Abstract
In this paper, a Stokes-Brinkman model is developed to study a transverse flow through a rough curved channel containing a porous medium. The analytical expression of the flow rate, for a given mean pressure drop, is obtained as a function of the parameters specifying the surface roughness, the channel permeability and the channel radius of curvature. The study shows that the transverse flow through the porous curved channel is decreased by the surface roughness. Further to that, we have shown that the surface roughness has greater impact on the rough curved channel flow, when compared to a straight rough channel flow with the same permeability. The Stokes and Darcian flow limits pertaining to the rough curved channel flow have been examined, and it is found that the roughness effect is relatively significant in the Stokes flow limit.
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Appendix
Appendix
The coefficients in Eqs. (14), (17), (18) and (21) are expressed as follows, where \({F}_{\beta y}\) is the derivative of the function \({F}_{\beta }\) with respect to the variable \(y\).
$${a}_{0}=\left(\delta \left(\mathrm{ln}\left(k-1\right)+\mathrm{ln}\left(k+1\right)\right)\left(k+1\right){K}_{1}\left(\delta \left(k+1\right)\right)+{K}_{0}\left(\delta \left(k-1\right)\right)-{K}_{0}\left(\delta \left(k+1\right)\right)\right)\left(k-1\right){I}_{1}\left(\delta \left(k-1\right)\right)-\left(\delta \left(\mathrm{ln}\left(k-1\right)-\mathrm{ln}\left(k+1\right)\right)\left(k-1\right){K}_{1}\left(\delta \left(k-1\right)\right)+{K}_{0}\left(\delta \left(k-1\right)\right)-{K}_{0}\left(\delta \left(k+1\right)\right)\right)\left(k+1\right){I}_{1}\left(\delta \left(k+1\right)\right)+\left(\left(k-1\right){K}_{1}\left(\delta \left(k-1\right)\right)-{K}_{1}\left(\delta \left(k+1\right)\right)(k+1)\right)\left({I}_{0}\left(\delta \left(k-1\right)\right)-{I}_{0}\left(\delta \left(k+1\right)\right)\right),$$
(A1)
$$ \begin{gathered} a_{1} = ((a((k - 1)^{{ - ak}} (k + 1)^{{ - ak}} - (k + 1)^{{ak}} (k - 1)^{{ - ak}} )(k + 1)k_{{ak + 1}} (\delta (k + 1)) \hfill \\ \quad \quad + \;2\alpha ( - K_{{ak}} (\delta (k + 1))(k - 1)^{{ak}} (k + 1)^{{ - ak}} + k_{{ak}} (\delta (k - 1)))) \hfill \\ \quad \quad \times \;(k - 1)I_{{ak + 1}} (\delta (k - 1)) - (k + 1) \times (\delta ((k - 1)^{{ak}} (k + 1)^{{ - ak}} (k - 1)^{{ - ak}} - (k + 1)^{{ - ak}} (k - 1)^{{ak}} ) \times (k - 1)k_{{ak + 1}} (\delta (k - 1)) \hfill \\ \quad \quad + \;2\alpha k(K_{{ak}} (\delta (k - 1))(k + 1)^{{ak}} (k - 1)^{{ - ak}} - K_{{ak}} (\delta (k + 1))))K_{{ak + 1}} \hfill \\ \quad \quad \times \;(\delta (k + 1)) - 2k(((k - 1)^{{ak}} (k + 1)^{{ - ak}} I_{{ak}} (\delta (k + 1)) - I_{{ak}} (\delta (k - 1))) \hfill \\ \quad \quad \times \;(k - 1)K_{{ak + 1}} (\delta (k - 1)) + K_{{ak + 1}} (\delta (k + 1))((k + 1)^{{ak}} (k - 1)^{{ - ak}} I_{{ak}} (\delta (k - 1)) - I_{{ak}} (\delta (k + 1)))(k + 1))\alpha )\delta , \hfill \\ \end{gathered} $$
(A2)
