Volume 20, Issue 6, June 2008

When the symmetry of axisymmetric Taylor vortexflow is broken, timeperiodic wavy vortexflow (WVF) appears and quite quickly becomes globally chaotic (in the Lagrangian sense) with increasing Reynolds number. Previously published simulations of WVF suggest that beyond a certain Re, nonmixing vortex cores reappear in the flow and grow in size with further increases in Re. This reappearance occurs well into the inertiadominated flow regime and coincides with a decrease in axial fluid dispersion and an increase in flow symmetry as measured by certain Eulerian symmetry measures. In this brief paper, we present experimental dyereaction visualization results from two WVF wave states in the region where vortex cores are predicted numerically. The experimental results show unambiguous visual evidence for the existence of vortex cores and provide visual agreement with the numerical results. They are significant in that experimental evidence for these structures in WVF has not been reported before. The results also suggest that vortextovortex transport occurs via sheetlike structures that are pulled from one vortex to another and become wrapped around the vortex cores before being stretched to the point at which molecular diffusion dominates.
 LETTERS


Signature of varicose wave packets in the viscous sublayer
View Description Hide DescriptionWaviness of the wallshear stress (WSS) field of a turbulent boundary layer is investigated experimentally. Arrays of flexible micropillars are used as sensors for WSS in the turbulent boundary layer of a flat plate in a zero pressure gradient flow channel. The pillars and flow conditions are chosen such that accurate dynamic measurements of the twodimensional (2D) WSS vectors at the sensor locations are obtained for the mean and the fluctuating part simultaneously. Timeresolved measurements reveal for the first time the orientation of the 2D WSS vectors and their spatiotemporal correlation. The results show trains or packets of varicose wavelike disturbances of the near wall flow with strong anticorrelation of the spanwise WSS component, which appear periodically and can be related to the signature of coherent packets of vortex loops. In addition these structures may represent the nearwall state of secondary varicose instability of sublayer streaks.

Path instability of rising spheroidal air bubbles: A shapecontrolled process
View Description Hide DescriptionThe conditions for which the paths of freely rising bubbles become oscillatory are studied experimentally using silicone oils with viscosities ranging from 0.5 to 9.4 times that of water. Since these fluids are nonpolar, as opposed to water, the gasliquid interfaces remain clean without the need of an ultrapure environment. We find the Reynolds number at incipient transition to vary from 70 to 470, for decreasing liquidviscosity. Correspondingly, the bubble aspect ratio remains almost constant, ranging only from 2.36 to 2.0 for the same set of conditions. Hence, we argue that the dominant parameter to trigger the instability is the bubble shape and not the Reynolds number. Since vorticitygenerated at the bubble surface is almost independent of the Reynolds number and mostly depends on the bubble aspect ratio in the parameter range covered by our experiments, present results strongly support the view that path instability is a direct consequence of the wake instability that occurs when this surfacevorticity exceeds a certain threshold.
 Top

 ARTICLES

 Interfacial Flows

Numerical study of incompressible fluid dynamics with nonuniform density by the immersed boundary method
View Description Hide DescriptionWe apply the immersed boundary method to the dynamics of an incompressible fluid with a nonuniform density. In order to take into account both the inertial and gravitational effects of the variable density, the penalty immersed boundary (pIB) method is used [Y. Kim and C. S. Peskin, Phys. Fluids19, 053103 (2007)]. Incompressible fluid motion with a nonuniform density has been extensively explored both experimentally and computationally. We show that the pIB method is a robust and efficient numerical tool for the simulation of fluids with variable density by conducting computations of some example problems: The falling of a heavier fluid surrounded by a lighter fluid and the Rayleigh–Taylor instabilities in two dimensions and three dimensions and the dynamic stabilization of the Rayleigh–Taylor instability.

Nonlinear free surface flows past a semiinfinite flat plate in water of finite depth
View Description Hide DescriptionWe consider the steady free surface twodimensional flow past a semiinfinite flat plate in water of a constant finite depth. The fluid is assumed to be inviscid, incompressible and the flow is irrotational; surface tension at the free surface is neglected. Our concern is with the periodic waves generated downstream of the plate edge. These can be characterized by a depthbased Froude number and the depth (draft) of the depressed plate. Previous analytical studies have been restricted to small values of and subcritical flows, when linearized theory can be used. Here, our main concern here is with the critical regime, , where we use a weakly nonlinear longwave analysis, For finite values of , we solve the fully nonlinear problem numerically using a boundary integral equation method. Our results confirm previous studies and extend these to the critical regime. As increases, our numerical results demonstrate that the downstream waves approach the highest possible wave. We also find some wavefree solutions, which when compared to the weakly nonlinear results are essentially just onehalf of a solitary wave.

