Thermocapillary flow between grooved superhydrophobic surfaces: transverse temperature gradients

Abstract

We consider the thermocapillary motion of a liquid layer which is bounded between two superhydrophobic surfaces, each made up of a periodic array of highly conducting solid slats, with flat bubbles trapped in the grooves between them. Following the recent analysis of the longitudinal problem (Yariv, J. Fluid Mech., vol. 855, 2018, pp. 574–594), we address here the transverse problem, where the macroscopic temperature gradient that drives the flow is applied perpendicular to the grooves, with the goal of calculating the volumetric flux between the two surfaces. We focus upon the situation where the slats separating the grooves are long relative to the groove-array period, for which case the temperature in the solid portions of the superhydrophobic plane is piecewise uniform. This scenario, which was investigated numerically by Baier et al. (Phys. Rev. E, vol. 82 (3), 2010, 037301), allows for a surprising analogy between the harmonic conjugate of the temperature field in the present problem and the unidirectional velocity in a comparable longitudinal pressure-driven flow problem over an interchanged boundary. The main body of the paper is concerned with the limit of deep channels, where the problem reduces to the calculation of the heat transport and flow about a single surface and the associated ‘slip’ velocity at large distance from that surface. Making use of Lorentz’s reciprocity, we obtain that velocity as a simple quadrature, providing the analogue to the expression obtained by Baier et al. (2010) in the comparable longitudinal problem. The rest of the paper is devoted to the diametric limit of shallow channels, which is analysed using a Hele-Shaw approximation, and the singular limit of small solid fractions, where we find a logarithmic scaling of the flux with the solid fraction. The latter two limits do not commute.

Collective viscous propulsion of a two-dimensional flotilla of Marangoni boats

“…Other mechanisms, such as capillary forces due to menisci with non-zero curvature 4 It has been demonstrated how a mixed boundary value problem arising in shear flow over a model of superhydrophobic surfaces 28,29 arises again in a model of a flotilla of Marangoni boats interacting through hydrodynamic effects caused by the diffusion of insoluble surfactant. The same mixed boundary value problem has also recently been found to have a surprising relevance to a temperature problem arising in thermocapillary flow over superhydrophobic surfaces 36 where an external temperature gradient drives the flow. This mixed boundary value problem emerges in other problems too 30,31 ; for example, the author has previously pointed out 37…”

Superhydrophobic annular pipes: a theoretical study

“…The new solutions found here might form the basis of a similar study of the pipe flow set-up which will help understand the kind of heat transfer optimization challenges considered by Lv & Zhang (2016). Another avenue of extension is to a study of thermocapillary effects (Yariv & Crowdy 2019) in annular SH pipes.…”

Thermocapillary stress and meniscus curvature effects on slip lengths in ridged microchannels

“…He also used matched asymptotic expansions to resolve the transverse flow and temperature fields in the small ridge-period-to-channel-height limit. The analysis Effects of thermocapillary stress and curvature on slip 894 A15-5 was extended to temperature gradients imposed transverse to the ridges in Yariv & Crowdy (2019) in several geometric limits (small solid fraction limit, and large-or small-ridge-period-to-channel height limits). Very recently Yariv & Crowdy (2020) extended the original longitudinal flow for θ = 0 • in Baier et al (2010) to the special case of θ = +90 • in the densely packed limit, and is the only such study to consider non-flat menisci.…”

“…The slip lengths were consistent with their numerical simulations, and it was found that imposing temperature gradients of the order of 10 C per cm may pump water through microchannels at mean velocities of the order of several mm per second. This type of thermocapillary-driven flow proposed by Baier et al has been further extended to internal flows for temperature gradients imposed either longitudinally or transverse to the ridges (Yariv 2018;Yariv & Crowdy 2019). For longitudinally imposed temperature gradients, Yariv (2018) developed a simple transformation to convert solutions from the literature for internal pressure-driven flows to those for thermocapillary-stress-driven internal (creeping) flows when the menisci are flat.…”

“…As all previous studies on thermocapillary stress have assumed a flat meniscus (except Yariv & Crowdy (2020) for thermocapillary-driven flow), in this paper we study the coupled e↵ects of thermocapillary stress and meniscus curvature on pressure-driven flow. Whereas (Baier et al 2010) and (Yariv 2018;Yariv & Crowdy 2019, 2020 impose a linear temperature gradient via the solid substrate to generate a flow in that direction, here (and in Hodes et al (2017)) the only condition imposed on the temperature field is a constant heat flux along the length of the ridges. A linear temperature gradient in the streamwise direction then arises naturally from advection due to the pressure gradient, and the fully-developed assumption.…”