$${a}_{2}=\left(-\left(k-1\right)\left(\delta \left(\mathrm{ln}\left(k-1\right)-\mathrm{ln}\left(k+1\right)\right)\left(k+1\right){K}_{1}\left(\delta \left(k+1\right)\right)+{K}_{0}\left(\delta \left(k-1\right)\right)-\;{K}_{0}\left(\delta \left(k+1\right)\right)\right){I}_{1}\left(\delta \left(k-1\right)\right)+\left(\delta \left(\mathrm{ln}\left(k-1\right)-\mathrm{ln}\left(k+1\right)\right)\left(k-1\right){K}_{1}\left(\delta \left(k-1\right)\right)+\;{K}_{0}\left(\delta \left(k-1\right)\right)-{K}_{0}\left(\delta \left(k+1\right)\right)\right)\left(k+1\right){I}_{1}\left(\delta \left(k+1\right)\right)-\left(\left(k-1\right){K}_{1}\left(\delta \left(k-1\right)\right)-\;{K}_{1}\left(\delta \left(k+1\right)\right)(k+1)\right)\left({I}_{0}\left(\delta \left(k-1\right)\right)-{I}_{0}\left(\delta \left(k+1\right)\right)\right)\delta \right).$$
(A3)
$${a}_{0}{c}_{0}=\left(\delta \left(\mathrm{ln}\left(k-1\right)+\mathrm{ln}\left(k+1\right)\right)\left(k+1\right){K}_{1}\left(\delta \left(k+1\right)\right)+{K}_{0}\left(\delta \left(k-1\right)\right)+{K}_{1}\left(\delta \left(k+1\right)\right)+\;{K}_{0}\left(\delta \left(k+1\right)\right)\right)\left(k-1\right){I}_{1}\left(\delta \left(k-1\right)\right)-\left(\delta \left(\mathrm{ln}\left(k-1\right)+\mathrm{ln}\left(k+1\right)\right)\left(k-1\right){K}_{1}\left(\delta \left(k+1\right)\right)+{K}_{1}\left(\delta \left(k-1\right)\right)+{K}_{1}\left(\delta \left(k+1\right)\right)+{K}_{0}\left(\delta \left(k-1\right)\right)+\;{K}_{1}\left(\delta \left(k+1\right)\right)+{K}_{0}\left(\delta \left(k+1\right)\right)\right)\times \left(k+1\right){I}_{1}\left(\delta \left(k+1\right)\right)+\left({I}_{0}\left(\delta \left(k-1\right)\right)+\;{I}_{0}\left(\delta \left(k+1\right)\right)\right)\left(\left(k-1\right){K}_{1}\left(\delta \left(k-1\right)\right)-{K}_{1}\left(\delta \left(k+1\right)\right)\left(k+1\right)\right),$$
(A4)
$$ a_{0} c_{1} = - 2\delta \left( {k - 1} \right)\left( {K_{1} \left( {\delta \left( {k + 1} \right)} \right)I_{1} \left( {\delta \left( {k - 1} \right)} \right) - I_{1} \left( {\delta \left( {k + 1} \right)} \right)K_{1} \left( {\delta \left( {k - 1} \right)} \right)} \right)\left( {k + 1} \right), $$
(A5)
$$ a_{0} c_{2} = 2\left( {k - 1} \right)K_{1} \left( {\delta \left( {k - 1} \right)} \right) - 2K_{1} \left( {\delta \left( {k + 1} \right)} \right)\left( {k + 1} \right), $$
(A6)
$$ a_{0} c_{3} = - 2\left( {k + 1} \right)I_{1} \left( {\delta \left( {k - 1} \right)} \right) + 2I_{1} \left( {\delta \left( {k + 1} \right)} \right)\left( {k + 1} \right), $$
(A7)
$${a}_{1}{c}_{4}=\delta {\left.{F}_{1y}\right|}_{y=1}\left(k-1\right)\left(k+1\right)\left({K}_{\alpha k}\left(\delta \left(k-1\right)\right){\left(k+1\right)}^{-\alpha k}-{K}_{\alpha k}\left(\delta \left(k+1\right)\right){\left(k-1\right)}^{-\alpha k}\right){I}_{\alpha k+1}\left(\delta \left(k-1\right)\right)-\delta {\left.