Longitudinal shear flow in the annular finned channel with slip condition at the external boundary

In the present study, in the Stokes approximation, we have solved the problem of the laminar shear flow of a viscous fluid in an annular channel with a finned external cylindrical surface for the translational motion of the inner cylinder. The solution of the boundaryvalue problem for the longitudinal velocity in the annular sector has been obtained by decomposition of the flow domain with subsequent representation of the velocity fields in the form of series expansions in eigenfunctions of the Laplace equation.
The velocity fields for the corresponding subdomains are:
\begin{equation} w_1(r,\phi)=\left(W-A_0\right)\frac{\ln r}{\ln r_1}+A_0+\sum_{n=1}^{\infty}A_n\cos(t_n\phi)\left[ -r_1^{-2t_n}r^{t_n}+r^{-t_n}\right] \, , \end{equation}

\begin{equation} w_2(r,\phi)=\sum_{n=1}^{\infty}B_n\cos(p_n\phi)\left[ r^{p_n}+r_2^{2p_n}r^{-p_n}\right] \, , \end{equation}
where
\begin{equation} p_n=\frac{\pi\left( 2n-1\right) }{2\phi_0}\, ,\, t_n=\frac{\pi n}{\phi_0}\, , \, n=1,2,3, \cdots \end{equation}
The unknown coefficients $A_n$ , $B_n$ can be obtained from the matching conditions on the subdomain boundary. To obtain an approximate solution, we restrict the number of terms in the series considered by the numbers $N$ and $M$ respectively. This mathematical model gives a good approximation for the longitudinal velocity field in channels with different number of fins and their height. The results can be used to simulate the effect of macroscopic hydrophobicity on a textured or porous boundary. The value of the longitudinal velocity, averaged at the level of the fins edges, can be a good approximation for the slip velocity at the interface of “liquidtextured surface”.