Perturbation solutions of weakly compressible Newtonian Poiseuille flows with Navier slip at the wall

Abstract

We consider both the planar and axisymmetric steady, laminar Poiseuille flows of a weakly compressible Newtonian fluid assuming that slip occurs along the wall following Navier’s slip equation and that the density obeys a linear equation of state. A perturbation analysis is performed in terms of the primary flow variables using the dimensionless isothermal compressibility as the perturbation parameter. Solutions up to the second order are derived and compared with available analytical results. The combined effects of slip, compressibility, and inertia are discussed with emphasis on the required pressure drop and the average Darcy friction factor.

Appendix

In the case of compressible, axisymmetric Newtonian Poiseuille flow with slip at the wall, the perturbation solution is as follows:

$$ \begin{array}{rll} u_z \!\left( {r,z} \right)&\!=\!&\frac{\overline B }{4B}\left( {B\!+\!2\!-\!Br^2} \right) \\ && \!+ \varepsilon \!\left[\!{ -\frac{{{{\bar{B}}^2}}}{{32B}} \!\left(\!{B\!+\!2\!-\!Br^2}\right)\! \left({1\!-\!z}\right)} \!+\! \frac{\alpha Re\overline B^4}{73728B^2} \!\left[\!- 2\!\left(\!{B^2 \!+\! 10B \!+\! 24} \right)\!+\!9\!\left(\!{B^2 \!+\! 6B \!+\! 8} \right) \! r^2 \!-\! 9 \!\left(\!{B^2\!+\!8B\!+\!16}\right)r^4 \!+\! 2\left( {B^2\!+\!4B} \right)r^6 \right] \!\right]\\ && +\ \varepsilon^2\left[\!{\frac{3\overline B^3}{512B} \left({B \!+\! 2 \!-\! Br^2} \right)\left( {1\!-\!z} \right)^2} - \frac{\alpha Re{{\bar{B}}^5} }{196608B^4} \right.\\ && \qquad\;\; \times \left[\!B^4 \!+\! 10B^3 \!+\! 72B^2 \!+\! 240B \!+\! 192 \!+\! 6\!\left(\!{B^4 \!+\! 8B^3 \!+\! 12B^2 \!-\! 16B}\right)\!r^2 \!-\! 9 \!\left(\!{B^4 \!+\! 6B^3 \!+\! 8B^2} \right)\!r^4 \!+\! 2\! \left(\!{B^4 \!+\! 4B^3} \right)\!r^6 \right] \\ && \qquad \quad \;\;\times \left({1\!-\!z}\right) +\frac{\alpha^2\overline B^4}{294912B^2}\left[B^2+32B-48-4\left( {7B^2+48B} \right)r^2 +27 \left({B^4+4B^3} \right)r^4\right] +\frac{\alpha^2Re^2\overline B^7}{459848300B^5} \\ &&\qquad \;\;\times \left[43B^5 \!+\! 774B^4 \!+\! 1328B^3 \!-\! 42720B^2 \!-\! 268800B \!-\! 460800 \! -\! 200\!\left(5B^5 \!+\! 80B^4 \!+\! 360B^3 \!-\! 96B^2 \!-\! 4032B \!-\! 6912 \right)r^2 \right.\\ &&\qquad \quad \;\; +100\left(33B^5 \!+\! 462B^4 \!+\! 2112B^3 \!+\! 2736B^2 \!-\! 3456B \!-\! 6912 \right)r^4 \!-\! 1200\left({3B^5 \!+\! 36B^4 \!+\! 148B^3 \!+\! 224B^2 \!+\! 64B} \right)r^6 \\ && \qquad\quad \;\;\left. \left. +1425\left({B^5+10B^4+32B^3+32B^2} \right)r^8-168 \left( {B^5+8B^4+16B^3} \right)r^{10} \right] \!\vphantom{\frac{3\overline B^2}{512B}}\right]+O\left( {\varepsilon^3} \right) \end{array} $$

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$$ u_r \left( r \right)=\varepsilon^2\frac{\alpha Re\overline B^5}{1179648B^2}r\left( {1-r^2} \right) \left[4\left( {B^2+10B+24} \right) -\left( {5B^2+32B+48} \right)r^2+\left( {B^2+4B} \right)r^4 \right]+O\left( {\varepsilon^3} \right) $$