{F}_{1y}\right|}_{y=-1}\left(k-1\right)\left(k+1\right)\left({K}_{\alpha k}\left(\delta \left(k-1\right)\right){\left(k+1\right)}^{-\alpha k}-{K}_{\alpha k}\left(\delta \left(k+1\right)\right){\left(k-1\right)}^{-\alpha k}\right){I}_{\alpha k+1}\left(\delta \left(k+1\right)\right)-\delta {\left.{F}_{1y}\right|}_{y=1}\left(k-1\right)\left(k+1\right)\left({I}_{\alpha k}\left(\delta \left(k-1\right)\right){\left(k-1\right)}^{-\alpha k}-{I}_{\alpha k}\left(\delta \left(k-1\right)\right){\left(k+1\right)}^{-\alpha k}\right){K}_{\alpha k+1}\left(\delta \left(k-1\right)\right)+\delta {\left.{F}_{1y}\right|}_{y=-1}\left(k-1\right)\left(k+1\right)\left({I}_{\alpha k}\left(\delta \left(k+1\right)\right){\left(k-1\right)}^{-\alpha k}-{I}_{\alpha k}\left(\delta \left(k-1\right)\right){\left(k+1\right)}^{-\alpha k}\right){K}_{\alpha k+1}\left(\delta \left(k+1\right)\right)+\;2k\left(\left(k+1\right){\left(k-1\right)}^{-\alpha k}{\left.{F}_{1y}\right|}_{y=1}-(k-1){\left(k+1\right)}^{-\alpha k}{\left.{F}_{1y}\right|}_{y=1}\right)\times \alpha \left({K}_{\alpha k}\left(\delta \left(k-1\right)\right){I}_{\alpha k}\left(\delta \left(k+1\right)\right)-{K}_{\alpha k}\left(\delta \left(k+1\right)\right){I}_{\alpha k}\left(\delta \left(k-1\right)\right)\right),$$
(A8)
$${a}_{1}{c}_{5}=-\delta \left(\left({\left.{F}_{1y}\right|}_{y=1}\left({K}_{\alpha k}\left(\delta \left(k-1\right)\right){\left(k+1\right)}^{\alpha k}-{K}_{\alpha k}\left(\delta \left(k+1\right)\right){\left(k-1\right)}^{\alpha k}\right){I}_{\alpha k+1}\left(\delta \left(k-1\right)\right)-{\left.{F}_{1y}\right|}_{y=-1}\left({K}_{\alpha k}\left(\delta \left(k-1\right)\right){\left(k+1\right)}^{\alpha k}-{K}_{\alpha k}\left(\delta \left(k+1\right)\right){\left(k-1\right)}^{\alpha k}\right){I}_{\alpha k+1}\left(\delta \left(k+1\right)\right)-\left({I}_{\alpha k}\left(\delta \left(k+1\right)\right){\left(k-1\right)}^{\alpha k}-{I}_{\alpha k}\left(\delta \left(k-1\right)\right){\left(k+1\right)}^{\alpha k}\right)\left({K}_{\alpha k+1}\left(\delta \left(k-1\right)\right){\left.{F}_{1y}\right|}_{y=1}-{K}_{\alpha k+1}\left(\delta \left(k+1\right)\right){\left.{F}_{1y}\right|}_{y=-1}\right)\right)\left(k-1\right)\left(k+1\right)\right),$$
(A9)
$${a}_{1}{c}_{6}=-\delta {\left.{F}_{1y}\right|}_{y=1}\left(k-1\right)\left(k+1\right)\left({\left(k-1\right)}^{\alpha k}{\left(k+1\right)}^{-\alpha k}-{\left(k+1\right)}^{\alpha k}{\left(k-1\right)}^{-\alpha k}\right){K}_{\alpha k+1}\left(\delta \left(k-1\right)\right)+\delta {\left.{F}_{1y}\right|}_{y=-1}\left(k-1\right)\left(k+1\right)\left({\left(k-1\right)}^{\alpha k}{\left(k+1\right)}^{-\alpha k}-{\left(k+1\right)}^{\alpha k}{\left(k-1\right)}^{-\alpha k}\right){K}_{\alpha k+1}\left(\delta \left(k+1\right)\right)-2\left(\left({\left(k+1\right)}^{\alpha k+1}{\left(k-1\right)}^{-\alpha k}{\left.{F}_{1y}\right|}_{y=1}-\left(k-1\right){\left.{F}_{1y}\right|}_{y=-1}\right){K}_{\alpha k}\left(\delta \left(k-1\right)\right)-\left(-{\left(k-1\right)}^{\alpha k+1}{\left(k+1\right)}^{-\alpha k}{\left.{F}_{1y}\right|}_{y=-1}+\left(k+1\right){\left.{F}_{1y}\right|}_{y=1}\right){K}_{\alpha k}\left(\delta \left(k+1\right)\right)\right)\alpha k,$$
