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# Analysis of magnetohydrodynamics transient flow in a horizontal annular duct

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## Abstract

The aim of this research is to investigate the key role of radial magnetic field with the transient flow of a viscous, incompressible, electrically-conducting fluid between two horizontal concentric cylinders. In this case, motion of such fluid creates a viscous force within the annulus region. The governing equation of the problem has been presented by using Brinkman model and solution is done by the separable variable’s method. The analytical solution of the problem is established in the form of Bessel, modified Bessel functions and eigenfunctions. Further, we have discussed the results in detail by using graphs and tables, to show the impacts of various parameters on the velocity profile. Moreover, the variation of velocity, skin-frictions, mass flux and the effect on ratio of radii i.e. outer radius to inner radius are thoroughly described with some non-dimensional parameters as Hartmann number, Reynolds number, Euler number and time. As a result, for higher value of Hartmann number, the solution reduces from unsteady to steady for shorter time scale, the steady solution can be obtained as limiting case of the solution. For some lower values of time, velocity rapidly enhances but after certain interval of time, the velocity does not change.

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## Abbreviations

$$a^{{\prime }}$$ :

$$b^{{\prime }}$$ :

$$B_{0}^{{\prime }}$$ :

Magnetic field

$$B{\kern 1pt} r$$ :

Brinkman number

$$Eu$$ :

Euler number

$$G$$ :

Negative applied pressure gradient $$\left( {{{ - \partial p} \mathord{\left/ {\vphantom {{ - \partial p} {\partial z}}} \right. \kern-0pt} {\partial z}}} \right)$$

$$Ha$$ :

Hartmann number

$$I_{n} \left( r \right)$$ :

Modified Bessel function of first kind of order n

$$J_{n} \left( r \right)$$ :

Bessel function of first kind of order n

$$Y_{n} \left( r \right)$$ :

Bessel function of second kind of order n

$$K_{n} \left( r \right)$$ :

Modified Bessel function of second kind of order n

p :

Pressure

$$\lambda$$ :

Ratio of radii of cylinder Outer and Inner

r :

$$r^{{\prime }}$$ :

$$\text{Re}$$ :

Reynolds number

t :

Time

$$t^{{\prime }}$$ :

Non-dimensional time

$$U$$ :

Non-dimensional fluid velocity in axial direction

$$u_{z}^{*}$$ :

Characteristics velocity

$$U_{\hbox{max} }$$ :

Non-dimensional maximum velocity in axial direction

$$U_{a}$$ :

Non-dimensional average velocity in axial direction

$$u_{z}^{{\prime }}$$ :

Fluid velocity in axial direction

$$u_{r}^{{\prime }}$$ :

$$u_{\theta }^{{\prime }}$$ :

Fluid velocity in angular direction

$$u$$ :

Non-dimensional velocity defined in Eq. (7)

$$z$$ :

Dimensionless axial coordinate

$$z^{{\prime }}$$ :

Axial coordinate

$$\alpha_{n}$$ :

Separation constant

$$\mu$$ :

Fluid viscosity

$$\rho$$ :

Fluid density

$$\sigma$$ :

Fluid conductivity

$$\theta^{{\prime }}$$ :

Angular coordinate

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## Acknowledgments

The Author (JPM) would like to acknowledge the financial support received from the University Grants Commission, New Delhi, India.

## Appendix

### Appendix

These constants which are mentioned below, we have used in the mathematical expressions:

