Rough curved microchannel slip flow

Abstract

This paper models a slip flow through a curved microchannel. The curved walls confining the flow have surface roughness of small amplitude. Through domain perturbation analysis, the analytical expression of the volumetric flow rate is obtained as a function of the Knudsen number of the slip flow, the channel radius of curvature and the parameters characterising the surface roughness. It is found that the surface roughness effect is decreased in the slip regime compared to the continuum case. The study further shows that the volumetric flow rate can be enhanced, depending on roughness parameters. The channel radius of curvature has a significant effect on the entire flow.

Appendices

Appendix A

1.1 Differential equations

Substituting Eq. (15) in Eq. (14), we have the ordinary differential equations with the boundary conditions at ${\mathcal{O}}$ (ε);

$$ \begin{aligned} &\left( {y + k} \right)^{2} \partial_{yy} g_{1} + \left( {y + k} \right)\partial_{y} g_{1} - \left( {\alpha^{2} \left( {y + k} \right)^{2} + 1} \right)g_{1} = 0, \hfill \\ &\left. {g_{1} } \right|_{y = 1} + \left. {\partial_{y} u^{\left( 0 \right)} } \right|_{y = 1} = - {\text{Kn}}\left[ {\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} g_{1} } \right) + \partial_{y} \left( {\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} u^{\left( 0 \right)} } \right)} \right)} \right]_{y = 1} , \hfill \\ &\left. {g_{1} } \right|_{y = - 1} + \cos \left( \varsigma \right)\left. {\partial_{y} u^{\left( 0 \right)} } \right|_{y = - 1} = {\text{Kn}}\left[ \left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} g_{1} } \right) \right.\\ &\quad \left.+ \cos \left( \varsigma \right)\partial_{y} \left( {\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} u^{\left( 0 \right)} } \right)} \right) \right]_{y = - 1} , \hfill \\ \end{aligned} $$

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$$ \left( {y + k} \right)^{2} \partial_{yy} g_{2} + \left( {y + k} \right)\partial_{y} g_{2} - \left( {\alpha^{2} \left( {y + k} \right)^{2} + 1} \right)g_{2} = 0, $$

$$ \left. {g_{2} } \right|_{y = 1} = - {\text{Kn}}\left[ {\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} g_{1} } \right)} \right]_{y = 1} . $$

$$ \begin{aligned}\left. {g_{2} } \right|_{y = - 1} + \sin \left( \varsigma \right)\left. {\partial_{y} u^{\left( 0 \right)} } \right|_{y = - 1} & = {\text{Kn}}\left[ \left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} g_{2} } \right) \right. \\ &\quad \left. + \sin \left( \varsigma \right)\partial_{y} \left( {\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} u^{\left( 0 \right)} } \right)} \right) \right]_{y = - 1} ,\end{aligned} $$

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Substituting Eq. (19) in Eq. (18), we have the ordinary differential equation with the boundary conditions at ${\mathcal{O}}$ ($\varepsilon^{2}$);

$$ \left( {y + k} \right)^{2} \partial_{yy} g_{3} + \left( {y + k} \right)\partial_{y} g_{3} - g_{3} = 0, $$

$$ \begin{aligned}& \left. {g_{3} } \right|_{y = 1} + \frac{1}{2}\left. {\partial_{y} g_{1} } \right|_{y = 1} + \frac{1}{4}\left. {\partial_{yy} u^{\left( 0 \right)} } \right|_{y = 1} \hfill \\& = - {\text{Kn}}\left[ {\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} g_{3} } \right) + \frac{1}{2}\partial_{y} \left( {\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} g_{1} } \right)} \right)} \right. \hfill \\ &\quad+ \frac{1}{4}\partial_{yy} \left( {\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} u^{\left( 0 \right)} } \right)} \right) \hfill \\&\quad \left. { - \alpha^{2} \left( {\frac{1}{2}g_{1} + \frac{1}{4}\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} u^{\left( 0 \right)} } \right)} \right)} \right]_{y = 1} , \hfill \\ \end{aligned} $$

