Poiseuille flow of a Bingham fluid in a channel with a superhydrophobic groovy wall

Abstract

Plane Poiseuille flow of a Bingham fluid in a channel armed with a superhydrophobic (SH) lower wall is analysed via a semi-analytical model, accompanied by complementary direct numerical simulations (DNS). The SH surface represents a groovy structure with air trapped inside its cavities. Therefore, the fluid adjacent to the wall undergoes stick–slip conditions. The model is developed based on introducing infinitesimal wall-induced perturbations into the motion equations, followed by Fourier series expansions, and solving the resulting equations as a boundary value problem. The Navier slip law accounts for the slip at the liquid/air interface (assuming the Cassie state). The presented analysis is fairly comprehensive, covering the creeping and inertial regimes for thick channels (via the semi-analytical and DNS solutions). The main dimensionless numbers are the Reynolds ( $Re$ ), Bingham ( $Bi$ ) and slip ( $b$ ) numbers, as well as the groove periodicity length ( $\ell$ ) and the slip area fraction ( $\varphi$ ). By increasing $Bi$ , the perturbation and slip velocity fields grow. As $Re$ increases, the perturbation and slip velocity fields become asymmetric. For certain flow parameters, an unyielded plug zone may appear on the SH wall liquid/air interface, while its formation is accelerated by inertial effects. The results classify the regimes of creeping and inertial flows via predicting the onset of the unyielded plug zone formation at the SH wall.

A comprehensive model for viscoplastic flows in channels with a patterned wall: longitudinal, transverse and oblique flows

“…where As demonstrated by Rahmani & Taghavi (2022), with an increase in the slip number, before any plug formation at the liquid/air interface, h only slightly deviates from h 0 .…”

“…In this work, semi-analytical and explicit-form solutions are developed for the Poiseuille flow of Bingham fluids in channels with a patterned wall, considering the limiting cases of longitudinal () and transverse () grooves (stripes), as well as the general case of the oblique () flow configuration. These solutions are developed for the thick channel limit (), where the lower yield surface (located at ) remains flat (Rahmani & Taghavi 2022). The solution can be derived by considering infinitesimal perturbations induced by the patterned wall (with respect to the no-slip flow) in the lower yielded zone ( and ) and solving for the leading-order terms.…”

“…2021). Very recently, some studies have begun to model viscoplastic material flows with SH wall surfaces, for both thick and thin channel limits, as well as creeping and inertial flows, albeit limited only to transverse groove orientations (Rahmani & Taghavi 2022, 2023; Rahmani et al. 2023; Rahmani, Larachi & Taghavi 2024).…”

Modeling of Shear Flows over Superhydrophobic Surfaces: From Newtonian to Non-Newtonian Fluids

“…Fluid rheology can affect the flow dynamics over slippery surfaces. − Unlike Newtonian fluids, the viscosity of non-Newtonian fluids is a function of the magnitude of the flow strain rate.. , Since slippery wall conditions influence the strain rate tensor, the flow viscosity field and, hence, the whole non-Newtonian flow dynamics is affected as well. , More complex slippery conditions, e.g., the superhydrophobicity, cause more complex dynamics for the non-Newtonian flows, ,− i.e., a feature that we will discuss further in this Review.…”

“…In this section, a number of existing studies dealing with non-Newtonian flows over superhydrophobic surfaces are discussed. These studies mostly address the shear-thinning fluids interactions with superhydrophobic surfaces, − ,, while more recently a few studies concern the problem of viscoplastic (yield stress) materials over these surfaces. ,, …”