Parametric effects of bubblemicrocantilever impacts in a confined channel
View Description Hide DescriptionIn this study, we have investigated the impact of bubbles with microcantilever obstacles in a confined channel. Static cantilevers of thickness were considered. Cantilevers were mounted perpendicular to the mean flow in a vertically oriented channel with width of , span of , and length of . Steady, fully developed upward flows with channel Reynolds numbers based on mean fluid velocity and hydraulic diameter of 800–2400 were considered. Bubbles of diameter of were introduced upstream of the test section, and impacts were observed by using a microscope equipped with a high frame rate camera. Observations were made in planes normal to the length of cantilevers backlit by using white light. Liquid density , interfacial surface tension, bubble velocity immediately prior to impact, bubble diameter , obstacle thickness , impact offset distance , channel width , and beam location are all potential influences on the result of bubblebeam impacts. Five dimensionless combinations of these quantities: Weber number , impact offset ( where is the impact offset distance), bubble diameter relative to beam thickness , bubble diameter relative to channel width , and beam offset from channel centerline relative to channel size (, where offset is the distance the beam is displaced from the channel centerline), were considered in the following ranges: , , , , and . Multiple types of interactions ranging from bouncing with little deformation to wrapping with substantial deformation to splitting into two were observed. Splitting required a minimum Weber number to occur, which was observed to be independent of for all cases considered. Impact offset and combined to affect impact outcome. For , the Weber number required for splitting was observed to increase with offset . As bubble diameter approached the channel width under laminar flow conditions, channel gradient effects could generate a substantial lift force toward the center of the channel for bubbles approaching offset from the channel centerline. This lift force caused bubbles to cross from one side of the beam to the other; this type of interaction increased the likelihood of splitting and resulted in a number of lowWe, high splitting cases. Bubble impacts with channel walls reduced this phenomenon for .

Unsteady propagation of a liquid plug in a liquidlined straight tube
View Description Hide DescriptionThis paper considers the propagation of a liquid plug driven by a constant pressure within a rigid axisymmetric tube whose inner surface is coated by a thin liquid film. The Navier–Stokes equations are solved using the finitevolume method and the SIMPLEST algorithm. The effects of precursor film thickness, initial plug length, pressure drop across the plug, and constant surface tension on the plug behavior and tube wall mechanical stresses are investigated. As a plug propagates through a liquidlined tube, the plug gains liquid from the leading front film, and it deposits liquid into the trailing film. If the trailing film is thicker (thinner) than the precursor film, the plug volume decreases (increases) as it propagates. For a decreasing volume, eventually the plug ruptures. Under a specific set of conditions, the trailing film thickness equals the precursor film thickness, which leads to steady state results. The plug speed decreases as the precursor film thins because the resistance to the moving front meniscus increases. As the pressure drop across the plug decreases, the plug speed decreases resulting in thinning of the trailing film. As the plug length becomes longer, the viscous resistance in the plug core region increases, which slows the plug and causes the trailing film to become even thinner. The magnitude of the pressure and shear stress at the tube inner wall is maximum in the front meniscus region, and it increases with a thinner precursor film. As the surface tension increases, the plug propagation speed decreases, the strength of the wall pressure in the front meniscus region increases, and the pressure gradient around the peak pressure becomes steeper.

On the nonaxisymmetric instability of round liquid jets
View Description Hide DescriptionA fully temporal linear stability analysis for round liquid jets is accomplished in this study. Threedimensional disturbances are considered and a detailed parametric investigation is performed. The results indicate that only the nonaxisymmetric mode with azimuthal wavenumber may prevail over the axisymmetric mode and appear to dominate the maximum growth rate. The domain in which the nonaxisymmetric modes may start to grow is also clarified. A comparison with experimental observations in the literature is also presented. The results provide a complete understanding of the stability characteristics of round liquid jets.

