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Compressibility effects on the Kelvin–Helmholtz and Rayleigh–Taylor instabilities between two immiscible fluids flowing through a porous medium

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Abstract

We examine analytically the Kelvin–Helmholtz stability of two compressible superposed viscous fluids flowing through porous medium. The dispersion relationship is derived and evaluated for special cases adopting the normal mode procedure. The results have been compared with the statics (RTI) and incompressible cases. It is shown that the behaviour of KHI tends to RTI behaviour if the difference in the initial velocity of two fluids (U2 − U1) is small. For incompressible KHI, the kinematic viscosity induces stability. The permeability has also stabilizing role on the perturbation’s growth, but the growth rate increases as permeability increases. The porosity has destabilizing effect on KHI for the case of small values of porosity, while for large values, it has stabilizing effect. The compressibility parameters (the specific heats ratio and pressure at equilibrium state) have stabilizing role on KHI. The behaviour of normalized growth rate of a compressible KHI model with the porosity effect in most cases capitulates to effect of porosity.

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Acknowledgements

The authors would like to thank Prof. Serge Gauthier for his valuable time for reading the manuscripts and providing valuable suggestions and his help to writing the paper, Dr. M. Elsayed (Department of Mathematics and Computer Science, Faculty of Science, South Valley University, Qena, Egypt) for assisting in the revising process. We would like also to thank the referees for their interest in this work and for their many constructive suggestions that improved the original manuscript.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Science, South Valley University, Qena, 83523, Egypt

    G. A. Hoshoudy

  2. Department of Mathematics, Babasaheb Bhimrao Ambedkar University, Lucknow, 226025, India

    Mukesh Kumar Awasthi

Authors
  1. G. A. Hoshoudy
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  2. Mukesh Kumar Awasthi
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Correspondence to G. A. Hoshoudy.

