Chiu-On Ng

Roll waves on a shallow layer of mud modelled as a power-law fluid

We give a theory of permanent roll waves on a shallow layer of fluid mud which is modelled as a power-law fluid. Based on the long-wave approximation, Karman’s momentum integral method is applied to derive the averaged continuity and the momentum equations. Linearized instability analysis of a uniform flow shows that the growth rate of unstable disturbances increases monotonically with the wavenumber, and therefore is insufficient to suggest a preferred wavelength for the roll wave. Nonlinear roll waves are obtained next as periodic shocks connected by smooth profiles with depth increasing monotonically from the rear to the front. Among all wavelengths only those longer than a certain threshold correspond to positive energy loss across the shock, and are physically acceptable. This threshold also implies a minimum discharge, viewed in the moving system, for the roll wave to exist. These facts suggest that a roll wave developed spontaneously from infinitesimal disturbances should have the shortest wavelength corresponding to zero dissipation across the shock, though finite dissipation elsewhere. The discontinuity at the wave front is a mathematical shortcoming needing a local requirement. Predictions for the spontaneously developed roll waves in a Newtonian case are compared with available experimental data. Longer roll waves, with dissipation at the discontinuous fronts, cannot be maintained if the uniform flow is linearly stable, when the fluid is slightly non-Newtonian. However, when the fluid is highly non-Newtonian, very long roll waves may still exist even if the corresponding uniform flow is stable to infinitesimal disturbances. Numerical results are presented for the phase speed, wave height and wavenumber, and wave profiles for a representative value of the flow index of fluid mud.

Some Applications of the Homogenization Theory

Publisher Summary This chapter describes some applications of the homogenization theory. The basic idea of the theory of homogenization has been employed for a long time. In the theory of wave propagation over slowly varying media, the familiar ray theory (geometrical optics approximation) is one such example. There, the method of multiple scales is employed to average over the locally periodic waves and find the slow variation of the wave envelope. Seepage through a porous media is one of the first examples to which the method of homogenization was applied. It is a good example to explain the role of physical scales and the mathematical procedure for three-dimensional problems. Furthermore, it can be used to illustrate the derivation of many properties of the constitutive coefficients. Examples of applications are particularly rich in the mechanics of composite media. The obvious example is to determine the relations between stresses and strains in a fiber-reinforced material. The problem of sound propagation through a liquid populated sparsely by bubbles is another interesting application of homogenization theory. The main objective is to find an effective equation for the propagation of sound whose wavelength is much greater than the bubble spacing, which is, in turn, much greater than the bubble radius.

Stokes shear flow over a grating: Implications for superhydrophobic slip

A semianalytical model based on the method of eigenfunction expansions and domain decomposition is developed for Stokes shear flow over a grating composed of a periodic array of parallel slats, with finite slippage on solid surfaces and infinite slippage on the bottom of troughs mimicking a no-shear liquid-gas interface penetrating into the space between slats. The model gives the macroscopic slip lengths for flow parallel or normal to the slats in terms of the microscopic slip length of the liquid-solid interface, area fraction of the no-shear liquid-gas interface, and depth of the liquid-gas interface in the grooves. When the no-shear interface lies flat on the top of the slats, the macroscopic slip lengths are the maximum and can be estimated with reasonably good accuracy by simple formulas. However, the slip lengths, particularly the transverse one, are very sensitive to penetration of the no-shear interface into the grooves. They can be reduced by a large factor when the interface just slightly gets ...

Dispersion in steady and oscillatory flows through a tube with reversible and irreversible wall reactions

An asymptotic analysis is presented for the advection–diffusion transport of a chemical species in flow through a small-diameter tube, where the flow consists of steady and oscillatory components, and the species may undergo linear reversible (phase exchange or wall retention) and irreversible (decay or absorption) reactions at the tube wall. Both developed and transient concentrations are considered in the analysis; the former is governed by the Taylor dispersion model, while the latter is required in order to formulate proper initial data for the developed mean concentration. The various components of the effective dispersion coefficient, valid when the developed state is attained, are derived as functions of the Schmidt number, flow oscillation frequency, phase partitioning and kinetics of the two reactions. Being more general than those available in the literature, this effective dispersion coefficient incorporates the combined effects of wall retention and absorption on the otherwise classical Taylor dispersion mechanism. It is found that if the phase exchange reaction kinetics is strong enough, the dispersion coefficient is probably to be increased by orders of magnitude by changing the tube wall from being non-retentive to being just weakly retentive.

Simulation of wave propagation over a submerged bar using the VOF method with a two-equation k-e turbulence modeling

A two-equation k–e turbulence model is used in this paper to simulate the propagation of cnoidal waves over a submerged bar, where the free surface is handled by the volume-offluid (VOF) method. Using a VOF partial-cell variable and a donor–acceptor method, the model is capable of treating irregular boundaries, including arbitrary bottom topography and internal obstacles, where the no-slip condition is satisfied. The model also allows the viscous sublayer to be modeled by a wall function approximation implemented in the grid nodes that are immediately adjacent to a wall boundary. The numerical model applied to the propagation of cnoidal waves over a submerged bar can produce results that are in general agreement with some laboratory measurements. Some remarks arising from the comparison between the computational and experimental results are presented. # 2003 Elsevier Ltd. All rights reserved.