$$ \begin{aligned} & \left. {g_{3} } \right|_{y = - 1} + \frac{1}{2}\cos \left( \varsigma \right)\left. {\partial_{y} g_{1} } \right|_{y = - 1} + \frac{1}{2}\sin \left( \varsigma \right)\left. {\partial_{y} g_{2} } \right|_{y = - 1} + \frac{1}{4}\left. {\partial_{yy} u^{\left( 0 \right)} } \right|_{y = - 1} \hfill \\ & = {\text{Kn}}\left[ {\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} g_{3} } \right) + \frac{1}{2}\cos \left( \varsigma \right)\partial_{y} \left( {\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} g_{1} } \right)} \right)} \right. \hfill \\ & \quad + \frac{1}{2}\sin \left( \varsigma \right)\partial_{y} \left( {\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} g_{2} } \right)} \right) + \frac{1}{4}\partial_{yy} \left( {\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} u^{\left( 0 \right)} } \right)} \right) \hfill \\ & \quad \left. { - \alpha^{2} \left( {\frac{1}{2}\cos \left( \varsigma \right)g_{1} + \frac{1}{2}\sin \left( \varsigma \right)g_{2} + \frac{1}{4}\left( {y + k} \right)\partial_{y} \left( {\left( {y + k} \right)^{ - 1} u^{\left( 0 \right)} } \right)} \right)} \right]_{y = - 1} . \hfill \\ \end{aligned} $$

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Appendix B

2.1 Coefficients

The coefficients for the perturbation solutions are expressed as follows

$$ B_{0} = 8k^{2} \left( {{\text{Kn}} + 1} \right) + 8\left( {6{\text{Kn}} - 1} \right),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad $$

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$$ \begin{aligned} B_{1} & = 4\left( {k^{2} - 1} \right)^{2} \left( { - \left( {\left( {k - 2{\text{Kn}} + 1} \right)K_{1} \left( {\alpha \left( {k + 1} \right)} \right) - \alpha {\text{Kn}}K_{0} \left( {\alpha \left( {k + 1} \right)} \right)\left( {k + 1} \right)} \right)\left( {k + 2{\text{Kn}}} \right.} \right. \hfill \\ &\qquad \left. { - 1} \right)I_{1} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \qquad + \left( {k - 2{\text{Kn}} + 1} \right)\left( {\left( {k + 2{\text{Kn}} - 1} \right)K_{1} \left( {\alpha \left( {k - 1} \right)} \right)} \right. \hfill \\ & \qquad \left. { + \alpha {\text{Kn}}K_{0} \left( {\alpha \left( {k - 1} \right)} \right)\left( {k - 1} \right)} \right)I_{1} \left( {\alpha \left( {k + 1} \right)} \right) \hfill \\ & \qquad + \alpha {\text{Kn}}\left( {I_{0} \left( {\alpha \left( {k + 1} \right)} \right)\left( {k + 1} \right)\left( {k + 2{\text{Kn}} - 1} \right)K_{1} \left( {\alpha \left( {k - 1} \right)} \right)} \right. \hfill \\ & \qquad + \left( {I_{0} \left( {\alpha \left( {k - 1} \right)} \right)\left( {k - 2{\text{Kn}} + 1} \right)K_{1} \left( {\alpha \left( {k + 1} \right)} \right)} \right. \hfill \\ & \qquad - \alpha {\text{Kn}}\left( {I_{0} \left( {\alpha \left( {k - 1} \right)} \right)K_{0} \left( {\alpha \left( {k + 1} \right)} \right) - I_{0} \left( {\alpha \left( {k + 1} \right)} \right)K_{0} \left( {\alpha \left( {k - 1} \right)} \right)} \right)\left( k \right. \hfill \\ & \qquad \left. {\left. {\left. {\left. { + 1} \right)} \right)\left( {k - 1} \right)} \right)} \right), \hfill \\ \end{aligned} $$