$$S = s_{1} + s_{3} + s_{5} - s_{2} - s_{4} - s_{6} ,$$
$$P = p_{1} - p_{2} - p_{3} ,$$
$$s_{1} = \frac{{Eu.\text{Re} }}{{Ha^{2} }}.\frac{{Y_{0} (\alpha_{n} \lambda )}}{{\alpha_{n} }}\left\{ {\lambda \;J_{1} (\alpha_{n} \lambda ) - J_{1} (\alpha_{n} )} \right\},$$
$$s_{2} = \frac{{Eu.\text{Re} }}{{Ha^{2} }}.\frac{{J_{0} (\alpha_{n} \lambda )}}{{\alpha_{n} }}\left\{ {\lambda \;Y_{1} (\alpha_{n} \lambda ) - Y_{1} (\alpha_{n} )} \right\},$$
$$s_{3} = L.\frac{{Eu.\text{Re} }}{{Ha^{2} }}.\frac{{Y_{0} (\alpha_{n} \lambda )}}{{Ha^{2} + \alpha_{n}^{2} }}I_{1} ,$$
$$s_{4} = L.\frac{{Eu.\text{Re} }}{{Ha^{2} }}.\frac{{J_{0} (\alpha_{n} \lambda )}}{{Ha^{2} + \alpha_{n}^{2} }}I_{2} ,$$
$$s_{5} = M.\frac{{Eu.\text{Re} }}{{Ha^{2} }}.\frac{{Y_{0} (\alpha_{n} \lambda )}}{{Ha^{2} + \alpha_{n}^{2} }}I_{3} ,$$
$$s_{6} = M.\frac{{Eu.\text{Re} }}{{Ha^{2} }}.\frac{{J_{0} (\alpha_{n} \lambda )}}{{Ha^{2} + \alpha_{n}^{2} }}I_{4} ,$$
I_{1} = \left\{ \begin{aligned} \lambda Ha.\;J_{0} (\alpha_{n} \lambda ).I_{1} (\lambda Ha) + \alpha_{n} \lambda J_{1} (\alpha_{n} \lambda ).I_{0} (\lambda Ha) \hfill \\ - Ha.J_{0} (\alpha_{n} ).I_{1} (Ha) - \alpha_{n} .J_{1} (\alpha_{n} )I_{0} (Ha) \hfill \\ \end{aligned} \right\},
I_{2} = \left\{ \begin{aligned} \lambda Ha.\;Y_{0} (\alpha_{n} \lambda ).I_{1} (\lambda Ha) + \alpha_{n} \lambda .Y_{1} (\alpha_{n} \lambda ).I_{0} (\lambda Ha) \hfill \\ - Ha.Y_{0} (\alpha_{n} ).I_{1} (Ha) - \alpha_{n} .Y_{1} (\alpha_{n} )I_{0} (Ha) \hfill \\ \end{aligned} \right\},
I_{3} = \left\{ \begin{aligned} \alpha_{n} \lambda .K_{0} (\lambda Ha).\;J_{1} (\alpha_{n} \lambda ) - \lambda Ha\;J_{0} (\alpha_{n} \lambda ).K_{1} (\lambda Ha) \hfill \\ - \alpha_{n} .K_{0} (Ha).\;J_{1} (\alpha_{n} ) + Ha\;J_{0} (\alpha_{n} ).K_{1} (Ha) \hfill \\ \end{aligned} \right\},
I_{4} = \left\{ \begin{aligned} \alpha_{n} \lambda .K_{0} (\lambda Ha).\;Y_{1} (\alpha_{n} \lambda ) - \lambda Ha\;Y_{0} (\alpha_{n} \lambda ).K_{1} (\lambda Ha) \hfill \\ - \alpha_{n} .K_{0} (Ha).\;Y_{1} (\alpha_{n} ) + Ha\;Y_{0} (\alpha_{n} ).K_{1} (Ha) \hfill \\ \end{aligned} \right\},
$$p_{1} = \frac{{\lambda^{2} }}{2}\left\{ {Y_{0} (\alpha_{n} \lambda ).\;J_{1} (\alpha_{n} \lambda ) - \;J_{0} (\alpha_{n} \lambda )Y_{1} (\alpha_{n} \lambda )} \right\}^{2} ,$$
$$p_{2} = \frac{1}{2}\left\{ {Y_{0} (\alpha_{n} \lambda ).\;J_{0} (\alpha_{n} ) - \;J_{0} (\alpha_{n} \lambda )Y_{0} (\alpha_{n} )} \right\}^{2} ,$$
$$p_{3} = \frac{1}{2}\left\{ {Y_{0} (\alpha_{n} \lambda ).\;J_{1} (\alpha_{n} ) - \;J_{0} (\alpha_{n} \lambda )Y_{1} (\alpha_{n} )} \right\}^{2} ,$$
$$\xi_{1} = L.Ha.I_{1} (Ha) - M.Ha.K_{1} (Ha),$$
$$\eta_{1} = Y_{0} (\alpha_{n} \lambda ).\alpha_{n} \;J_{1} (\alpha_{n} ) - J_{0} (\alpha_{n} \lambda ).\alpha_{n} Y_{1} (\alpha_{n} ),$$
$$\xi_{2} = L.Ha.I_{1} (\lambda Ha) - M.Ha.K_{1} (\lambda Ha),$$
$$\eta_{2} = Y_{0} (\alpha_{n} \lambda ).\alpha_{n} \;J_{1} (\alpha_{n} \lambda ) - J_{0} (\alpha_{n} \lambda ).\alpha_{n} Y_{1} (\alpha_{n} \lambda ),$$
$$\xi_{3} = \lambda I_{1} (\lambda Ha) - I_{1} (Ha),$$
$$\eta_{3} = \lambda K_{1} (\lambda Ha) - K_{1} (Ha),$$
$$\xi_{4} = \lambda J_{1} (\alpha_{n} \lambda ) - J_{1} (\alpha_{n} ),$$
$$\eta_{4} = \lambda Y_{1} (\alpha_{n} \lambda ) - Y_{1} (\alpha_{n} ).$$

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