(A10)
$${a}_{1}{c}_{7}=-\delta {\left.{F}_{1y}\right|}_{y=1}\left(k-1\right)\left(k+1\right)\left({\left(k-1\right)}^{\alpha k}{\left(k+1\right)}^{-\alpha k}-{\left(k+1\right)}^{\alpha k}{\left(k-1\right)}^{-\alpha k}\right){I}_{\alpha k+1}\left(\delta \left(k-1\right)\right)+\delta {\left.{F}_{1y}\right|}_{y=-1}\left(k-1\right)\left(k+1\right)\left({\left(k-1\right)}^{\alpha k}{\left(k+1\right)}^{-\alpha k}-{\left(k+1\right)}^{\alpha k}{\left(k-1\right)}^{-\alpha k}\right){I}_{\alpha k+1}\left(\delta \left(k+1\right)\right)+2\left(\left({\left(k+1\right)}^{\alpha k+1}{\left(k-1\right)}^{-\alpha k}{\left.{F}_{1y}\right|}_{y=1}-\left(k-1\right){\left.{F}_{1y}\right|}_{y=-1}\right){I}_{\alpha k}\left(\delta \left(k-1\right)\right)-\left(-{\left(k-1\right)}^{\alpha k+1}{\left(k+1\right)}^{-\alpha k}{\left.{F}_{1y}\right|}_{y=-1}+\left(k+1\right){\left.{F}_{1y}\right|}_{y=1}\right){I}_{\alpha k}\left(\delta \left(k+1\right)\right)\right)\alpha k,$$
(A11)
$${a}_{1}{c}_{8}={\left.{F}_{2y}\right|}_{y=-1}\left(k-1\right)\times \left(\delta \left(k+1\right)\left({K}_{\alpha k}\left(\delta \left(k+1\right)\right){\left(k-1\right)}^{-\alpha k}-{K}_{\alpha k}\left(\delta \left(k-1\right)\right){\left(k+1\right)}^{-\alpha k}\right){I}_{\alpha k+1}\left(\delta \left(k+1\right)\right)+\;\delta \left({I}_{\alpha k}\left(\delta \left(k+1\right)\right){\left(k-1\right)}^{-\alpha k}-{K}_{\alpha k}\left(\delta \left(k-1\right)\right){\left(k+1\right)}^{-\alpha k}\right)\left(k+1\right){K}_{\alpha k+1}\left(\delta \left(k+1\right)\right)+\;2\alpha k{\left(k+1\right)}^{-\alpha k}\left({K}_{\alpha k}\left(\delta \left(k+1\right)\right){I}_{\alpha k}\left(\delta \left(k-1\right)\right)-{K}_{\alpha k}\left(\delta \left(k-1\right)\right){I}_{\alpha k}\left(\delta \left(k+1\right)\right)\right)\right),$$
(A12)
$${a}_{1}{c}_{9}=\delta \left(-\left(\left({K}_{\alpha k}\left(\delta \left(k+1\right)\right){\left(k-1\right)}^{\alpha k}-{K}_{\alpha k}\left(\delta \left(k-1\right)\right){\left(k+1\right)}^{\alpha k}\right){I}_{\alpha k+1}\left(\delta \left(k+1\right)\right)+\;{K}_{\alpha k+1}\left(\delta \left(k+1\right)\right)\left({I}_{\alpha k}\left(\delta \left(k+1\right)\right){\left(k-1\right)}^{\alpha k}-{I}_{\alpha k}\left(\delta \left(k-1\right)\right){\left(k+1\right)}^{\alpha k}\right)\right){\left.{F}_{2y}\right|}_{y=-1}\left(k-1\right)\left(k+1\right)\right),$$
(A13)
$${a}_{1}{c}_{10}={\left.{F}_{2y}\right|}_{y=-1}\left(k-1\right)\times \left(\delta \left(k+1\right)\left({\left(k-1\right)}^{\alpha k}{\left(k+1\right)}^{-\alpha k}-{\left(k+1\right)}^{\alpha k}{\left(k-1\right)}^{-\alpha k}\right){K}_{\alpha k+1}\left(\delta \left(k+1\right)\right)-\;2\alpha k\left({\left(k+1\right)}^{-\alpha k}{\left(k-1\right)}^{\alpha k}{K}_{\alpha k}\left(\delta \left(k+1\right)\right)-{K}_{\alpha k}\left(\delta \left(k-1\right)\right)\right)\right),$$
(A14)