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$$ \begin{array}{rll} p\left( {r,z} \right)&=&\frac{\overline B }{8}\left( {1-z} \right)+\varepsilon \left[-\frac{\overline B^2}{128}\left( {1-z} \right)^2+\frac{\alpha Re\overline B^4}{16384B^3} \left( B^3+8B^2+24B+32 \right)\left( {1-z} \right)+\frac{\alpha^2\overline B^2}{768} \left({1-r^2} \right) \right]\\&& +\ \varepsilon^2\left[ -\frac{\overline B^3}{1024}\left( {1-z} \right)^3-\frac{\alpha Re\overline B ^5}{65536B^3} \left({B^3+8B^2+24B+32} \right)\left( {1-z} \right)^2 -\frac{\alpha^2\overline B^4}{147456B^2} \right.\\ && \qquad\;\; \times \left[ 29B^2+168B+228-9\left( {B^2+4B} \right)r^2 \right]\left( {1-z} \right) +\frac{\alpha^2Re^2\overline B^7}{113246208B^6} \\ && \qquad \;\;\times \left( 2B^6+32B^5+267B^4+1332B^3 +3672B^2+5184B+3456\right) \left({1-z} \right) +\frac{\alpha^3Re\overline B^5}{14155776B^3} \\ && \qquad\;\; \times \left. \left[ 19B^3+202B^2+576B+288-18 \left({3B^3+24B^2+52B+16} \right)r^2+45\left( {B^3+6B^2+8B} \right)r^4\right.\right.\\ && \qquad\qquad \;\!\left.\left. -10\left({B^3+4B^2} \right)r^6 \right] \vphantom{\frac{\overline B^2}{128}}\right]+O\left( {\varepsilon^3} \right) \end{array} $$

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$$ \rho \left( {r,z} \right) = 1 + \varepsilon \frac{{\bar{B}}}{8}\left( {1 - z} \right) + {\varepsilon ^2}\left[ { - \frac{{{{\bar{B}}^2}}}{{128}}{{\left( {1 - z} \right)}^2} + \frac{{\alpha Re{{\bar{B}}^4}}}{{16384{B^3}}}\left( {{B^3} + 8{B^2} + 24B + 32} \right)\left( {1 - z} \right)\frac{{{\alpha ^2}{{\overline \beta }^2}}}{{768}}\left( {1 - {r^2}} \right)} \right] + O\left( {{\varepsilon ^3}} \right) $$

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In the above solution, z* is scaled by the tube length L*, r* by the tube radius R*, the axial velocity by $U^\ast =\dot{{M}}^\ast /\left( {\pi \rho _0^\ast R^{\ast 2}} \right)$, the radial velocity by U ^∗ R ^∗/L ^∗, and the pressure by 8η ^∗ L ^∗ U ^∗/R ^∗ 2. The dimensionless numbers are defined as follows:

$$ \begin{array}{rll} \alpha &\equiv& \frac{R^\ast }{L^\ast },\quad Re \equiv \frac{\rho_0^\ast U^\ast R^\ast }{\eta^\ast },\quad \varepsilon \equiv \frac{8\kappa^\ast \eta^\ast L^\ast U^\ast }{R^{\ast 2}},\\ B&\equiv& \frac{\beta^\ast R^\ast }{\eta^\ast },\quad \overline B \equiv \frac{8B}{B+4} \end{array} $$

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The volumetric flow rate,

$$ Q\left( z \right)\equiv 2\int_0^1 {u_z \left( {r,z} \right)rdr} $$

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is given by

$$ \begin{array}{rll} Q\left( z \right)&\!=\!&1\!-\!\varepsilon \frac{\overline B }{8}\left( {1-z} \right)\\ && +\thinspace\varepsilon^2\!\left[\!\frac{3\overline B^2}{128}\!\left({1\!-\!z}\right)^2\!-\! \frac{\alpha Re\overline B^3}{2048B^2}\!\left({B^2\!+\!4B\!+\!8}\right)\left( {1\!-\!z} \right)\right.\\ && \qquad \left. -\frac{\alpha^2\overline B^3}{9216B}\left({B+3} \right) \right]+O\left( {\varepsilon^3} \right) \end{array} $$

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For ε = 0 the standard fully-developed Poiseuille flow solution with slip at the wall is recovered with

$$ u_z \left( r \right)=\frac{\overline B }{2B}+\frac{\overline B }{4}\left( {1-r^2} \right) $$

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