quad -2\left( r^2 \log (r)-\lambda _2^2 \log (\lambda _2)\right) -2\left( \lambda _2^2-\lambda ^2\right) \log \left( \frac{\lambda _1}{r_\mathrm {i}}\right) \nonumber \\&\left. \quad -(2\varLambda )^\frac{1}{2} \left\{ 2\int _{\lambda _2}^{r}\left( \int _{r}^{r_\mathrm {o}} \frac{1}{\sqrt{r(r^2-\lambda ^2)}}\mathrm {d}r\right) r \,\mathrm {d}r -\left( \lambda _2^2-\lambda ^2\right) \int _{r_\mathrm {i}}^{\lambda _1} \frac{1}{\sqrt{r(\lambda ^2-r^2)}} \mathrm {d}r \right\} \right] \nonumber \\&\quad +\frac{D_1(t)}{r} \quad \text{ if } \; \lambda _2\le r\le r_\mathrm {o}. \end{aligned}$$
(A.2c)

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Debnath, S., Saha, A.K., Mazumder, B.S. et al. Transport of a reactive solute in a pulsatile non-Newtonian liquid flowing through an annular pipe. J Eng Math 116, 1–22 (2019). https://doi.org/10.1007/s10665-019-09999-1

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