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$$ \begin{aligned} B_{2}& = 4\left( {k - 1} \right)^{2} \left( { - \left( {\left( {k - 2{\text{Kn}} + 1} \right)K_{1} \left( {\alpha \left( {k + 1} \right)} \right) - \alpha {\text{Kn}}K_{0} \left( {\alpha \left( {k + 1} \right)} \right)\left( {k + 1} \right)} \right)\left( {k + 2{\text{Kn}}} \right.} \right. \hfill \\ & \qquad \left. { - 1} \right)I_{1} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \qquad + \left( {k - 2{\text{Kn}} + 1} \right)\left( {\left( {k + 2{\text{Kn}} - 1} \right)K_{1} \left( {\alpha \left( {k - 1} \right)} \right) + \alpha {\text{Kn}}K_{0} \left( {\alpha \left( {k - 1} \right)} \right)} \right.\left( k \right. \hfill \\ & \qquad \left. { - 1} \right)I_{1} \left( {\alpha \left( {k + 1} \right)} \right) \hfill \\ & \qquad + \alpha {\text{Kn}}\left( {I_{0} \left( {\alpha \left( {k + 1} \right)} \right)\left( {k + 1} \right)\left( {k + 2{\text{Kn}} - 1} \right)K_{1} \left( {\alpha \left( {k - 1} \right)} \right) + \left( k \right.} \right. \hfill \\ & \qquad \left. { - 1} \right)\left( {I_{0} \left( {\alpha \left( {k - 1} \right)} \right)\left( {k - 2{\text{Kn}} + 1} \right)K_{1} \left( {\alpha \left( {k + 1} \right)} \right)} \right. \hfill \\ & \qquad - \alpha {\text{Kn}}\left( {\left( {I_{0} \left( {\alpha \left( {k - 1} \right)} \right)K_{0} \left( {\alpha \left( {k + 1} \right)} \right) - I_{0} \left( {\alpha \left( {k + 1} \right)} \right)K_{0} \left( {\alpha \left( {k - 1} \right)} \right)} \right)} \right.\left( k \right. \hfill \\ & \qquad \left. {\left. {\left. { + 1} \right)} \right)} \right), \hfill \\ \end{aligned} $$

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$$ B_{3} = 16k\left( {k^{2} - 1} \right)^{2} \left( {\left( {{\text{Kn}} + 1} \right)k^{2} + 3{\text{Kn}} - 1} \right) $$

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$$ \begin{aligned} A_{0} B_{0} & = - \left( {k - 1} \right)^{3} \left( {1 + k - 2{\text{Kn}}} \right)\ln \left( {k - 1} \right) \\ &\qquad + \left( {2{\text{Kn}} + k - 1} \right)\left( {\left( {k + 1} \right)^{3} \ln \left( {k + 1} \right) - 2k\left( {1 + k - 2{\text{Kn}}} \right)} \right), \end{aligned} $$

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$$ A_{1} B_{0} = \left( {\left( {k^{2} - 1} \right)\ln \left( {k - 1} \right) - k^{2} \ln \left( {k + 1} \right) - 2k{\text{Kn}} + \ln \left( {k + 1} \right)} \right)\left( {k^{2} - 1} \right)^{2} , $$

(34)

$$ \begin{aligned} A_{2} B_{1} & = \left( {k - 1} \right)^{2} \left( {k + 2{\text{Kn}} - 1} \right)\left( {2k\left( {k + 1} \right)^{3} \ln \left( {k + 1} \right) + k^{4} + k^{3} \left( {3 - 4A_{0} } \right)} \right. \hfill \\ & \quad + 3k^{2} \left( {1 - 4A_{0} } \right) + k\left( {1 - 12A_{0} + 4A_{1} } \right) - 4A_{0} \hfill \\ & \quad \left. { - 4\left( {4{\text{Kn}} - 1} \right)A_{1} } \right)K_{1} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \quad - \cos \left( \varsigma \right)\left( {2k\left( {k - 1} \right)^{3} \ln \left( {k - 1} \right) + k^{4} - k^{3} \left( {3 + 4A_{0} } \right) + 3k^{2} \left( {1 + 4A_{0} } \right)} \right. \hfill \\ & \quad \left. { - k\left( {1 + 12A_{0} - 4A_{1} } \right) + 4A_{0} + 4\left( {4{\text{Kn}} - 1} \right)A_{1} } \right)\left( {k - 2{\text{Kn}}} \right. \hfill \\ & \quad \left. { + 1} \right)\left( {k + 1} \right)^{2} K_{1} \left( {\alpha \left( {k + 1} \right)} \right) \hfill \\ & \quad + \left( {\left( {k - 1} \right)^{3} } \right.\left( {2k\left( {k + 1} \right)^{3} \ln \left( {k + 1} \right) + k^{4} + k^{3} \left( {3 - 4A_{0} } \right) + 3k^{2} \left( {1 - 4A_{0} } \right)} \right. \hfill \\ & \quad \left. { + k\left( {1 - 12A_{0} + 4A_{1} } \right) - 4A_{0} - 4\left( {4{\text{Kn}} - 1} \right)A_{1} } \right)K_{0} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \quad + \cos \left( \varsigma \right)\left( {2k\left( {k - 1} \right)^{3} \ln \left( {k - 1} \right) + k^{4} - k^{3} \left( {3 + 4A_{0} } \right) + 3k^{2} \left( {1 + 4A_{0} } \right)} \right. \hfill \\ & \quad \left. { - k\left( {1 + 12A_{0} - 4A_{1} } \right) + 4A_{0} + 4\left( {4{\text{Kn}} - 1} \right)A_{1} } \right)\left. {\left( {k + 1} \right)^{3} K_{0} \left( {\alpha \left( {k + 1} \right)} \right)} \right)\alpha {\text{Kn}}, \hfill \\ \end{aligned} $$

