A mathematical model is presented for the problem of apparent slip arising from Stokes shear flow over a composite surface featuring mixed boundary conditions on the microscale. The surface can be composed of a bidimensional array of solid areas placed on an otherwise no-shear surface corresponding to an envelope over the tops of posts, or no-shear areas placed on an otherwise solid surface corresponding to an envelope over

The addition of high molecular components to a base oil increases its extensional and shear viscosity. Although the extensional viscosity affected the ease with which the oil could be injected, the results showed that it was the shear viscosity that determined the relative velocity between the oil and the wall of the vitreous cavity, and thus the propensity to emulsify.

Mathematical models are developed for heat conduction in creeping flow of a liquid over a microstructured superhydrophobic surface, where because of hydrophobicity, a gas is trapped in the cavities of the microstructure. As gas is much lower in thermal conductivity than liquid, an interfacial temperature slip between the liquid and the surface will develop on the macroscale. In this note, the temperature jump coefficient is numerically determined for several types of superhydrophobic surfaces: a surface with parallel grooves, and surfaces with two-dimensionally distributed patches corresponding to the top of circular or square posts, and circular or square holes. These temperature jump coefficients are found to have a nearly constant ratio with the corresponding velocity slip lengths.

A perturbation analysis is carried out to the second order to give effective equations for Darcy-Brinkman flow through a porous channel with slightly corrugated walls. The flow is either parallel or normal to the corrugations, and the corrugations of the two walls are either in phase or half-period out of phase. The present study is based on the assumptions that the corrugations are periodic sinusoidal waves of small amplitude, and the channel is filled with a sparse porous medium so that the flow can be described by the Darcy-Brinkman model, which approaches the Darcian or Stokes flow limits for small or large permeability of the medium. The Reynolds number is also assumed to be so low that the nonlinear inertia can be ignored. The effects of the corrugations on the flow are examined, quantitatively and qualitatively, as functions of the flow direction, the phase difference, and the wavelength of the corrugations, as well as the permeability of the channel. It is found that the corrugations will have greater effects when it is nearer the Stokes' flow limit than the Darcian flow limit, and when the wavelength is shorter. For the same wavelength and phase difference, cross flow is more affected than longitudinal flow by the corrugations. Opposite effects can result from 180 • out-of-phase corrugations, depending on the flow direction, the wavelength, as well as the permeability.

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A third-order asymptotic solution in Lagrangian description for nonlinear water waves propagating over a sloping beach is derived. The particle trajectories are obtained as a function of the nonlinear ordering parameter 3 and the bottom slope a to the third order of perturbation. A new relationship between the wave velocity and the motions of particles at the free surface profile in the waves propagating on the sloping bottom is also determined directly in the complete Lagrangian framework. This solution enables the description of wave shoaling in the direction of wave propagation from deep to shallow water, as well as the successive deformation of wave profiles and water particle trajectories prior to breaking. A series of experiments are conducted to investigate the particle trajectories of nonlinear water waves propagating over a sloping bottom. It is shown that the present third-order asymptotic solution agrees very well with the experiments.

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Analytical solutions are derived for steady fully-developed buoyancy-driven flow in a vertical annular microchannel, of which either the inner or outer wall exhibits superhydrophobic velocity slip and temperature jump, and the inner wall is maintained either at constant wall temperature or constant heat flux. For the four possible cases of hydrodynamic and thermal boundary conditions, we determine the flow rate as a function of the core size, slip length and temperature jump coefficient. Asymptotic limits are obtained for very large slip and temperature jump. The effects of slip and temperature jump on two issues, namely, the optimum core radius for maximum flow rate and the singular increase of the flow rate for very small core radius, are investigated in particular.