Mass transport in gravity waves revisited

On re-examining the problem of mass transport due to partially standing waves in a domain that can be closed or open ended, it is pointed out that several aspects of the existing theories need to be clarified. Particular attention is paid to the free surface setup, whose expression is shown to be different when described by the Eulerian and Lagrangian approaches. In the Lagrangian system, it is the horizontal gradient of the setup, rather than the mean pressure gradient alone, that is implied by the condition of no net flux in a closed domain. In connection with this, one may determine for an unbounded domain the streamline value along the free surface, which cannot be fixed in the stream function formulation alone, by requiring the setup due to the unidirectional part of the mass transport to be zero. Three components of the free surface setup, which are of different orders of magnitude and arise owing to various mechanisms, are obtained in the process of deriving the solutions for the mass transport velocity in the boundary layers and the fluid core. Copyright 2004 by the American Geophysical Union.published_or_final_versio

Starting flow in channels with boundary slip

This paper contains a collection of problems and their analytical solutions for starting flows under the effect of boundary slip in channels of various geometries. The start-up flows examined in this work include: (1) plane Couette flow between two plates, (2) rotary Couette flow between two coaxial cylinders, (3) Poiseuille flow through a parallel-plate channel, (4) Poiseuille flow through a rectangular channel, (5) Poiseuille flow through a circular channel, and (6) Poiseuille flow through an annulus. It is first shown that, using a depletion layer to model the effective slip, the slip length may attain its steady state much faster than the starting flow will do. This supports the use of a constant Navier slip length in the present problems. Transient solutions are derived in the form of eigenfunction expansions, where the eigenvalues are determined, by solving the characteristic equations numerically, as a function of the channel geometry and the slip lengths. From the leading eigenvalue, it is found that the boundary slip will in general lengthen the transient period of a starting flow.postprin

How does wall slippage affect hydrodynamic dispersion?

How hydrodynamic dispersion is affected by wall slip remains to be fully understood. An attempt is made in this article looking into this issue for dispersion in some elementary pressure-driven flows. Both the long-time Taylor-Aris dispersion and the early-phase convection-dominated dispersion are investigated, analytically and numerically, respectively. The mean and the variance of the residence time distribution are also examined. In the basic case where the walls of a parallel-plate channel have equal slip lengths, the slip is in general to reduce the spread of a solute cloud in a finite channel by either increasing the convection speed or decreasing the dispersivity. However, the decreasing effect of slip on dispersion can be diminished or even reversed by unequal slip lengths and/or phase exchange with the wall. The convection-dominated regime is investigated, following a recently proposed transport-based method, to determine how the mean residence time and variance of elution profiles may change with axial positions depending on the slip. © 2010 The Author(s).published_or_final_versionSpringer Open Choice, 31 May 201

Effective slip for flow in a rotating channel bounded by stick-slip walls

This paper aims to look into how system rotation may modify the role played by boundary slip in controlling flow through a rotating channel bounded by stick-slip walls. A semianalytical model is developed for pressure-driven flow in a slit channel that rotates about an axis perpendicular to its walls, which are superhydrophobic surfaces patterned with periodic alternating no-shear and no-slip stripes. The cases where the flow is driven by a pressure gradient parallel or normal to the stripes are considered. The effects of the no-shear area fraction on the velocities and effective slip lengths for the primary and secondary flows are investigated as functions of the rotation rate and the channel height. It is mathematically proved that the secondary flow rate is exactly the same in the two cases, irrespective of whether the primary flow is parallel or normal to the wall stripes. For any rotation speed, there is an optimal value of the no-shear area fraction at which the primary flow rate is maximum. This is a consequence of two competing effects: the no-shear part of the wall may serve to reduce the wall resistance, thereby enhancing the flow especially at low rotation, but it also weakens the formation of the near-wall Ekman layer, which is responsible for pumping the flow especially at high rotation. Wall slip in a rotating environment is to affect flow in the Ekman layer, but not flow in the geostrophic core.postprin

Mass transport and set-ups due to partial standing surface waves in a two-layer viscous system

This is a theoretical study on the mass transport due to partially reflected long surface waves in a two-layer viscous system, which can be closed or open at its far end. Based on Lagrangian coordinates, a perturbation analysis is carried out to the second order to find the mean Lagrangian drifts in the two layers, where the lower fluid is taken to be much more viscous than the upper one. The free-surface and interfacial set-ups are also found as part of the solutions. A single analytical expression is obtained for the mass transport velocity in each layer, incorporating all the cases where the wave can be progressive, standing or partially standing, and the domain can be closed or open so that a return current may or may not exist. Through some numerical calculations, the patterns of flow in the recirculation cells due to the standing component of the wave, and in the unidirectional drifts due to the progressive component of the wave in a closed system are shown to vary with the lower-layer fluid viscosity. It is possible that, under some specific conditions, the mass transport in the core region of the upper layer is completely quiescent despite the existence of some strong drifts in the lower layer. The mean flow structures in the two layers can also respond rather differently to a change in the reflection coefficient in the presence or the absence of the return current. © 2004 Cambridge University Press.published_or_final_versio