Appendix

Appendix

$$ \frac{{\rho^{(0)} }}{{\varepsilon^{2} }}\left( {\varepsilon \frac{{\partial u_{x}^{(1)} }}{\partial t} + u_{x}^{(0)} (y)\frac{{\partial u_{x}^{(1)} }}{\partial x} + u_{y}^{(1)} \frac{{\partial u_{x}^{(0)} (y)}}{\partial y}} \right) = - \frac{{\partial p^{(1)} }}{\partial x} - \frac{{\rho^{(0)} \nu_{0} }}{\rm K}u_{x}^{(1)} , $$
(25)
$$ \frac{{\rho^{(0)} }}{{\varepsilon^{2} }}\left( {\varepsilon \frac{{\partial u_{y}^{(1)} }}{\partial t} + u_{x}^{(0)} (y)\frac{{\partial u_{y}^{(1)} }}{\partial x}} \right) = - \frac{{\partial p^{(1)} }}{\partial y} - \frac{{\rho^{(0)} \nu_{0} }}{\rm K}u_{y}^{(1)} - \rho^{(1)} \,g, $$
(26)
$$ \frac{{\partial \rho^{(1)} }}{\partial t} = - \frac{1}{\varepsilon }\left\{ {\rho^{(0)} (y)\left( {\frac{{\partial u_{x}^{(1)} }}{\partial x} + \frac{{\partial u_{y}^{(1)} }}{\partial y}} \right) + u_{x}^{(0)} (y)\frac{{\partial \rho^{(1)} }}{\partial x} + u_{y}^{(1)} \frac{{\partial \rho^{(0)} (y)}}{\partial y}} \right\}, $$
(27)
$$ \frac{{\partial p^{(1)} }}{\partial t} = - \frac{1}{\varepsilon }\left\{ {\bar{\gamma }p^{(0)} (y)\left( {\frac{{\partial u_{x}^{(1)} }}{\partial x} + \frac{{\partial u_{y}^{(1)} }}{\partial y}} \right) + u_{x}^{(0)} (y)\frac{{\partial p^{(1)} }}{\partial x} + u_{y}^{(1)} \frac{{\partial p^{(0)} (y)}}{\partial y}} \right\}, $$
(28)
$$ \frac{{\partial p^{(0)} (y)}}{\partial y} = - \rho^{(0)} (y)g. $$
(29)
$$ \left\{ {\frac{{\rho^{(0)} \nu_{0} }}{\rm K} + \frac{{\rho^{(0)} }}{{\varepsilon^{2} }}\left( {\varepsilon n + iku_{x}^{(0)} (y)} \right)} \right\}u_{x}^{(1)} = - ikp^{(1)} - \frac{{\rho^{(0)} }}{{\varepsilon^{2} }}u_{y}^{(1)} (x,y,t)\frac{{\partial u_{x}^{(0)} (y)}}{\partial y}, $$
(30)
$$ \left\{ {\frac{{\rho^{(0)} \nu_{0} }}{\rm K} + \frac{{\rho^{(0)} }}{{\varepsilon^{2} }}\left( {\varepsilon n + iku_{x}^{(0)} (y)} \right)} \right\}u_{y}^{(1)} = - \frac{{\partial p^{(1)} }}{\partial y} - \rho_{1} g, $$
(31)
$$ \rho^{(1)} = - \frac{1}{{\varepsilon \,n + ik\,u_{x}^{(0)} (y)}}\left\{ {\rho^{(0)} (y)\left( {iku_{x}^{(1)} + \frac{{\partial u_{y}^{(1)} }}{\partial y}} \right) + u_{y}^{(1)} \frac{{\partial \rho^{(0)} (y)}}{\partial y}} \right\}, $$
(32)
$$ p^{(1)} = - \frac{1}{{\varepsilon n + iku_{x}^{(0)} (y)}}\left\{ {\bar{\gamma }p^{(0)} (y)\left( {iku_{x}^{(1)} + \frac{{\partial u_{y}^{(1)} }}{\partial y}} \right) - \rho^{(0)} (y)gu_{y}^{(1)} } \right\}. $$
(33)
$$ \lambda_{1} = \frac{{g\gamma_{1} }}{{2c_{1}^{2} }} + k\left[ \begin{aligned} &\left( {\frac{{g\gamma_{1} }}{{2kc_{1}^{2} }}} \right)^{2} + \frac{{\frac{{g^{2} }}{{c_{1}^{2} }}\left( {\gamma_{1} - 1} \right)}}{{\left\{ {\frac{{\nu_{01} }}{{k_{1} }} + \frac{1}{{\varepsilon_{1}^{2} }}\left( {\varepsilon_{1} n + ik\bar{U}_{1} } \right)} \right\}\left\{ {\varepsilon_{1} n + ik\bar{U}_{1} } \right\}}} \hfill \\&\quad + \left[ {\left\{ {\varepsilon_{1} n + ik\bar{U}_{1} } \right\}\left( {\frac{1}{{k^{2} c_{1}^{2} }}} \right)\left\{ {\frac{{\nu_{01} }}{{k_{1} }} + \frac{1}{{\varepsilon_{1}^{2} }}\left( {\varepsilon_{1} n + ik\bar{U}_{1} } \right)} \right\} + 1} \right] \hfill \\ \end{aligned} \right]^{{\frac{1}{2}}} , $$
(34)