$${a}_{1}{c}_{11}={\left.{F}_{2y}\right|}_{y=-1}\left(k-1\right)\times \left(\delta \left(k+1\right)\left({\left(k-1\right)}^{\alpha k}{\left(k+1\right)}^{-\alpha k}-{\left(k+1\right)}^{\alpha k}{\left(k-1\right)}^{-\alpha k}\right){I}_{\alpha k+1}\left(\delta \left(k+1\right)\right)+\;2\alpha k\left({\left(k+1\right)}^{-\alpha k}{\left(k-1\right)}^{\alpha k}{I}_{\alpha k}\left(\delta \left(k+1\right)\right)-{I}_{\alpha k}\left(\delta \left(k-1\right)\right)\right)\right),$$
(A15)
$${a}_{2}{c}_{12}=\left(k-1\right)\delta \left(\delta \left({\left.{F}_{3}\right|}_{y=-1}\mathrm{ln}\left(k+1\right)-{\left.{F}_{3}\right|}_{y=1}\mathrm{ln}\left(k-1\right)\right)\left(k+1\right){K}_{1}\left(\delta \left(k+1\right)\right)+\;\left({\left.{F}_{3y}\right|}_{y=1}\left(k+1\right)\mathrm{ln}\left(k+1\right)-{\left.{F}_{3}\right|}_{y=1}\right){K}_{0}\left(\delta \left(k-1\right)\right)-\;\left({\left.{F}_{3y}\right|}_{y=1}\left(k+1\right)\mathrm{ln}\left(k-1\right)-{\left.{F}_{3}\right|}_{y=-1}\right){K}_{0}\left(\delta \left(k+1\right)\right)\right)\times {I}_{1}\left(\delta \left(k-1\right)\right)-\;\left(\delta \left({\left.{F}_{3}\right|}_{y=-1}\mathrm{ln}\left(k+1\right)-{\left.{F}_{3}\right|}_{y=1}\mathrm{ln}\left(k-1\right)\right)\left(k-1\right){K}_{1}\left(\delta \left(k-1\right)\right)+\;\left({\left.{F}_{3y}\right|}_{y=-1}\left(k-1\right)\mathrm{ln}\left(k+1\right)-{\left.{F}_{3}\right|}_{y=1}\right){K}_{0}\left(\delta \left(k-1\right)\right)-\;\left({\left.{F}_{3}\right|}_{y=-1}\left(k-1\right)\mathrm{ln}\left(k-1\right)-{\left.{F}_{3}\right|}_{y=-1}\right){K}_{0}\left(\delta \left(k+1\right)\right)\right)\times \delta \left(k+1\right){I}_{1}\left(\delta \left(k+1\right)\right)-\;\left(\left(-{\left.{F}_{3y}\right|}_{y=1}\left(k+1\right)\mathrm{ln}\left(k+1\right)+{\left.{F}_{3}\right|}_{y=1}\right){I}_{0}\left(\delta \left(k-1\right)\right)+{I}_{0}\left(\delta \left(k+1\right)\right)\times\;\left({\left.{F}_{3y}\right|}_{y=1}\left(k+1\right)\mathrm{ln}\left(k-1\right)-{\left.{F}_{3}\right|}_{y=-1}\right)\right)\times \left(k-1\right)\delta {K}_{1}\left(\delta \left(k-1\right)\right)+\;\left(\left(-{\left.{F}_{3y}\right|}_{y=-1}\left(k-1\right)\mathrm{ln}\left(k+1\right)+{\left.{F}_{3}\right|}_{y=1}\right){I}_{0}\left(\delta \left(k-1\right)\right)+\left({\left.{F}_{3y}\right|}_{y=-1}\left(k-1\right)\mathrm{ln}\left(k-1\right)-{\left.{F}_{3}\right|}_{y=-1}\right){I}_{0}\left(\delta \left(k-1\right)\right)\right)\times\; \delta \left(k+1\right){K}_{1}\left(\delta \left(k+1\right)\right)+\left({I}_{0}\left(\delta \left(k+1\right)\right){K}_{0}\left(\delta \left(k-1\right)\right)-{K}_{0}\left(\delta \left(k+1\right)\right){I}_{0}\left(\delta \left(k-1\right)\right)\right)\times\; \left(\left({\left.{F}_{3y}\right|}_{y=-1}-{\left.{F}_{3y}\right|}_{y=1}\right)k-{\left.{F}_{3y}\right|}_{y=-1}-{\left.{F}_{3y}\right|}_{y=1}\right),$$
(A16)
