Darcy–Brinkman Micropolar Fluid Flow through Corrugated Micro-Tube with Stationary Random Model

Abstract

A boundary perturbation method is presented to investigate the modified micropolar Brinkman model. A slightly micro-corrugated cylindrical tube is filled with a porous medium in which a micropolar fluid flows through its pores. Stationary random model is used to mimic the surface roughness of the tube. The amplitude of the corrugations, which are perpendicular to the flow, is assumed to be small compared to the mean tube radius. The obtained solution is used for the evaluation of the influence of the corrugations on pressure gradient and the flow rate. The corrugation function depending on wavelength, permeability of the porous medium and micropolarity parameter is constructed. The limiting cases of Stokesian and Darcian micropolar fluid flows are investigated.

APPENDIX

$${{\gamma }_{1}} = - {{\psi }_{{0,rr}}}(1) = \frac{{2{{\alpha }^{2}}}}{{{{\Delta }_{0}}}}\left( {\alpha _{2}^{2} - \alpha _{1}^{2}} \right){{\alpha }_{3}},$$

( (A.1)

$$\begin{gathered} {{\gamma }_{2}} = - \psi _{{0,rrr}}^{{(2)}}(1) + 2\psi _{{0,rr}}^{{(2)}}(1) - (2 - {{\alpha }^{2}})\psi _{{0,r}}^{{(2)}}(1) \\ - \,\,{{\alpha }^{2}}\psi _{0}^{{\left( 2 \right)}}\left( 1 \right) = \frac{2}{{{{\Delta }_{0}}}}\left( {\alpha _{1}^{2} - {{\alpha }^{2}}} \right) \\ \times \,\,\left( {\alpha _{2}^{2} - {{\alpha }^{2}}} \right)\left( {{{\alpha }_{1}}{{\alpha }_{4}} - {{\alpha }_{2}}{{\alpha }_{5}}} \right), \\ \end{gathered} $$

(A.2)

$$\begin{gathered} {{\gamma }_{3}} = - \left( {\frac{1}{2}{{\psi }_{{0,rrr}}}(1) + {{\phi }_{{1,rr}}}(1)} \right) = \frac{1}{{{{\Delta }_{1}}}}\left[ {\left( {\beta _{2}^{{}}{{\beta }_{5}} - {{\beta }_{1}}{{\beta }_{4}}} \right){{\gamma }_{2}}} \right. \\ \left. { + \,\,\left( {\alpha _{1}^{2} - \alpha _{2}^{2}} \right){{\beta }_{3}}{{\gamma }_{1}}} \right]{{\alpha }^{2}}{{I}_{1}}\left( \lambda \right) - \left( {{{\gamma }_{1}} + {{\gamma }_{2}}} \right) + \frac{1}{{{{\Delta }_{0}}}} \\ \times \,\,\left[ {\alpha _{1}^{3}\left( {\alpha _{2}^{2} - {{\alpha }^{2}}} \right){{\alpha }_{4}} - \alpha _{2}^{3}\left( {\alpha _{1}^{2} - {{\alpha }^{2}}} \right){{\alpha }_{5}}} \right. \\ \left. { - \,\,{{\alpha }^{2}}\left( {\alpha _{1}^{2} - \alpha _{2}^{2}} \right){{\alpha }_{3}}} \right], \\ \end{gathered} $$

(A.3)