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$$ \begin{aligned} A_{3} B_{1} & = - \left( {k - 1} \right)^{2} \left( {k + 2{\text{Kn}} - 1} \right)\left( {2k\left( {k + 1} \right)^{3} \ln \left( {k + 1} \right) + k^{4} + k^{3} \left( {3 - 4A_{0} } \right)} \right. \hfill \\ & \quad + 3k^{2} \left( {1 - 4A_{0} } \right) + k\left( {1 - 12A_{0} + 4A_{1} } \right) - 4A_{0} \hfill \\ & \quad \left. { - 4\left( {4{\text{Kn}} - 1} \right)A_{1} } \right)I_{1} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \quad + \cos \left( \varsigma \right)\left( {2k\left( {k - 1} \right)^{3} \ln \left( {k - 1} \right) + k^{4} - k^{3} \left( {3 + 4A_{0} } \right) + 3k^{2} \left( {1 + 4A_{0} } \right)} \right. \hfill \\ & \quad \left. { - k\left( {1 + 12A_{0} - 4A_{1} } \right) + 4A_{0} + 4\left( {4{\text{Kn}} - 1} \right)A_{1} } \right)\left( {k - 2{\text{Kn}}} \right. \hfill \\ & \quad \left. { + 1} \right)\left( {k + 1} \right)^{2} I_{1} \left( {\alpha \left( {k + 1} \right)} \right) \hfill \\ & \quad + \alpha {\text{Kn}}\left( {\left( {k - 1} \right)^{3} } \right.\left( {2k\left( {k + 1} \right)^{3} \ln \left( {k + 1} \right) + k^{4} + k^{3} \left( {3 - 4A_{0} } \right)} \right. \hfill \\ & \quad + 3k^{2} \left( {1 - 4A_{0} } \right) + k\left( {1 - 12A_{0} + 4A_{1} } \right) - 4A_{0} \hfill \\ & \quad \left. { - 4\left( {4{\text{Kn}} - 1} \right)A_{1} } \right)I_{0} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \quad + \cos \left( \varsigma \right)\left( {2k\left( {k - 1} \right)^{3} \ln \left( {k - 1} \right) + k^{4} - k^{3} \left( {3 + 4A_{0} } \right) + 3k^{2} \left( {1 + 4A_{0} } \right)} \right. \hfill \\ & \quad \left. {\left. { - k\left( {1 + 12A_{0} - 4A_{1} } \right) + 4A_{0} + 4\left( {4{\text{Kn}} - 1} \right)A_{1} } \right)\left( {k + 1} \right)^{3} I_{0} \left( {\alpha \left( {k + 1} \right)} \right)} \right), \hfill \\ \end{aligned} $$

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$$ \begin{aligned} & A_{4} B_{2} = \left( 2k\left( {k - 1} \right)^{3} { \ln }\left( {k - 1} \right) + k^{4} - k^{3} \left( {3 + 4A_{0} } \right) + 3k^{2} \left( {1 + 4A_{0} } \right) \right. \\ &\qquad\qquad \quad\left. - k\left( {1 + 12A_{0} - 4A_{1} } \right) + 16{\text{Kn}}A_{1} + 4\left( {A_{0} - A_{1} } \right) \right) \\ & \qquad\quad\qquad\left( { - \left( {k - 2{\text{Kn}} + 1} \right)K_{1} \left( {\alpha \left( {k + 1} \right)} \right) + \alpha {\text{Kn}}\left( {k + 1} \right)K_{0} \left( {\alpha \left( {k + 1} \right)} \right)} \right){ \sin }\left( \varsigma \right), \\ \end{aligned} $$

