Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges for Arbitrary Meniscus Curvature

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.

Nusselt numbers for Poiseuille flow over isoflux parallel ridges accounting for meniscus curvature

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 dewetted (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 with the ridge period are compared with numerical solutions of the dual series equations. 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 are useful for engineering applications.

APPLICATION OF DUAL SERIES EQUATIONS TO WAVY CONTACT BETWEEN PIEZOELECTRIC MATERIALS AND AN ELASTIC SOLID

In this paper, the wavy contact between piezoelectric materials and an isotropic solid is considered. The Papkovich–Neuber potentials for the isotropic solid and three harmonic functions for piezoelectric materials are also presented. The stated problem is reduced to a pair of dual series equations and then recast as an integral equation of the Abel type. Employing the product relation for trigonometric functions and the Mehler integral yields an exact solution of the reduced Abel type integral equation. The relationship between contact length and the level of loading, and the distribution of the surface normal stress are given in terms of elementary functions. The derived results agree well with the previous ones for the purely elastic solid. It is found that a critical loading exists for the disturbance. For limiting cases, such as the low level of loading case and full contact case, corresponding contact behaviors are presented. Numerical analyses are done to reveal the influence of the level of loading on the contact behaviors.

Numerical Analysis of Quasistatic Contact between Rubber Roller and Rigid Roller

A quasistatic and steady state contact problems of rubber roller and rigid roller is modeled by using a stress function in the form of series. The model is derived from the paper feed unit of duplicating machine. The point interpolation meshfree method (PIM) is applied to obtain the solution of the dual series equations resulting from the boundary condition. Stress and deformation for the rubber roller are obtained based on elasticity theory. Effects of rubber thickness and normal load are observed through the numerical examples. The observation show that the contact zone is changing with the normal load F and the rubber thickness has effect on the stress distribution. In addition, the study shows that the PIM method is an efficient and promising method for simulating the problem.

INVERSE SQUARE ROOT MOMENT SINGULARITIES IN RECTANGULAR PLATES CONTAINING AN INTERNAL LINE SUPPORT

Two cases of a rectangular plate having moment singularities at the ends of a partial internal line support are analytically investigated. The bending of the plate by uniform loading is formulated in terms of dual-series equations. Application of the finite Hankel integral transform reduces the dual-series equations to a Fredholm integral equation of the second kind that can be solved by standard techniques. Numerical results are given for the deflections and bending moments along the line outside of an internal line support and the change in strain energy due to the presence of a partial support.