$$\begin{gathered} {{\gamma }_{4}} = \frac{1}{2}\left[ { - \psi _{{0,rrrr}}^{{(2)}}(1) + 3\psi _{{0,rrr}}^{{(2)}}(1) - \left( {6 - {{\alpha }^{2}}} \right)\psi _{{0,rr}}^{{(2)}}(1)} \right. \\ \left. { + \,\,2\left( {3 - {{\alpha }^{2}}} \right)\psi _{{0,r}}^{{(2)}}(1) + 2{{\alpha }^{2}}\psi _{0}^{{\left( 2 \right)}}(1)} \right] - \phi _{{1,rrr}}^{{(2)}}(1) \\ + \,\,2\phi _{{1,rr}}^{{(2)}}(1) - \left[ {2 - \left( {{{\lambda }^{2}} + {{\alpha }^{2}}} \right)} \right]\phi _{{1,r}}^{{(2)}}(1) - \left( {{{\lambda }^{2}} + {{\alpha }^{2}}} \right) \\ \times \,\,\phi _{1}^{{\left( 2 \right)}}(1) = - \frac{1}{{{{\Delta }_{0}}}}\left( {\alpha _{1}^{2} - {{\alpha }^{2}}} \right)\left( {\alpha _{2}^{2} - {{\alpha }^{2}}} \right) \\ \times \,\,\left[ {\left( {\alpha _{2}^{2} - \alpha _{1}^{2}} \right){{\alpha }_{3}} + {{\alpha }_{1}}{{\alpha }_{4}} - {{\alpha }_{2}}{{\alpha }_{5}}} \right] \\ + \,\,\frac{{{{\gamma }_{1}}}}{{{{\Delta }_{1}}}}\left( {\alpha _{1}^{2} - {{\alpha }^{2}}} \right)\left( {\alpha _{2}^{2} - {{\alpha }^{2}}} \right){{I}_{1}}(\lambda )\left( {{{\beta }_{2}}{{\beta }_{5}} - {{\beta }_{1}}{{\beta }_{4}}} \right) + \frac{{{{\gamma }_{2}}}}{{{{\Delta }_{1}}}} \\ \times \,\,\left[ {\left( {\alpha _{2}^{2} - {{\alpha }^{2}}} \right)\left( {{{\beta }_{1}}{{I}_{1}}(\lambda )\left( {{{\beta }_{4}} - {{\beta }_{2}}{{\beta }_{6}}} \right) + {{\beta }_{2}}{{\beta }_{5}}\lambda {{I}_{0}}\left( \lambda \right)} \right)} \right. \\ - \,\,\left( {\alpha _{1}^{2} - {{\alpha }^{2}}} \right)\left( {{{\beta }_{2}}{{I}_{1}}\left( \lambda \right)\left( {{{\beta }_{5}} - {{\beta }_{1}}{{\beta }_{6}}} \right) + {{\beta }_{1}}{{\beta }_{4}}\lambda {{I}_{0}}\left( \lambda \right)} \right), \\ \left. { - \,\,\left( {\alpha _{2}^{2} - \alpha _{1}^{2}} \right)\lambda {{I}_{0}}\left( \lambda \right){{\beta }_{3}}} \right],\,\,{{\beta }_{6}} = {{I}_{0}}\left( {{{\beta }_{1}}} \right){{I}_{0}}({{\beta }_{2}}), \\ \end{gathered} $$

(A.4)

$$\begin{gathered} \gamma _{3}^{*} = \frac{1}{{{{\Delta }_{0}}}}\left[ {\alpha _{1}^{3}\left( {\alpha _{2}^{2} - {{\alpha }^{2}}} \right){{\alpha }_{4}} - \alpha _{2}^{3}\left( {\alpha _{1}^{2} - {{\alpha }^{2}}} \right){{\alpha }_{3}}} \right. \\ \left. { - \,\,{{\alpha }^{2}}\left( {\alpha _{1}^{2} - \alpha _{2}^{2}} \right){{\alpha }_{3}}} \right] - \frac{1}{{{{\Delta }_{0}}}}\left[ {\left( {\alpha _{2}^{2} - \alpha _{1}^{2}} \right)\left( {{{\alpha }^{2}}{{\gamma }_{1}} - 2{{\gamma }_{2}}} \right)} \right. \\ \left. { \times \,\,I_{1}^{2}({{\alpha }_{1}}) + {{\alpha }_{1}}{{\alpha }_{2}}{{\gamma }_{2}}\left( {{{\alpha }_{2}}{{\alpha }_{4}} - {{\alpha }_{1}}{{\alpha }_{5}}} \right)} \right], \\ \end{gathered} $$

(A.5)

$$\begin{gathered} \gamma _{4}^{*} = \frac{1}{{{{\Delta }_{0}}}}\left( {\alpha _{2}^{2} - {{\alpha }^{2}}} \right)\left( {\alpha _{1}^{2} - {{\alpha }^{2}}} \right) \\ \times \,\,\left( {{{\alpha }_{1}}{{\alpha }_{4}} - {{\alpha }_{2}}{{\alpha }_{5}} - \left( {\alpha _{2}^{2} - \alpha _{1}^{2}} \right){{\alpha }_{3}}} \right) \\ - \,\,\frac{{{{\gamma }_{1}}}}{{{{\Delta }_{0}}}}\left( {\alpha _{1}^{2} - {{\alpha }^{2}}} \right)\left( {\alpha _{2}^{2} - {{\alpha }^{2}}} \right)\left( {{{\alpha }_{1}}\,{{\alpha }_{4}} - {{\alpha }_{2}}{{\alpha }_{5}}} \right) \\ - \,\,\frac{{{{\gamma }_{2}}}}{{{{\Delta }_{0}}}}\left[ {{{\alpha }_{1}}\left( {{{\alpha }_{2}}\left( {\alpha _{2}^{2} - \alpha _{1}^{2}} \right){{I}_{0}}({{\alpha }_{2}}){{I}_{0}}({{\alpha }_{1}})} \right.} \right. \\ \left. {\left. { + \,\,2\left( {\alpha _{1}^{2} - {{\alpha }^{2}}} \right){{\alpha }_{4}}} \right) - 2{{\alpha }_{2}}\left( {\alpha _{2}^{2} - {{\alpha }^{2}}} \right){{\alpha }_{5}}} \right]. \\ \end{gathered} $$