Mass transport in water waves over a thin layer of soft viscoelastic mud

A theory is presented for the mass transport induced by a small-amplitude progressive wave propagating in water over a thin layer of viscoelastic mud modelled as a Voigt medium. Based on a sharp contrast in length scales near the bed, the boundary-layer approximation is applied to the Navier–Stokes equations in Lagrangian form, which are then solved for the first-order oscillatory motions in the mud and the near-bed water layers. On extending the analysis to second order for the mass transport, it is pointed out that it is inappropriate, as was done in previous studies, to apply the complex viscoelastic parameter to a higher-order analysis, and also to suppose that a Voigt body can undergo continuous steady motion. In fact, the time-mean motion of a Voigt body is only transient, and will stop after a time scale given by the ratio of the viscosity to the shear modulus. Once the mud has attained its steady deformation, the mass transport in the overlying water column can be found as if it were a single-layer system. It is found that the near-bed mass transport has non-trivial dependence on the mud depth and elasticity, which control the occurrence of resonance. Even when the resonance is considerably damped by viscosity, the mass transport in water over a viscoelastic layer can be dramatically different, in terms of magnitude and direction, from that over a rigid bed. Wave–mud interaction is one of the key mechanisms controlling the transport of sediments in coastal and estuarine waters. In the presence of cohesive sediments or marine muds, which are composed primarily of very fine particles and act as an effective energy dissipator, wave damping is enhanced; surface waves can be attenuated appreciably in a finite number of wave periods or wavelengths. Frequency modulation may also occur to waves propagating over a muddy bottom. Meanwhile, the sediments will undergo various processes such as resuspension, fluidization and mass transport under the forcing of surface waves. The erosion, transport and deposition of cohesive sediments are known to pose various problems that are of concern to the environmental as well as coastal engineering. These problems have motivated extensive studies into the mechanics of mud in a wave-dominated marine environment. Cohesive sediments are made up of very tiny particles, typically less than O(10) µm in size, and their properties are much determined by the micro-structure which is dominated by many physico-chemical effects. Since the mineralogical make-up is so complex and susceptible to change in response to the local geo-environment, a mud tends to exhibit vastly different rheological behaviours depending on the site conditions. For the sake of analysis, it is common in the modelling of wave–seabed

Effective slip for Stokes flow over a surface patterned with two- or three-dimensional protrusions

Effective slip lengths are obtained, using semi-analytic methods, for Stokes flows over a surface that is patterned with a periodic array of two-dimensional (2D) cylindrical or 3D spherical protrusions. The protruding surface can be perfect- or non-slipping, corresponding to a bubble mattress or a rough boundary. For longitudinal and transverse flows over cylindrical bumps and 3D flow over a square array of spherical bumps, the effective slip length is obtained as a function of the protrusion angle, the area fraction of surface covered by protrusions and the partial slip length of the protruding surface. The results are compared with analytical dilute limits in order to ascertain the range of validity of these limits. Phenomenological equations are also derived to enable a quick evaluation of the slip length for some particular values of the protrusion angle at which the slip length is maximum in magnitude.

Effects of kinetic sorptive exchange on solute transport in open-channel flow

A theory is presented for the transport in open-channel flow of a chemical species under the influence of kinetic sorptive exchange between phases that are dissolved in water and sorbed onto suspended sediments. The asymptotic method of homogenization is followed to deduce effective transport equations for both phases. The transport coefficients for the solute are shown to be functions of the local sediment concentration and therefore vary with space and time. The three important controlling parameters are the suspension number, the bulk solid–water distribution ratio and the sorption kinetics parameter. It is illustrated with a numerical example that when values of these parameters are sufficiently high, the advection and dispersion of the solute cloud can be dominated by the sorption effects. The concentration distribution can exhibit an appreciable deviation from Gaussianity soon after discharge, which develops into a long tailing as the solute cloud gradually moves ahead of the sediment cloud.

Aggregate Diffusion Model Applied to Soil Vapor Extraction in Unidirectional and Radial Flows

An aggregate diffusion model for soil vapor extraction is applied in this paper. The model accounts for the diffusion within water-saturated aggregates and convective diffusion in air-filled macropores. Focusing only on the dominant and measurable mechanisms, we represent the kinetic term for the gas aggregate exchange by a convolution integral. The nondimensional transport equation involves only three basic parameters: (1) Peclet number, (2) ratio of advection time to aggregate diffusion time, and (3) ratio of mass within aggregate to mass in gas. Two examples of soil vapor extraction based on the model are studied. The first problem is concerned with vapor transport in a one-dimensional column, and the theory is compared with published experiments. The second example deals with vapor extraction by a well centered at a radially symmetric contaminated zone. Effects of the physical parameters on contaminant removal are discussed.

On the effects of liquid-gas interfacial shear on slip flow through a parallel-plate channel with superhydrophobic grooved walls

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 sma...