We investigate forced convection in a parallel plate-geometry microchannel with superhydrophobic walls consisting of a periodic array of ridges aligned parallel to the direction of a Poiseuille flow. In the de-wetted (Cassie) state, the liquid contacts the channel walls only at the tips of the ridges, where we apply a constant heat flux boundary condition. The subsequent hydrodynamic and thermal problems within the liquid are then analysed accounting for curvature of the liquid-gas interface (meniscus) using boundary perturbation, assuming a small deflection from flat. The effects of this surface deformation on both the effective hydrodynamic slip length and the Nusselt number are computed analytically in the form of eigenfunction expansions, reducing the problem to a set of dual series equations for the expansion coefficients which must, in general, be solved numerically. The Nusselt number quantifies the convective heat transfer, the results for which are completely captured in a single figure, presented as a function of channel geometry at each order in the perturbation. Asymptotic solutions for channel heights large compared to ridge period are compared to numerical solutions of the dual series. The asymptotic slip length expressions are shown to consist of only two terms, with all other terms exponentially small. As a result these expressions are accurate even for heights as low as half the ridge period, and hence useful for engineering applications.

[1] Collisionless magnetic reconnection requires the violation of ideal MHD by various kinetic-scale effects whose relative importance is uncertain. Recent research has highlighted the potential importance of wave-particle interactions by showing that Vlasov simulations of unstable ion-acoustic waves predict an anomalous resistivity that can be at least an order of magnitude higher than a popular analytical quasi-linear estimate. Here, we investigate the nonlinear evolution of the ion-acoustic instability and its resulting anomalous resistivity by examining the properties of a statistical ensemble of Vlasov simulations. The simulations differ in their initial electric noise field but are otherwise identical with a Maxwellian electron-ion plasma of low number density and low electron to ion temperature ratio, appropriate to collisionless space plasmas. By studying the evolution of an ensemble of 104 Vlasov simulations with reduced mass ratio m i /m e = 25, we show that (1) the probability distribution of anomalous resistivity values produced during the linear, quasi-linear, and nonlinear evolution of the ion-acoustic instability is approximately Gaussian, (2) the ensemble mean of the ion-acoustic resistivity during the nonlinear regime is higher than estimates at quasi-linear saturation, and (3) the ensemble standard deviation is comparable to the ensemble mean. We argue that the large variability during the nonlinear phase is due to electron and ion bounce motion which is sensitive to the initial conditions. We demonstrate that the results are essentially similar for a real mass ratio simulation.Citation: Petkaki, P., M. P. Freeman, T. Kirk, C. E. J. Watt, and R. B. Horne (2006), Anomalous resistivity and the nonlinear evolution of the ion-acoustic instability,

Abstract

We numerically compute Nusselt numbers for laminar, hydrodynamically, and thermally fully developed Poiseuille flow of liquid in the Cassie state through a parallel plate-geometry microchannel symmetrically textured by a periodic array of isoflux ridges oriented parallel to the flow. Our computations are performed using an efficient, multiple domain, Chebyshev collocation (spectral) method. The Nusselt numbers are a function of the solid fraction of the ridges, channel height to ridge pitch ratio, and protrusion angle of menisci. Significantly, our results span the entire range of these geometrical parameters. We quantify the accuracy of two asymptotic results for Nusselt numbers corresponding to small meniscus curvature, by direct comparison against the present results. The first comparison is with the exact solution of the dual series equations resulting from a small boundary perturbation (Kirk et al., 2017, “Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges Accounting for Meniscus Curvature,” J. Fluid Mech., 811, pp. 315–349). The second comparison is with the asymptotic limit of this solution for large channel height to ridge pitch ratio.

We compute the apparent hydrodynamic slip length for (laminar and fully developed) Poiseuille flow of liquid through a heated parallel-plate channel. One side of the channel is textured with parallel (streamwise) ridges and the opposite one is smooth. On the textured side of the channel, the liquid is in the Cassie state. No-slip and constant heat flux boundary conditions are imposed at the solid–liquid interfaces along the tips of the ridges, and the menisci between ridges are considered to be flat and adiabatic. The smooth side of the channel is subjected to no-slip and adiabatic boundary conditions. We account for the streamwise and transverse thermocapillary stresses along menisci. When the latter is sufficiently small, Stokes flow may be assumed. Then, our solution is based upon a conformal map. When, additionally, the ratio of channel height to half of the ridge pitch is of order 1 or larger, an accurate but less cumbersome solution follows from a matched asymptotic expansion. When inertial effects are relevant, the slip length is numerically computed. Setting the thermocapillary stress equal to zero yields the slip length for an adiabatic flow.