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$$ \begin{aligned} & A_{5} B_{2} = \left( 2k\left( {k - 1} \right)^{3} { \ln }\left( {k - 1} \right) + k^{4} - k^{3} \left( {3 + 4A_{0} } \right) + 3k^{2} \left( {1 + 4A_{0} } \right) \right. \\ &\qquad\qquad \quad\left. - k\left( {1 + 12A_{0} - 4A_{1} } \right) + 16{\text{Kn}}A_{1} + 4\left( {A_{0} - A_{1} } \right) \right) \\ & \qquad\quad\qquad\left( {\left( {k - 2{\text{Kn}} + 1} \right)I_{1} \left( {\alpha \left( {k + 1} \right)} \right) + \alpha {\text{Kn}}\left( {k + 1} \right)I_{0} \left( {\alpha \left( {k + 1} \right)} \right)} \right){ \sin }\left( \varsigma \right), \\ \end{aligned} $$

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$$ \begin{aligned} A_{6} B_{3} & = - 2\alpha \left( {k - 1} \right)^{3} \left( {k + 1} \right)^{5} \left( {k + 2{\text{Kn}} - 1} \right)\left( {A_{2} { \cos }\left( \varsigma \right) + A_{4} \sin \left( \varsigma \right)} \right)I_{0} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \quad + 2\alpha \left( {k - 1} \right)^{3} \left( {k + 1} \right)^{5} \left( {k + 2{\text{Kn}} - 1} \right)\left( {A_{3} { \cos }\left( \varsigma \right) + A_{5} \sin \left( \varsigma \right)} \right)K_{0} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \quad + 2\left( {k - 1} \right)^{2} \left( {k + 1} \right)^{5} \left( {k + 4{\text{Kn}} - 1} \right)\left( {A_{2} { \cos }\left( \varsigma \right) + A_{4} \sin \left( \varsigma \right)} \right)I_{1} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \quad + 2\left( {k - 1} \right)^{2} \left( {k + 1} \right)^{5} \left( {k + 4{\text{Kn}} - 1} \right)\left( {A_{3} { \cos }\left( \varsigma \right) + A_{5} \sin \left( \varsigma \right)} \right)K_{1} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \quad + 2\alpha A_{2} \left( {k + 1} \right)^{3} \left( {k - 1} \right)^{5} \left( {k - 2{\text{Kn}} + 1} \right)I_{0} \left( {\alpha \left( {k + 1} \right)} \right) \hfill \\ & \quad - 2\alpha A_{3} \left( {k + 1} \right)^{3} \left( {k - 1} \right)^{5} \left( {k - 2{\text{Kn}} + 1} \right)K_{0} \left( {\alpha \left( {k + 1} \right)} \right) \hfill \\ & \quad - 2A_{2} \left( {k + 1} \right)^{2} \left( {k - 1} \right)^{5} \left( {k - 4{\text{Kn}} + 1} \right)I_{1} \left( {\alpha \left( {k + 1} \right)} \right) \hfill \\ & \quad - 2A_{3} \left( {k + 1} \right)^{2} \left( {k - 1} \right)^{5} \left( {k - 4{\text{Kn}} + 1} \right)K_{1} \left( {\alpha \left( {k + 1} \right)} \right) \hfill \\ & \quad + k\left( {2\alpha^{2} k^{9} {\text{Kn}} + 2k^{7} \left( {1 - 2\alpha^{2} {\text{Kn}}} \right) + 4\alpha^{2} k^{6} {\text{Kn}}A_{1} - 6k^{5} \left( {1 - \alpha^{2} {\text{Kn}}} \right)} \right. \hfill \\ & \quad + 4k^{4} A_{1} \left( {\left( {\alpha^{2} - 6} \right){\text{Kn}} - 4} \right) + 2k^{3} \left( {3 - 2\alpha^{2} {\text{Kn}}} \right) - 20k^{2} {\text{Kn}}A_{1} \left( {\alpha^{2} + 12} \right) \hfill \\ & \quad \left. { - k\left( {2 - \alpha^{2} {\text{Kn}}} \right) + \left( {16 + 12\left( {\alpha^{2} - 10} \right){\text{Kn}}} \right)A_{1} } \right), \hfill \\ \end{aligned} $$