$$ \lambda_{2} = \frac{{g\gamma_{2} }}{{2c_{2}^{2} }} - k\left[ \begin{aligned} &\left( {\frac{{g\gamma_{2} }}{{2kc_{2}^{2} }}} \right)^{2} + \frac{{\frac{{g^{2} }}{{c_{2}^{2} }}\left( {\gamma_{2} - 1} \right)}}{{\left\{ {\frac{{\nu_{02} }}{{k_{2} }} + \frac{1}{{\varepsilon_{2}^{2} }}\left( {\varepsilon_{2} n + ik\bar{U}_{2} } \right)} \right\}\left\{ {\varepsilon_{2} n + ik\bar{U}_{2} } \right\}}} \hfill \\&\quad + \left[ {\left\{ {\varepsilon_{2} n + ik\bar{U}_{2} } \right\}\left( {\frac{1}{{k^{2} c_{2}^{2} }}} \right)\left\{ {\frac{{\nu_{02} }}{{k_{2} }} + \frac{1}{{\varepsilon_{2}^{2} }}\left( {\varepsilon_{2} n + ik\bar{U}_{2} } \right)} \right\} + 1} \right] \hfill \\ \end{aligned} \right]^{{\frac{1}{2}}} $$
(35)
$$ \bar{\lambda }_{1} = \frac{{m\alpha_{1} }}{2} + \left[ \begin{aligned} &\left( {\frac{{m\alpha_{1} }}{2}} \right)^{2} + \frac{{\left( {\gamma_{1} - 1} \right)m\alpha_{1} }}{{\gamma_{1} \left\{ {\varepsilon_{1} \bar{n} + iU_{1} } \right\}\left\{ {\frac{{\nu_{1} }}{{k_{1} }} + \frac{1}{{\varepsilon_{1}^{2} }}\left( {\varepsilon_{1} \bar{n} + iU_{1} } \right)} \right\}}} \hfill \\&\quad + \left[ {\left( {\frac{{m\alpha_{1} }}{{\gamma_{1} }}} \right)\left\{ {\varepsilon_{1} \bar{n} + iU_{1} } \right\}\left\{ {\frac{{\nu_{1} }}{{k_{1} }} + \frac{1}{{\varepsilon_{1}^{2} }}\left( {\varepsilon_{1} \bar{n} + iU_{1} } \right)} \right\} + 1} \right] \hfill \\ \end{aligned} \right]^{{\frac{1}{2}}} , $$
(36)
$$ \bar{\lambda }_{2} = \frac{{m\alpha_{2} }}{2} - \left[ \begin{aligned} &\left( {\frac{{m\alpha_{2} }}{2}} \right)^{2} + \frac{{\left( {\gamma_{2} - 1} \right)m\alpha_{2} }}{{\gamma_{2} \left\{ {\varepsilon_{2} \bar{n} + iU_{2} } \right\}\left\{ {\frac{{\nu_{2} }}{{k_{2} }} + \frac{1}{{\varepsilon_{2}^{2} }}\left( {\varepsilon_{2} \bar{n} + iU_{2} } \right)} \right\}}} \hfill \\&\quad + \left[ {\left( {\frac{{m\alpha_{2} }}{{\gamma_{2} }}} \right)\left\{ {\varepsilon_{2} \bar{n} + iU_{2} } \right\}\left\{ {\frac{{\nu_{2} }}{{k_{2} }} + \frac{1}{{\varepsilon_{2}^{2} }}\left( {\varepsilon_{2} \bar{n} + iU_{2} } \right)} \right\} + 1} \right] \hfill \\ \end{aligned} \right]^{{\frac{1}{2}}} , $$
(37)
$$ \bar{\lambda }_{3} = \frac{{m\alpha_{1} }}{2} + \left[ {\left( {\frac{{m\alpha_{1} }}{2}} \right)^{2} + \frac{{\left( {\gamma_{1} - 1} \right)m\alpha_{1} }}{{\gamma_{1} \bar{n}^{2} }} + \left( {\frac{{m\alpha_{1} }}{{\gamma_{1} }}} \right)\bar{n}^{2} + 1} \right]^{{\frac{1}{2}}} , $$
(38)
$$ \bar{\lambda }_{4} = \frac{{m\alpha_{2} }}{2} - \left[ {\left( {\frac{{m\alpha_{2} }}{2}} \right)^{2} + \frac{{\left( {\gamma_{2} - 1} \right)m\alpha_{2} }}{{\gamma_{2} \bar{n}^{2} }} + \left( {\frac{{m\alpha_{2} }}{{\gamma_{2} }}} \right)\bar{n}^{2} + 1} \right]^{{\frac{1}{2}}} . $$
(39)
$$ \bar{\lambda }_{5} = \frac{{m\alpha_{1} }}{2} + \left[ {\left( {\frac{{m\alpha_{1} }}{2}} \right)^{2} + \frac{{\left( {\gamma_{1} - 1} \right)m\alpha_{1} }}{{\gamma_{1} \bar{n}\left\{ {\frac{{\nu_{1} \varepsilon_{1} }}{{k_{1} }} + \bar{n}} \right\}}} + \left[ {\left( {\frac{{m\alpha_{1} }}{{\gamma_{1} }}} \right)\bar{n}\left\{ {\frac{{\nu_{1} \varepsilon_{1} }}{{k_{1} }} + \bar{n}} \right\} + 1} \right]} \right]^{{\frac{1}{2}}} , $$
(40)
$$ \bar{\lambda }_{6} = \frac{{m\alpha_{2} }}{2} - \left[ {\left( {\frac{{m\alpha_{2} }}{2}} \right)^{2} + \frac{{\left( {\gamma_{2} - 1} \right)m\alpha_{2} }}{{\gamma_{2} \bar{n}\left\{ {\frac{{\nu_{2} \varepsilon_{2} }}{{k_{2} }} + \bar{n}} \right\}}} + \left[ {\left( {\frac{{m\alpha_{2} }}{{\gamma_{2} }}} \right)\bar{n}\left\{ {\frac{{\nu_{2} \varepsilon_{2} }}{{\bar{k}_{2} }} + \bar{n}} \right\} + 1} \right]} \right]^{{\frac{1}{2}}} . $$