Substrate design or reconstruction from free surface data for thin film flows
View Description Hide DescriptionThis paper addresses two inverse problems relevant to the flow of thin liquid films on uneven surfaces. The first one consists in controlling the free surface profile by a suitable design of the substrate and the second involves the reconstruction of the substrate topography from a known free surface profile. These problems are treated in the classical lubrication approximation framework and admit, for planar flows, a surprisingly simple closeform solution, which is successfully tested on a range of problems.
 Viscous and NonNewtonian Flows

Two spheres in a free stream of a secondorder fluid
View Description Hide DescriptionThe forces acting on two fixed spheres in a secondorder uniform flow are investigated. When , where and are fluid parameters related to the first and second normal stress coefficients, the velocity field for a secondorder fluid is the same as the one predicted by the Stokes equations while the pressure is modified. The Stokes solutions given by Stimson and Jeffery [Proc. R. Soc. London, Ser. A111, 110 (1926)] for the case when the flow direction is along the line of centers and Goldman et al. [Chem. Eng. Sci.21, 1151 (1966)] for the case when the flow direction is perpendicular to the line of centers are utilized and the stresses and the forces acting on the particles in a secondorder fluid are calculated. For flow along the line of centers or perpendicular to it, the net force is in the direction that tends to decrease the particle separation distance. For the case of flow at arbitrary angle, unequal forces are applied to the spheres perpendicularly to the line of centers. These forces result in a change of orientation of the sedimenting spheres until the line of centers aligns with the flow direction. In addition, the potential flow of a secondorder fluid past two fixed spheres in a uniform flow is investigated. The normal stress at the surface of each sphere is calculated and the viscoelasticeffects on the normal stress for different separation distances are analyzed. The contribution of the potential flow of a secondorder fluid to the force applied to the particles is an attractive force. Our explanations of the aggregation of particles in viscoelastic fluids rest on three pillars; the first is a viscoelastic “pressure” generated by normal stresses due to shear. Second, the total time derivative of the pressure is an important factor in the forces applied to moving particles. The third is associated with a change in the normal stress at points of stagnation which is a purely extensional effect unrelated to shearing.

Stokesian dynamics of close particles
View Description Hide DescriptionStokesian dynamics simulations of close particles are reported, taking into account lubrication forces and manybody hydrodynamic interactions between spheres. A periodic trajectory of three particles maintaining a permanent proximity to each other has been found and analyzed. This solution is used as a benchmark to study the accuracy and stability of various numerical integration schemes. In particular, different methods of preventing unphysical overlaps of the particles are considered and potential artifacts discussed.

Viscosity stratification and the horizontal scale of enddriven cellular flow
View Description Hide DescriptionWe analyze isothermal viscositystratified creeping flow in a horizontal semiinfinite slot. The slot top and bottom are tractionfree, and a cellular motion is driven by a given uniform shear stress applied at the vertical end. The viscosity is a prescribed function of depth: A thin lowviscosity channel of uniform viscosity and thickness overlies a thick sublayer of viscosity and thickness . By coupling a lubrication analysis of the channel to an exact solution for the sublayer, we show that in the limit of , with fixed, the flow has horizontal scale given for by . The analysis is motivated by the longstanding suggestion that the presence of a lowviscosity layer could explain the existence of long enddriven cells beneath the Pacific and the IndoAustralian plates. Our formula supports that suggestion; applied to published viscosity profiles, it predicts that viscosity stratification could allow a cell length of up to seven times that for isoviscous flow.

Surface tank treading: Propulsion of Purcell’s toroidal swimmer
View Description Hide DescriptionIn this work we address the “smoking ring” propulsion technique, originally proposed by Purcell [Am. J. Phys.45, 3 (1977)] for selflocomotion at low Reynolds numbers. We first consider selflocomotion of a doughnutshaped swimmer powered by surface rotation. Different modes of surfacemotion are assumed and propulsion velocity and swimming efficiency are determined. The swimmer is propelled against the direction of its outer surfacemotion, the inner surface having very little effect. The simplest swimming mode corresponding to constant angular velocity can achieve propulsion speeds of up to 66% of the surface tanktreading velocity and swimming efficiency of up to 13%. Higher efficiency is possible for more complicated modes powered by nonuniform twirling of extensible surface. A practical design of a necklaceshaped swimmer motivated by Purcell’s idea is proposed and demonstrated numerically. Finally, the explicit solution is found for the twodimensional swimmer composed of two counterrotating disks, using complex variable techniques.

