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We present solutions for the boundary layer between two uniform shear flows flowing in the same direction. In the upper layer, the flow has shear strength a, fluid density ρ1, and kinematic viscosity ν1, while the lower layer has shear strength b, fluid density ρ2, and kinematic viscosity ν2. Similarity transformations reduce the boundary-layer equations to a pair of ordinary differential equations governed by three dimensionless parameters: the shear strength ratio γ = b/a, the density ratio ρ = ρ2/ρ1, and the viscosity ratio ν = ν2/ν1. Further analysis shows that an affine transformation reduces this multi-parameter problem to a single ordinary differential equation which may be efficiently integrated as an initial-value problem. Solutions of the original boundary-value problem are shown to agree with the initial-value integrations, but additional dual and quadruple solutions are found using this method. We argue on physical grounds and through bifurcation analysis that these additional solutions are not tenable. The present problem is applicable to the trailing edge flow over a thin airfoil with camber.

This paper presents a continued study to our previous work on electroosmotic (EO) flow in a channel with vertical baffle plates by further investigating EO flow through an array of baffle plates arranged in parallel to the channel walls. The flow may be driven either in the direction along or in the direction transverse to the plates, thus distinguishing the longitudinal EO pumping (LEOP) and the transverse EO pumping (TEOP). In both types of EO pumping, it is more interesting to examine the cases when the baffle plates develop a higher zeta potential (denoted by α) than that on the channel walls (β). This semi-analytical study enables us to compare between LEOP and TEOP in the pumping efficiency under similar conditions. The TEOP case is more difficult to solve due to the higher order governing partial differential equations caused by the induced non-uniform pressure gradient distribution. In particular, we examine how the EO pumping rates deviate from those predicted by the Helmholtz-Smoluchowski veloci...

The present study is aimed to investigate the effects of wavy roughness on electro-osmotic (EO) flow over a wavy plate. The waviness of the plate is modeled by the product of two cosinoidal functions, and the roughness (e) is defined to be the ratio of the wavy amplitude to the Debye length. The effects are examined with respect to the roughness e and different wave numbers (α and β) of the plate waviness. The analysis of the EO flow over the wavy plate is carried out for the applied electric potential, the potential for the electric double layer, as well as the EO flow velocity and pressure field under the Debye-Huckel approximation by using a boundary perturbation method. It is found that the velocity component along the direction of the applied electric field is modified by a second-order term of the roughness, though the same velocity component near the wavy wall exhibits periodic behaviors in phase with the plate waviness. The mean flow rate deficit (e2μ2) due to the surface roughness presents a soph...

The viscous flow in a minute channel with three-dimensional bumpy walls is analyzed by perturbations using the small amplitude to channel width ratio. It is found that the bumpiness causes changes in velocity, shear stress, and vorticity. The channel resistance may be increased or decreased by the bumps. There exist optimum protrusion aspect ratios for maximum flow.

Stokes shear flow over a surface with evenly spaced, finite-depth rectangular grooves is solved analytically by eigenfunction expansions and matching. Macroscopically the rough surface is equivalent to a smooth surface with partial slip. The slip coefficients are determined for shear flow both along and transverse to the grooves. The partial slip condition is then applied to the flow through a channel with nonaligned grooves on the walls.

The relative sliding motion of a smooth plate over a periodically finned plate is studied under the Stokes limit. The resulting harmonic or biharmonic equations with mixed boundary conditions are solved by eigenfunction expansions and collocation. The drag is determined as a function of fin height and fin spacing.

The Stokes flow through a row of equally spaced circular cylinders with partial slip on the boundaries is studied by domain decomposition and matching. The pressure drop or resistance across the grid is found for various slip and normalized cylinder radii. Asymptotic formulas are found for sparcely and closely spaced cylinders.

Thermoconvective instability of a rectangular box containing a fluid-saturated porous media is studied. The box top is maintained at constant temperature and the bottom is heated by constant flux.

