Viscoplastic flows in superhydrophobic channels with oriented grooves: From anisotropic slip to secondary flow

In this work, Poiseuille flows of viscoplastic fluids in typically thin channels equipped with a superhydrophobic groovy wall are numerically studied. The orientation of the groove relative to the applied pressure gradient can vary, and this orientation is measured via the groove orientation angle ($\theta$). In particular, longitudinal ($\theta =0$), oblique ($0<\theta <90^{\circ }$), and transverse ($\theta =90^{\circ }$) flow configurations are considered. The Bingham constitutive equation is employed to model the viscoplastic rheology, within the framework of the Papanastasiou regularization method. Assuming that air (gas) fills the groove completely and that the formed liquid/air interface remains flat while pinned at the groove edges, the viscoplastic fluid slippage is modeled on the liquid/air interface using the Navier slip law. Due to the anisotropic slip dynamics for the oblique flow configuration, a secondary flow is generated normal to the direction of the pressure gradient, offering unique flow features. Our work systematically analyzes the effects of the flow parameters, i.e., the groove orientation angle ($\theta$), the Bingham ($B$) and slip ($b$) numbers, the groove periodicity length ($\ell$), and the slip area fraction ($\phi$) on the flow variables of interest, i.e., the main and secondary velocity fields, the unyielded center plug zone, the effective slip length tensor ($\chi$), the secondary flow index ($I_S$), the slip angle difference ($\theta -s$), and the pressure drop ($\Delta P$). It is demonstrated that $\chi$’s shear component, $I_S$, and $\theta -s$ are maximum at intermediate $\theta$, the value of which generally decreases with $B$. In addition, the center plug is unbroken for the longitudinal flow while it breaks with an increase in $\theta$ for sufficiently large $b$.