(41)
$$ \bar{\lambda }_{7} = \frac{{m\alpha_{1} }}{2} + \left[ {\left( {\frac{{m\alpha_{1} }}{2}} \right)^{2} + \frac{{\left( {\gamma_{1} - 1} \right)m\alpha_{1} }}{{\gamma_{1} \left( {\bar{n} + iU_{1} } \right)^{2} }} + \left( {\frac{{m\alpha_{1} }}{{\gamma_{1} }}} \right)\left( {\bar{n} + iU_{1} } \right)^{2} + 1} \right]^{{\frac{1}{2}}} , $$
(42)
$$ \bar{\lambda }_{8} = \frac{{m\,\alpha_{2} \,}}{2\,} - \left[ {\left( {\frac{{m\alpha_{2} }}{2\,}} \right)^{2} + \frac{{\left( {\gamma_{2} - 1} \right)\,\,m\,\,\alpha_{2} }}{{\gamma_{2} \,\left( {\bar{n}\, + i\,U_{{2_{\begin{subarray}{l} \\ \end{subarray} } }} } \right)^{2} }} + \,\,\left( {\frac{{m\,\alpha_{2} }}{{\gamma_{2} }}} \right)\,\left( {\bar{n}\, + i\,U_{{2_{\begin{subarray}{l} \\ \end{subarray} } }} } \right)^{2} + 1\,} \right]^{{\frac{1}{2}}} , $$
(43)
$$ \zeta_{1} = \left( {\frac{{m\,\alpha_{1} }}{{\gamma_{1} }}} \right)\,\,\left( {\frac{{m\,\alpha_{2} }}{{\gamma_{2} }}} \right)\,\,\alpha_{2} ,\,\,\,\,\,\,\,\zeta_{2} = \,\,\left( {\frac{{m\,\alpha_{1} }}{{\gamma_{1} }}} \right)\,\,\left( {\frac{{m\,\,\alpha_{2} }}{{\gamma_{2} }}} \right)\,\,\alpha_{1} ,\, $$
(44)
$$ \zeta_{3} = \left( {\frac{{m\alpha_{2} }}{{\gamma_{2} }}} \right)\alpha_{2} ,\quad \zeta_{4} = \left( {\frac{{m\alpha_{1} }}{{\gamma_{1} }}} \right)\alpha_{1} , $$
(45)
$$ \zeta_{5} = 2\left\{ \begin{aligned} &\left( {\frac{{m\alpha_{1} }}{{\gamma_{1} }}} \right)\left\{ { - \frac{{m\alpha_{2} }}{2} - \frac{1}{2}\left( {\frac{{m\alpha_{2} }}{2}} \right)^{2} - 1} \right\}\alpha_{2} \hfill \\&\quad - \left( {\frac{{m\alpha_{2} }}{{\gamma_{2} }}} \right)\left\{ { - \frac{{m\alpha_{1} }}{2} + \frac{1}{2}\left( {\frac{{m\alpha_{1} }}{2}} \right)^{2} + 1} \right\}\alpha_{1} + \frac{{m\alpha_{1} }}{{\gamma_{1} }}\frac{{m\alpha_{2} }}{{\gamma_{2} }}A \hfill \\ \end{aligned} \right\}, $$
(46)
$$ \zeta_{6} = 2\left[ {\left\{ { - \frac{{m\alpha_{2} }}{2} - \frac{1}{2}\left( {\frac{{m\alpha_{2} }}{2}} \right)^{2} - 1} \right\}\alpha_{2} - \frac{1}{2}\left\{ {\frac{{m\alpha_{2} }}{{\gamma_{2} }}\frac{{\left( {\gamma_{1} - 1} \right)m\alpha_{1} }}{{\gamma_{1} }}} \right\}\alpha_{1} + \frac{{m\alpha_{2} }}{{\gamma_{2} }}A} \right], $$
(47)
$$ \zeta_{7} = 2\left[ { - \frac{1}{2}\frac{{m\alpha_{1} }}{{\gamma_{1} }}\frac{{\left( {\gamma_{2} - 1} \right)m\alpha_{2} }}{{\gamma_{2} }}\alpha_{2} - \left\{ { - \frac{{m\alpha_{1} }}{2} + \frac{1}{2}\left( {\frac{{m\alpha_{1} }}{2}} \right)^{2} + 1} \right\}\alpha_{1} + \frac{{m\alpha_{1} }}{{\gamma_{1} }}A} \right], $$
(48)
$$ \zeta_{8} = \frac{{\left( {\gamma_{2} - 1} \right)m\alpha_{2} }}{{\gamma_{2} }}\alpha_{2} + \frac{{\left( {\gamma_{1} - 1} \right)m\alpha_{1} }}{{\gamma_{1} }}\alpha_{1} - 2A. $$
(49)

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Hoshoudy, G.A., Awasthi, M.K. Compressibility effects on the Kelvin–Helmholtz and Rayleigh–Taylor instabilities between two immiscible fluids flowing through a porous medium. Eur. Phys. J. Plus 135, 169 (2020). https://doi.org/10.1140/epjp/s13360-020-00160-x

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