(A.6)

$$\begin{gathered} \gamma _{3}^{'} = \frac{{\gamma _{1}^{'}}}{{\Delta _{1}^{'}}}\left[ {\lambda \left( {2\beta {\kern 1pt} '\beta _{0}^{'} + \left( {2{{\ell }^{2}} - 3h} \right)\beta _{1}^{'}} \right)\lambda _{1}^{2} + h\lambda \lambda _{0}^{2}\beta _{1}^{'}} \right. \\ \left. { - \,\,2\left( {h + {{\lambda }^{2}}} \right){{\lambda }_{1}}{{\lambda }_{0}}\beta _{1}^{'}} \right] + \frac{{\gamma _{2}^{'}}}{{\Delta _{1}^{'}}}\left[ {\lambda \left( {2\beta {\kern 1pt} '\beta _{0}^{'} - {{\ell }^{2}}\beta _{1}^{'}} \right)\lambda _{1}^{2}} \right. \\ \left. { + \,\,\left( {{{\ell }^{2}}\lambda {{\lambda }_{0}} - 2\beta {\kern 1pt} {{'}^{2}}{{\lambda }_{1}}} \right){{\lambda }_{0}}\beta _{1}^{'}} \right] \\ - \,\,\frac{4}{{\Delta _{0}^{'}}}\left[ {(3h - {{\ell }^{2}}){{I}_{1}}(\ell ) - {{\ell }^{3}}{{I}_{0}}(\ell )} \right], \\ \end{gathered} $$

(A.7)

$$\begin{gathered} \gamma _{4}^{'} = \frac{1}{{\Delta _{1}^{'}}}\gamma _{2}^{'}\left[ \begin{gathered} 2\left[ {\left( {{{\lambda }^{2}} + h} \right)\beta _{1}^{'} + \left( {{{\lambda }^{2}} - h} \right)\beta {\kern 1pt} '\beta _{0}^{'}} \right]{{\lambda }_{1}}{{\lambda }_{0}} \hfill \\ + \lambda \left[ {h\beta _{1}^{'} - \left( {2 + h} \right)\beta {\kern 1pt} '\beta _{0}^{'}} \right]\lambda _{1}^{2} \hfill \\ + \lambda \left[ {\beta {\kern 1pt} 'h\beta _{0}^{'} - \left( {2{{\lambda }^{2}} + h} \right)\beta _{1}^{'}} \right]\lambda _{0}^{2} \hfill \\ \end{gathered} \right] \\ + \,\,\frac{1}{{\Delta _{1}^{'}}}\gamma _{1}^{'}2\lambda h{{I}_{1}}(\lambda )\left[ {\beta {{\lambda }_{1}}\beta _{0}^{'} - \lambda {{\lambda }_{0}}\beta _{1}^{'}} \right] \\ + \,\,\frac{{4h\ell }}{{\Delta _{0}^{'}}}\left[ {\ell {{I}_{1}}(\ell ) - {{I}_{2}}(\ell )} \right], \\ \end{gathered} $$

(A.8)

$$\begin{gathered} \gamma _{1}^{'} = - \frac{8}{{\Delta _{0}^{'}}}\left( {h - {{\ell }^{2}}} \right){{I}_{1}}(\ell ),\,\,\,\gamma _{2}^{'} = \frac{{8\ell h}}{{\Delta _{0}^{'}}}{{I}_{2}}(\ell ), \\ \Delta _{0}^{'} = 4\ell {{I}_{0}}(\ell ) - \left( {8 + h} \right){{I}_{1}}(\ell ), \\ \end{gathered} $$

(A.9)

$$\begin{gathered} \Delta _{1}^{'} = 2\left( {{{\lambda }^{2}} + h} \right){{\lambda }_{1}}{{\lambda }_{0}}\beta _{1}^{'} \\ + \,\,\lambda \left( {h\beta _{1}^{'} - 2\beta {\kern 1pt} '\beta _{0}^{'}} \right)\lambda _{1}^{2} - \lambda h\beta _{1}^{'}\lambda _{0}^{2}, \\ \end{gathered} $$

(A.10)

$$\begin{gathered} \beta {\kern 1pt} ' = \sqrt {{{\lambda }^{2}} + {{\ell }^{2}}} ,\,\,\,{{\lambda }_{1}} = {{I}_{1}}(\lambda ),\,\,\,{{\lambda }_{0}} = {{I}_{0}}(\lambda ), \\ \beta _{0}^{'} = {{I}_{0}}(\beta {\kern 1pt} '),\,\,\,\beta _{1}^{'} = {{I}_{1}}(\beta {\kern 1pt} '). \\ \end{gathered} $$

(A.11)