Channel flow of a viscoplastic fluid with the presence of a superhydrophobic wall

Plane Poiseuille flow of a Bingham fluid in channels with a superhydrophobic (SH) groovy wall is studied, via semi-analytical and numerical simulations. The lower wall is the SH groovy wall, where air is trapped inside the grooves, assuming that the liquid/air interface remains flat while attached to the groove edges. At the liquid/air interface, the Bingham fluid slips over the SH wall, for modeling of which the Navier slip law is used. We consider a transverse flow configuration where the groove direction is normal to the flow stream. Perturbing the flow governing equations over the no-slip flow solution, employing the Fourier expansion method and then solving for the linear terms, we obtain a semi-analytical solution, allowing us to solve for the flow velocity profiles for different flow parameters. We also perform complementary numerical simulations, in order to verify the semi-analytical solution results and to gain further insights about the flow dynamics and regimes. We evaluate the effects of the key dimensionless flow parameters, i.e. the Reynolds (R), Bingham (B) and slip (b) numbers, the channel thickness (ell), and the slip area fraction (varphi), on the flow variables (such as the flow velocity and the effective slip length).