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$$ \begin{aligned} A_{7} B_{3} & = 2\alpha \left( {k + 1} \right)^{2} \left( {k - 1} \right)^{3} \left( {k + 2{\text{Kn}} - 1} \right)\left( {k - 2{\text{Kn}} + 1} \right)\left( {A_{2} { \cos }\left( \varsigma \right) + A_{4} { \sin }\left( \varsigma \right)} \right)I_{0} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \quad - 2\alpha \left( {k + 1} \right)^{2} \left( {k - 1} \right)^{3} \left( {k + 2{\text{Kn}} - 1} \right)\left( {k - 2{\text{Kn}} + 1} \right)\left( {A_{3} { \cos }\left( \varsigma \right) + A_{5} \sin \left( \varsigma \right)} \right)K_{0} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \quad - 2\left( {k + 1} \right)^{2} \left( {k - 1} \right)^{2} \left( {k + 4{\text{Kn}} - 1} \right)\left( {k - 2{\text{Kn}} + 1} \right)\left( {A_{2} { \cos }\left( \varsigma \right) + A_{4} \sin \left( \varsigma \right)} \right)I_{1} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \quad - 2\left( {k + 1} \right)^{2} \left( {k - 1} \right)^{2} \left( {k + 4{\text{Kn}} - 1} \right)\left( {k - 2{\text{Kn}} + 1} \right)\left( {A_{3} { \cos }\left( \varsigma \right) + A_{5} \sin \left( \varsigma \right)} \right)K_{1} \left( {\alpha \left( {k - 1} \right)} \right) \hfill \\ & \quad - 2\alpha A_{2} \left( {k - 1} \right)^{2} \left( {k + 1} \right)^{3} \left( {k + 2{\text{Kn}} - 1} \right)\left( {k - 2{\text{Kn}} + 1} \right)I_{0} \left( {\alpha \left( {k + 1} \right)} \right) \hfill \\ & \quad + 2\alpha A_{3} \left( {k - 1} \right)^{2} \left( {k + 1} \right)^{3} \left( {k + 2{\text{Kn}} - 1} \right)\left( {k - 2{\text{Kn}} + 1} \right)K_{0} \left( {\alpha \left( {k + 1} \right)} \right) \hfill \\ & \quad + 2A_{2} \left( {k - 1} \right)^{2} \left( {k + 1} \right)^{2} \left( {k + 2{\text{Kn}} - 1} \right)\left( {k - 4{\text{Kn}} + 1} \right)I_{1} \left( {\alpha \left( {k + 1} \right)} \right) \hfill \\ & \quad + 2A_{3} \left( {k - 1} \right)^{2} \left( {k + 1} \right)^{2} \left( {k + 2{\text{Kn}} - 1} \right)\left( {k - 4{\text{Kn}} + 1} \right)K_{1} \left( {\alpha \left( {k + 1} \right)} \right) \hfill \\ & \quad - k\left( {\alpha^{2} k^{7} {\text{Kn}} + k^{5} {\text{Kn}}\left( {\alpha^{2} \left( {4{\text{Kn}} - 3} \right) - 2} \right) + 4\alpha^{2} k^{4} {\text{Kn}}A_{1} } \right. \hfill \\ & \quad + k^{3} {\text{Kn}}\left( {4 + \alpha^{2} \left( {3 - 8{\text{Kn}}} \right)} \right) - 8k^{2} A_{1} \left( {1 + \left( {\alpha^{2} + 2} \right){\text{Kn}}} \right) \hfill \\ & \quad \left. { + k{\text{Kn}}\left( {\alpha^{2} \left( {4{\text{Kn}} - 1} \right) - 2} \right) + 96{\text{Kn}}^{2} A_{1} + 4{\text{Kn}}\left( {\alpha^{2} - 20} \right)A_{1} + 8A_{1} } \right). \hfill \\ \end{aligned} $$

(40)