$$ a_{2} c_{{13}} = \delta \left( { - (k - 1)(k + 1) \times \left( {\left( {\delta \left( {\left. {F_{3} } \right|_{{y = - 1}} - \left. {F_{3} } \right|_{{y = 1}} } \right)K_{1} \left( {\delta \left( {k + 1} \right)} \right) - \left. {F_{{3y}} } \right|_{{y = 1}} \left( {K_{0} \left( {\delta \left( {k + 1} \right)} \right) - K_{0} \left( {\delta \left( {k - 1} \right)} \right)} \right)} \right)I_{1} \left( {\delta \left( {k - 1} \right)} \right) + \;\left( { - \delta \left( {\left. {F_{3} } \right|_{{y = - 1}} - \left. {F_{3} } \right|_{{y = 1}} } \right)K_{1} \left( {\delta \left( {k + 1} \right)} \right) + \;\left. {F_{{3y}} } \right|_{{y = - 1}} \left( {K_{0} \left( {\delta \left( {k + 1} \right)} \right) - \;K_{0} \left( {\delta \left( {k - 1} \right)} \right)} \right)} \right)I_{1} \left( {\delta \left( {k + 1} \right)} \right) + \left( {I_{0} \left( {\delta \left( {k - 1} \right)} \right) - I_{0} \left( {\delta \left( {k + 1} \right)} \right)} \right)\left( {K_{1} \left( {\delta \left( {k - 1} \right)} \right)\left. {F_{{3y}} } \right|_{{y = 1}} - \left. {F_{{3y}} } \right|_{{y = - 1}} K_{1} \left( {\delta \left( {k - 1} \right)} \right)} \right)} \right)} \right), $$
(A17)
$${a}_{2}{c}_{14}=\delta \left(k-1\right)\left({\left.{F}_{3y}\right|}_{y=1}\left(k+1\right)\mathrm{ln}\left(k-1\right)-{\left.{F}_{3y}\right|}_{y=1}\left(k+1\right)\mathrm{ln}\left(k+1\right)-{\left.{F}_{3}\right|}_{y=-1}+{\left.{F}_{3}\right|}_{y=1}\right){K}_{1}\left(\delta \left(k-1\right)\right)-\left({\left.{F}_{3y}\right|}_{y=-1}\left(k-1\right)\mathrm{ln}\left(k-1\right)-{\left.{F}_{3y}\right|}_{y=-1}\left(k-1\right)\mathrm{ln}\left(k+1\right)-{\left.{F}_{3}\right|}_{y=-1}+{\left.{F}_{3}\right|}_{y=1}\right)\delta \left(k+1\right){K}_{1}\left(\delta \left(k+1\right)\right)+\left(\left({\left.{F}_{3y}\right|}_{y=-1}-{\left.{F}_{3y}\right|}_{y=1}\right)k-{\left.{F}_{3y}\right|}_{y=-1}-{\left.{F}_{3y}\right|}_{y=1}\right)\left({K}_{0}\left(\delta \left(k+1\right)\right)-{K}_{0}\left(\delta \left(k-1\right)\right)\right),$$
(A18)
$${a}_{2}{c}_{15}=\delta \left(k-1\right)\left({\left.{F}_{3y}\right|}_{y=1}\left(k+1\right)\mathrm{ln}\left(k-1\right)-{\left.{F}_{3y}\right|}_{y=1}\left(k+1\right)\mathrm{ln}\left(k+1\right)-{\left.{F}_{3}\right|}_{y=-1}+{\left.{F}_{3}\right|}_{y=1}\right){I}_{1}\left(\delta \left(k-1\right)\right)-\left({\left.{F}_{3y}\right|}_{y=-1}\left(k-1\right)\mathrm{ln}\left(k-1\right)-{\left.{F}_{3y}\right|}_{y=-1}\left(k-1\right)\mathrm{ln}\left(k+1\right)-{\left.{F}_{3}\right|}_{y=-1}+{\left.{F}_{3}\right|}_{y=1}\right)\delta \left(k+1\right){I}_{1}\left(\delta \left(k+1\right)\right)+\left(\left({\left.{F}_{3y}\right|}_{y=-1}-{\left.{F}_{3y}\right|}_{y=1}\right)k-{\left.{F}_{3y}\right|}_{y=-1}-{\left.{F}_{3y}\right|}_{y=1}\right)\left({I}_{0}\left(\delta \left(k+1\right)\right)-{I}_{0}\left(\delta \left(k-1\right)\right)\right).$$
(A19)
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Okechi, N.F. Stokes-Brinkman Flow in a Rough Curved Channel. Transp Porous Med 139, 513–526 (2021). https://doi.org/10.1007/s11242-021-01677-0
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DOI: https://doi.org/10.1007/s11242-021-01677-0