Analytical solution of electroosmotic flow in a semicircular microchannel
View Description Hide DescriptionThe electroosmotic flow through a microchannel with a semicircular cross section is studied under the Debye–Huckel approximation. Analytical series solutions are found for two basic cases. The solutions for the two basic cases considered can be superposed to yield solutions for any combination of constant zeta potentials on the flat or curved wall boundaries. Moreover, in the limit of a thin electric double layer (small Debye length compared to the nominal dimension), a method of solution is shown for variable zeta potentials by using the Smoluchowski slip approximation.

Viscous flow over outflow slits covered by an anisotropic Brinkman medium: A model of flow above interendothelial cell clefts
View Description Hide DescriptionAn analytic series solution is presented for the shear driven flow of a viscous fluid over an infinite series of outflow slits covered by a Brinkman medium with an anisotropic Darcy permeability. The solution is used to model the cellular scale flow of water over and within the endothelial glycocalyx, when the transmural water flux through the vascular endothelium is only allowed to pass via interendothelial cell clefts. Results are presented illustrating the effect of both the glycocalyx properties and the applied shearing rate (imposed by vascular scale fluid dynamics) on several relevant measures of the velocity field, including the wall normal velocity and the shear rate evaluated at the luminal surface of the glycocalyx.
 Particulate, Multiphase, and Granular Flows

A method for determining Stokes flow around particles near a wall or in a thin film bounded by a wall and a gasliquid interface
View Description Hide DescriptionA method for determining Stokes flow around particles near a wall or in a thin film bounded by a wall on one side and a nondeformable gasliquid interface on the other side is developed. The noslip boundary conditions at the wall are satisfied by constructing an image system based on Lamb’s multipoles. Earlier results for the image systems for the flow due to a point force or a force dipole are extended to image systems for force or source multipoles of arbitrary orders. For the case of a film, the image system consists of an infinite series of multipoles on both sides of the film. Accurate evaluation of the flow due to these images is discussed, including the use of Shanks transforms. The method is applied to several problems including chains of particles, radially expanding particles, drops, and porous particles.

Planar collapse of a granular column: Experiments and discrete element simulations
View Description Hide DescriptionThe collapse of a granular column is an intriguingly simple tabletop experiment which exhibits a host of interesting phenomena. Here, we introduce a planar version in which the collapsing column is only one particle deep perpendicular to the plane of motion to make observations of the internal motion possible. This configuration also particularly lends itself to comparison with discrete element simulations which are performed in tandem. Our experiments confirm that this planar system displays all the same features as collapsing cylinders and rectangular blocks. In particular, the dominant dependence on the initial parameters of the column runout is through a power law of the initial heighttowidth aspect ratio. Discrete element simulations, which are found to reproduce the experimental behavior very well, are then used to analyze the velocity field of the collapse process. A predominantly linear velocity profile is found in a moving layer over an evolving static pile. The timedependent strain rate in this moving layer is in reasonable correspondence with a strain rate prediction for flow down a fixed slope by Rajchenbach [Phys. Rev. Lett.90, 144302 (2003)].

Dipolophoresis of nanoparticles
View Description Hide DescriptionA general methodology is presented for evaluating the dielectrophoretic velocity of a freely suspended spherical colloid in an electrolyte solutes under the action of a nonuniform electric field and for a Debye layer of arbitrary thickness. The nonlinear induced charge electrophoretic problem is first considered. General analytic expressions are derived for the mobility of an uncharged particle in the form of products between adjacent modes of the ambient electric field demonstrating a symmetrybreakingtype phenomena. It is shown that the mobility of a conducting (i.e., ideally polarizable) spherical particle vanishes for a quadratic electric field in the limiting case of a thin Debye layer. For an infinitely thick Debye layer it attains asymptotically a positive finite value. Yet, there is another value of a finite Debye length for which the mobility changes sign. This interesting nonintuitive effect may have implications to separation of particles by size. The linear case of a uniformly charged colloid is obtained as a special limit and the classical mobility expressions of Henry [Proc. R. Soc. (London)4, 106 (1931)], Smulokowski [Handbook of Electricity and Magnetism, edited by L. Graetz (Barth, Leipzig, 1921), p. 2], and Huckel [Physik25, 204 (1924)] for a spherical colloid are readily recovered. The formulation is based on an extension of Teubner’s integral approach for nonuniform electric fields and on utilizing a variant of the Lorentz reciprocal theorem for Stokes flows. As an example demonstrating the effect of nearby boundaries, the method is finally applied for the radially symmetric case involving a freely suspended colloid within a hollow spherical capsule filled with electrolyte. It can be considered as an extension of Henry’s unbounded solution for a confined three dimensional embodiment.
 Laminar Flows