Chemical transport associated with discharge of contaminated fine particles to a steady open-channel flow

In this paper, an analytical study on the advective-dispersive transport of a chemical contaminant resulting from the discharge of contaminated fine solid particles into a two-dimensional, steady and uniform turbulent open-channel flow is presented. Because of sorptive exchange, the transport of the chemical cloud is affected by that of the suspended particulates. Such a relationship has so far not been explicitly established by intuitive arguments. The effective transport equations are formally derived by an extended method of homogenization. It is found that over a long time scale the fall velocity will delay the sediment advection, and the advection velocity and dispersion coefficient for the chemical transport will change with space and time according to the local sediment concentration. Numerical results confirm that the centers of mass of the sediment and dissolved phase clouds are not advancing at the same speed, and the dispersion of the chemical is enhanced by the local retardation factor. © 2000 American Institute of Physics. [S1070-6631(00)01501-4].published_or_final_versio

Dispersion in electroosmotic flow generated by oscillatory electric field interacting with oscillatory wall potentials

An analytical study is presented in this article on the dispersion of a neutral solute released in an oscillatory electroosmotic flow (EOF) through a two-dimensional microchannel. The flow is driven by the nonlinear interaction between oscillatory axial electric field and oscillatory wall potentials. These fields have the same oscillation frequency, but with disparate phases. An asymptotic method of averaging is employed to derive the analytical expressions for the steady-flow-induced and oscillatory-flow-induced components of the dispersion coefficient. Dispersion coefficients are functions of various parameters representing the effects of electric double-layer thickness (Debye length), oscillation parameter, and phases of the oscillating fields. The time-harmonic interaction between the wall potentials and electric field generates steady as well as time-oscillatory components of electroosmotic flow, each of which will contribute to a steady component of the dispersion coefficient. It is found that, for a thin electric double layer, the phases of the oscillating wall potentials will play an important role in determining the magnitude of the dispersion coefficient. When both phases are zero (i.e., full synchronization of the wall potentials with the electric field), the flow is nearly a plug flow leading to very small dispersion. When one phase is zero and the other phase is π, the flow will be sheared to the largest possible extent at the center of the channel, and such a sharp velocity gradient will lead to the maximum possible dispersion coefficient. © 2011 The Author(s).published_or_final_versionSpringer Open Choice, 21 Feb 201

Electro-osmotic flow through a thin channel with gradually varying wall potential and hydrodynamic slippage

The lubrication approximation is applied to electro-osmotic flow through a thin parallel-plate channel under the combined effect of charge and hydrodynamic slippage modulation on the walls. The walls are periodically patterned for the charge and slip distributions, with a wavelength much longer than the channel height. It is shown that the phase of the wall patterns will play a significant role in determining the section-averaged velocity as well as the local convection pattern, both quantitatively and qualitatively. The effect of the phase on the flow will be dramatically different, depending on whether the electric field is applied along or perpendicular to the varying direction of the patterns. The possibility of generating a net flow in a direction perpendicular to the applied field is demonstrated. © 2012 The Japan Society of Fluid Mechanics and IOP Publishing Ltd.postprin

Lagrangian transport induced by peristaltic pumping in a tube

This is an analytical study, based on equations of motion in Lagrangian form, for the steady Lagrangian fluid transport induced by long peristaltic waves (which can be progressive, purely or partially standing) of small amplitude traveling on the boundary of a flexible tube. The first-order oscillatory viscous flow and the higher-order time-mean Lagrangian drifts (or steady streaming) are obtained as functions of the wave properties. Two cases are considered. Firstly, the wave frequency is slow such that the steady-streaming Reynolds number (Re s) is very small and the viscous diffusion is significant across the entire flow region. The time-mean flow can be found in the secondorder problem. Secondly, high-frequency pumping is considered such that Re s = O(1). Under this condition, the flow domain is divided into a thin Stokes boundary layer near the wall and the inviscid core region. The steady streaming in the core region is to be found in the fourth-order problem. Based on the Lagrangian coordinates, all the solutions are analytically expressed. Results are generated to illustrate the effects of wave properties on the Lagrangian transport. The phenomenon of reflux, a backward time-mean flow, is examined in particular. © 2011 The Japan Society of Fluid Mechanics and IOP Publishing Ltd.postprin

Electro-osmotic flow through a rotating microchannel

Paper no. 306An analytical model is presented for electro-osmotic flow through a wide rectangular microchannel rotating about an axis perpendicular to its own. The flow is driven by a steady electric field applied along the channel axis, where the upper and lower walls are charged with uniform but possibly disparate zeta potentials. The aim is to understand the interaction between Coriolis force, pressure gradient, viscous force, and the Lorentz force.postprin