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Crowdyet al.(2023Phys. Rev. Fluids, vol. 8, 094201), recently showed that liquid suspended in the Cassie state over an asymmetrically spaced periodic array of alternating cold and hot ridges such that the menisci spanning the ridges are of unequal length will be pumped in the direction of the thermocapillary stress along the longer menisci. Their solution, applicable in the Stokes flow limit for a vanishingly small thermal Péclet number, provides the steady-state temperature and velocity fields in a semi-infinite domain above the superhydrophobic surface, including the uniform far-field velocity, i.e. pumping speed, the key engineering parameter. Here, a related problem in a finite domain is considered where, opposing the superhydrophobic surface, a flow of liquid through a microchannel is bounded by a horizontally mobile smooth wall of finite mass subjected to an external load. A key assumption underlying the analysis is that, on a unit area basis, the mass of the liquid is small compared with that of the wall. Thus, as shown, rather than the heat equation and the transient Stokes equations governing the temperature and flow fields, respectively, they are quasi-steady and, as a result, governed by the Laplace and Stokes equations, respectively. Under the further assumption that the ridge period is small compared with the height of the microchannel, these equations are resolved using matched asymptotic expansions which yield solutions with exponentially small asymptotic errors. Consequently, the transient problem of determining the velocity of the smooth wall is reduced to an ordinary differential equation. This approach is used to provide a theoretical demonstration of the conversion of thermal energy to mechanical work via the thermocapillary stresses along the menisci. 

A mechanism for a microfluidic pump that leverages alternating adverse and favorable thermocapillary stresses along menisci in a (periodically) fully developed transverse flow in a microchannel is exemplified. The transverse ridges are the interdigitated teeth of cold and hot (isothermal) “combs” and free surface menisci span the interstitial regions between them. The teeth are asymmetrically positioned so that the widths of adjacent menisci differ. This architecture is essentially that of the theoretical pump proposed by Adjari [Phys. Rev. E 61, R45(R) (2000)] but exploits thermocapillarity rather than electro-osmotic slip to drive unidirectional pumping. A theoretical model of the multiphysics pumping mechanism is given that is solved in closed form. Two explicit formulas for the pumping speed are provided. One is derived from the exact solution to the full problem; the other follows from the reciprocal theorem for Stokes flow combined with an exact solution to a distinct problem resolving apparent slip over superhydrophobic surfaces [D. G. Crowdy, Phys. Fluids 23, 072001 (2011)]. A conceptual design of the pump is also outlined; this involves no moving parts, requires no external driving pressure, and pumps a continuous stream of liquid through a microchannel, as opposed to a series of discrete droplets. Since there is only a periodic component of the pressure field the microchannel could be made arbitrarily long and the menisci, which would be essentially flat, are more robust than for conventional pressure-driven flow. 

Abstract We develop a two-dimensional model for the transient diffusion of gas from the cavities in ridge-type structured surfaces to a quiescent liquid suspended above them in the Cassie state to predict the location of the liquid vapor-interface (meniscus) as a function of time. The transient diffusion equation is numerically solved by a Chebyshev collocation (spectral) method coupled to the Young-Laplace equation and the ideal gas law. We capture the effects of variable meniscus curvature and, subsequently, when applicable, movement of triple contact lines. Results are presented for the evolution of the dissolved gas concentration field in the liquid and, when applicable, the time it takes for a meniscus to depin and that for longevity, i.e., the onset of the Cassie to Wenzel state transition. Two configurations are examined; viz., one where an impermeable membrane pressurizes the liquid above the ridges and one where hydrostatic pressure is considered and the top of the liquid is exposed to non-condensable gas.