Computation of the effective slip of rough hydrophobic surfaces via homogenization

Abstract

Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx)We present a quantitative analysis of the effect of rough hydrophobic surfaces on viscous newtonian flows. We use a model introduced by Ybert and coauthors in Ref. 20, in which the rough surface is replaced by a flat plane with alternating small areas of slip and noslip. We investigate the averaged slip generated at the boundary, depending on the ratio between these areas. This problem reduces to the homogenization of a non-local system, involving the Dirichlet to Neumann map of the Stokes operator, in a domain with small holes. Pondering on the works of Allaire (see Ref. 2, 3) we compute accurate scaling laws of the averaged slip for various types of roughness (riblets, patches). Numerical computations complete and confirm the analysis.

Understanding the fouling/scaling resistance of superhydrophobic/omniphobic membranes in membrane distillation

“…However, significant amounts of saline brine is generated accompanied with organic contaminants; although the absolute volume is small, the energy consumption is high to reach near zero liquid discharge (nZLD), such as electrodialysis (ED) [6,7], mechanical vapor compression (MVC) [8][9][10], or the footprint is large as evaporation pond [10]. Integrated forward osmosis (FO) and membrane distillation (MD) was reported recently for treating shale-gas-flow-back water for reuse purpose [11], although technically viable, cost of FO-MD remains prohibited high.…”

On the Deviation of Computed Permeability Induced by Unresolved Morphological Features of the Pore Space

“…The boundary conditions at the pore interface Σ are split according to the normal and tangential parts of the velocity and ensure, respectively, the impermeability of the interface and the slip along the tangential directions. In literature, two formulations can be found for the slip condition on the tangential velocity components: using either the velocity gradient whose formalism was detailed in [18] and [32], or the Navier condition involving the shear tensor, as considered in (3).…”

“…defining the solid as the domain below the graph of this function (that is to say the point (x, y, z) such that z < φ(x, y)), and the fluid domain above its graph. The usual approach when modeling a roughness or its homogenization, is to set a Navier/Robin boundary condition that quantifies the slip at a given height relative to the top, mean line or bottom of the roughness pattern [32]. This leads to the relation (1) on the tangential velocity introduced previously:…”

Boundary Layer for a Non-Newtonian Flow Over a Rough Surface

“…• The limit wall law is a homogeneous Dirichlet condition. Let us point out that this feature persists even starting from a microscopic pure slip condition, under some non-degeneracy of the roughness: [9,5,6].…”