Twodimensional buoyancy driven thermal mixing in a horizontally partitioned adiabatic enclosure
View Description Hide DescriptionThe dynamics of the transient, twodimensional buoyancy driven thermal mixing of two fluid masses at different temperatures, initially at rest and confined to separate portions of a horizontally partitioned adiabatic enclosure, is investigated numerically within the framework of the Boussinesq approximation. The fluids are allowed to mix through a centrally located opening or vent in the partition. Apart from the geometric parameters, the dynamics is governed by the Rayleigh (Ra) and Prandtl (Pr) numbers. Spanning the range at and unity aspect ratios of the vent and the enclosures, the dominant spatial and temporal flow structures, in the asymptotic approach of the system towards a state of thermomechanical equilibrium, have been identified. These dominant modes have been utilized to classify the flowdynamics observed at different Ra into three distinct flow regimes. An approach utilizing new scalar norms to quantify the instantaneous state of mixing and to track the mixing process in time has been utilized to identify the flow modes favoring or opposing the mixing process. It is shown that the flow mode comprising of counterflowing streams in the vent yields the highest mixing rate. It is also shown that this flow mode results in a large buildup of enstrophy in the system. For , an increase in Ra brings about an increase in the overall mixing rate. However, for , there exists a vortex trapped in the vent for a significant length of time, preventing the two fluid masses to mix directly, thereby slowing down the overall mixing rate in comparison to the flows for .

Nonmixing vortex cores in wavy Taylor vortex flow
View Description Hide DescriptionWhen the symmetry of axisymmetric Taylor vortexflow is broken, timeperiodic wavy vortexflow (WVF) appears and quite quickly becomes globally chaotic (in the Lagrangian sense) with increasing Reynolds number. Previously published simulations of WVF suggest that beyond a certain Re, nonmixing vortex cores reappear in the flow and grow in size with further increases in Re. This reappearance occurs well into the inertiadominated flow regime and coincides with a decrease in axial fluid dispersion and an increase in flow symmetry as measured by certain Eulerian symmetry measures. In this brief paper, we present experimental dyereaction visualization results from two WVF wave states in the region where vortex cores are predicted numerically. The experimental results show unambiguous visual evidence for the existence of vortex cores and provide visual agreement with the numerical results. They are significant in that experimental evidence for these structures in WVF has not been reported before. The results also suggest that vortextovortex transport occurs via sheetlike structures that are pulled from one vortex to another and become wrapped around the vortex cores before being stretched to the point at which molecular diffusion dominates.

Streaming potential and electroviscous effects in periodical pressuredriven microchannel flow
View Description Hide DescriptionAn analytical solution for pressuredriven periodical electrokineticflows in a twodimensional uniform microchannel is presented based on the Poisson–Boltzmann equation for electrical double layer and the Navier–Stokes equations for incompressible viscous fluid. The analytical results indicate that the periodical streaming potential strongly depends on the periodical Reynolds number which is a function of the frequency, the channel size, and the kinetic viscosity of fluids. For , the streaming potential behaves similarly to that of steady flow, whereas it decreases rapidly with Re as . In addition, the electroviscous force affects greatly both the periodical flow and streaming potential, particularly when the nondimensional electrokinetic diameter is small. The electroviscous force has been found to depend on three factors: first, the electroviscous parameter, which is defined as the ratio of the maximum electroviscous force to the pressure gradient; second, the distribution parameter describing the distribution of the electroviscous force over the cross section of the microchannel; third, the coupling coefficient, which is a function of both the periodical Reynolds number and electroviscous parameter, determining both the amplitude attenuation and phase offset of the electroviscous force.