Linear instability of lid- and pressure-driven flows in channels textured with longitudinal superhydrophobic grooves

Abstract

It is known that an increased flow rate can be achieved in channel flows when smooth walls are replaced by superhydrophobic surfaces. This reduces friction and increases the flux for a given driving force. Applications include thermal management in microelectronics, where a competition between convective and conductive resistance must be accounted for in order to evaluate any advantages of these surfaces. Of particular interest is the hydrodynamic stability of the underlying basic flows, something that has been largely overlooked in the literature, but is of key relevance to applications that typically base design on steady states or apparent-slip models that approximate them. We consider the global stability problem in the case where the longitudinal grooves are periodic in the spanwise direction. The flow is driven along the grooves by either the motion of a smooth upper lid or a constant pressure gradient. In the case of smooth walls, the former problem (plane Couette flow) is linearly stable at all Reynolds numbers whereas the latter (plane Poiseuille flow) becomes unstable above a relatively large Reynolds number. When grooves are present our work shows that additional instabilities arise in both cases, with critical Reynolds numbers small enough to be achievable in applications. Generally, for lid-driven flows one unstable mode is found that becomes neutral as the Reynolds number increases, indicating that the flows are inviscidly stable. For pressure-driven flows, two modes can coexist and exchange stability depending on the channel height and slip fraction. The first mode remains unstable as the Reynolds number increases and corresponds to an unstable mode of the two-dimensional Rayleigh equation, while the second mode becomes neutrally stable at infinite Reynolds numbers. Comparisons of critical Reynolds numbers with the experimental observations for pressure-driven flows of Daniello et al. (Phys. Fluids, vol. 21, issue 8, 2009, p. 085103) are encouraging.

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

“…In the present study, we have assumed a flat liquid/air interface at the SH walls, following the recent works in the relevant literature (Yu, Teo & Khoo 2016;Crowdy 2017a;Hodes et al 2017;Patlazhan & Vagner 2017;Arenas et al 2019;Fairhall, Abderrahaman-Elena & García-Mayoral 2019;Schnitzer & Yariv 2019;Vagner & Patlazhan 2019;Landel et al 2020;Nayak et al 2020;Baier & Hardt 2021;Tomlinson & Papageorgiou 2022). This simplifying assumption has enabled us to tackle the problem of a viscoplastic fluid flow over the groovy SH wall, while addressing the complexities arising due to the yield stress effects, in a systematic way.…”

“…2020; Nayak et al. 2020; Baier & Hardt 2021; Tomlinson & Papageorgiou 2022). This simplifying assumption has enabled us to tackle the problem of a viscoplastic fluid flow over the groovy SH wall, while addressing the complexities arising due to the yield stress effects, in a systematic way.…”

“…2017; Arenas et al. 2019; Tomlinson & Papageorgiou 2022); (ii) the interface is deformed towards the groove or towards the main flow while it is pinned at the groove edges (Crowdy 2017 b ; Game, Hodes & Papageorgiou 2019); (iii) the liquid partially fills the groove and the liquid/air interface is in contact with the side and bottom walls of the groove (and the interface is not anymore pinned to the groove edges) (Giacomello et al. 2012; Papadopoulos et al.…”

“…However, in some conditions, the trapped air escapes and the liquid penetrates into the groove, fills the cavity and forms the Wenzel state (Lee et al 2016;Hardt & McHale 2022). In this context, there are several possible scenarios for the liquid/air interface, summarized as follows: (i) the liquid/air interface is nearly flat and pinned at the edges of the groove (ideal Cassie state) (Hodes et al 2017;Arenas et al 2019;Tomlinson & Papageorgiou 2022); (ii) the interface is deformed towards the groove or towards the main flow while it is pinned at the groove edges (Crowdy 2017b;Game, Hodes & Papageorgiou 2019); (iii) the liquid partially fills the groove and the liquid/air interface is in contact with the side and bottom walls of the groove (and the interface is not anymore pinned to the groove edges) (Giacomello et al 2012;Papadopoulos et al 2013;He et al 2021). The occurrence of these situations is mostly dependent on the flow parameters and the surface properties, including the system pressure, surface tension and the fluid's rheology (Tsai et al 2009;Papadopoulos et al 2013;Annavarapu et al 2019;Rofman et al 2020;He et al 2021).…”

Turbulent transition in a channel with superhydrophobic walls: anisotropic slip and shear misalignment effects

“…In particular, it has been shown in Min & Kim (2005) that slip boundary conditions have a strong influence on the linear growth of modal disturbances, but a very weak influence on the maximum transient energy growth of perturbations at subcritical Reynolds numbers, concluding that slip boundary conditions are not likely to have a significant effect on the transition to turbulence in channel flows. By means of global stability analyses, Tomlinson & Papageorgiou (2022) have shown that in the presence of SH grooves, additional instabilities may arise, with critical Reynolds numbers small enough to be achievable in applications.…”