Comparisons between slip lengths predicted by a liquid-gas coupled model and that by an idealized zero-gas-shear model are presented in this paper. The problem under consideration is pressure-driven flow of a liquid through a plane channel bounded by two superhydrophobic walls which are patterned with longitudinal or transverse gas-filled grooves. Effective slip arises from lubrication on the liquid-gas interface and intrinsic slippage on the solid phase of the wall. In the mathematical models, the velocities are analytically expressed in terms of eigenfunction series expansions, where the unknown coefficients are determined by the matching of velocities and shear stresses on the liquid-gas interface. Results are generated to show the effects due to small but finite gas viscosity on the effective slip lengths as functions of the channel height, the depth of grooves, the gas area fraction of the wall, and intrinsic slippage of the solid phase. Conditions under which even a gas/liquid viscosity ratio as small as 0.01 may have appreciable effects on the slip lengths are discussed.

The electro-osmotic flow through a wavy microchannel is studied under the Debye–Huckel approximation. An analytic solution by perturbation with appropriate averaging is carried out up to the second-order in terms of the small amplitude of corrugation. It is shown that the wavelength and phase difference of the corrugations can be utilized to control the flow relative to the case of flat walls. In particular, for thick electric double layers the electro-osmotic flow can be enhanced at long-wavelength corrugations because of the coherence between the electric potential and the wall shape function. Notably, these findings are not restricted to small amplitudes of corrugation. By applying the Ritz method to solve for the electro-osmotic flow, it is found that the enhancement becomes even greater (up to 30%) with increases in corrugation. Moreover, the nonlinear Poisson–Boltzmann equation is solved by finite difference to study the electro-osmotic flow in terms of the relative strength of the zeta potential. The issue of overlapped electric double layers when they are very thick is also discussed. The relative flow rate is shown to increase under the following conditions: (i) completely out-of-phase corrugations with long wavelength and large amplitude, (ii) small zeta potential, and (iii) slight overlapping of electric double layers.

A stability analysis determining the onset of oscillatory convection in a bounded cylindrical cavity containing a viscoelastic fluid saturated porous medium is performed for insulated sidewalls, isothermal top wall, and bottom heated by constant heat flux or constant temperature. The dependence of the critical Darcy-Rayleigh number and the preferred convective motion on both the aspect ratio of the cylinder and the viscoelastic parameters is established. The differences between bottom constant heat flux heating and bottom constant temperature heating are compared.

The Stokes flow through a tube with a bumpy wall is solved through a perturbation in the small amplitude of the three-dimensional bumps. The solution can be expressed in terms of modified Bessel functions. For given area or volume of a bump, there exists an optimum circumferential wave number and an optimum aspect ratio for which the flow rate is maximized.

An exact similarity solution of the Navier–Stokes equations is found. The solution represents the three‐dimensional fluid motion caused by the stretching of a flat boundary.

An exact solution of the unsteady Navier–Stokes equations is found. The solution describes the stagnation region of an accelerating circular jet impinging obliquely on a flat plate.

Two wavy plates, separated by a viscous fluid, move relative to each other in a direction normal to the corrugations. Assuming small‐amplitude corrugations and low Reynolds number, the unsteady Navier–Stokes equations are solved by perturbations. It is found not only that the mean drag is increased, but that an unsteady drag and an unsteady lift are also induced by the corrugations. These fluid‐induced vibrations, predicted for the first time, may be important in the design of grooved sliders.

Jets of different fluid properties impinge head‐on. The region near the stagnation point is investigated. Similarity solutions of the Navier–Stokes equations are matched at the interface. It is found that the flow field depends heavily on the ratios of viscosity and density.

Two concentric corrugated cylinders are separated by a viscous fluid. When the inner cylinder is rotated, unsteady and steady torque and forces may be generated. Assuming low Reynolds numbers and small amplitude corrugations, the Navier–Stokes equations are solved by perturbations to second order. The torque and forces are dependent on the gap width, the amplitude of corrugations, and especially the number of waves of the corrugations.

It is shown that the general pair correlation function of Balescu and Taylor reduces directly to the correct expression in the case of equilibrium.