Abstract
In this paper, the problem of transient unidirectional flow in a circular porous medium is examined. The governing equation of fluid flow is modelled by employing the Darcy–Brinkman model. Using separation of variables technique, the analytical solution of modelled equation is established in the form of Bessel and modified Bessel functions. Moreover, the steady solution can be obtained as limiting case of the solution. Finally, the impact of relevant parameters on the transient velocity of fluid is also analyzed by the graphical and tabular illustrations.
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Abbreviations
- \( a \) :
-
Radius of cylinder
- \( Br \) :
-
Brinkman number
- \( Da \) :
-
Darcy number
- \( Eu \) :
-
Euler number
- \( G \) :
-
Negative applied pressure gradient \( \left( {{{ - \partial p} \mathord{\left/ {\vphantom {{ - \partial p} {\partial z}}} \right. \kern-0pt} {\partial z}}} \right) \)
- \( I_{n} \left( r \right) \) :
-
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 :
-
Permeability
- \( M \) :
-
Viscosity ratio parameter, \( {{\mu_{e} } \mathord{\left/ {\vphantom {{\mu_{e} } \mu }} \right. \kern-0pt} \mu } \)
- p :
-
Pressure
- r :
-
Radial coordinate in non-dimensional form
- \( r' \) :
-
Radial coordinate
- \( \text{Re} \) :
-
Reynolds number
- \( s \) :
-
\( {1 \mathord{\left/ {\vphantom {1 {\sqrt {MDa} }}} \right. \kern-0pt} {\sqrt {MDa} }} \)
- \( t \) :
-
Time
- \( t' \) :
-
Non-dimensional time
- \( U \) :
-
Non-dimensional fluid velocity in axial direction
- \( \overline{U} \) :
-
Non-dimensional velocity defined in Eq. (6)
- \( U^{*} \) :
-
Non-dimensional steady velocity in axial direction
- \( U_{\hbox{max} } \) :
-
Non-dimensional maximum velocity in axial direction
- \( U_{a} \) :
-
Non-dimensional average velocity in axial direction
- \( u_{z}^{*} \) :
-
Characteristics velocity
- \( u^{\prime}_{z} \) :
-
Fluid velocity in axial direction
- \( u^{\prime}_{r} \) :
-
Fluid velocity in radial direction
- \( u^{\prime}_{\theta } \) :
-
Fluid velocity in angular direction
- \( z \) :
-
Dimensionless axial coordinate
- \( z' \) :
-
Axial coordinate
- \( \alpha_{n} \) :
-
Separation constant
- \( \lambda_{n} \) :
-
Solution of Bessel function of first kind of zero order \( \left( {{{\alpha_{n} } \mathord{\left/ {\vphantom {{\alpha_{n} } {\sqrt M }}} \right. \kern-0pt} {\sqrt M }}} \right) \)
- \( \mu \) :
-
Fluid viscosity
- \( \mu_{e} \) :
-
Effective viscosity in Brinkman term
- \( \rho \) :
-
Fluid density
- \( \theta ' \) :
-
Angular coordinate
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Acknowledgements
The author (SLY) would like to thank the Council of Scientific and Industrial Research, New Delhi, India, for financial support in the form of a Junior Research Fellowship.
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Yadav, S.L., Singh, A.K. Transient Flow in a Circular Cylinder Filled with Porous Material. Int. J. Appl. Comput. Math 4, 145 (2018). https://doi.org/10.1007/s40819-018-0579-6
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DOI: https://doi.org/10.1007/s40819-018-0579-6