Volume 24, Issue 5, May 2012
Index of content:

The effect of weak vertical motion on the dynamics of materials that are limited to move on the oceansurface is an unresolved problem with important environmental and ecological implications (e.g., oil spills and larvae dispersion). We investigate this effect by introducing into the classical horizontal timeperiodic doublegyre model vertical motion associated with diurnal convection. The classical model produces chaotic advection on the surface. In contrast, the weak vertical motion simplifies this chaotic surfacemixing pattern for a wide range of parameters. Melnikov analysis is employed to demonstrate that these conclusions are general and may be applicable to realistic cases. This counter intuitive result that the very weak nocturnal convection simplifies oceansurfacemixing has significant outcomes.
 ARTICLES


Biofluid Mechanics

Dynamics of freely swimming flexible foils
View Description Hide DescriptionWe use modeling and simulations guided by initial experiments to study thin foils which are oscillated at the leading edge and are free to move unidirectionally under the resulting fluid forces. We find resonantlike peaks in the swimming speed as a function of foil length and rigidity. We find good agreement between the inviscid model and the experiment in the foil motions (particularly the wavelengths of their shapes), the dependences of their swimming speeds on foil length and rigidity, and the corresponding flows. The model predicts that the foil speed is proportional to foil length to the −1/3 power and foil rigidity to the 2/15 power. These scalings give a good collapse of the experimental data.

Selfpropulsion in viscoelastic fluids: Pushers vs. pullers
View Description Hide DescriptionWe use numerical simulations to address locomotion at zero Reynolds number in viscoelastic (Giesekus) fluids. The swimmers are assumed to be spherical, to selfpropel using tangential surface deformation, and the computations are implemented using a finite element method. The emphasis of the study is on the change of the swimming kinematics, energetics, and flow disturbance from Newtonian to viscoelastic, and on the distinction between pusher and puller swimmers. In all cases, the viscoelastic swimming speed is below the Newtonian one, with a minimum obtained for intermediate values of the Weissenberg number, We. An analysis of the flow field places the origin of this swimming degradation in nonNewtonian elongational stresses. The power required for swimming is also systematically below the Newtonian power, and always a decreasing function of We. A detail energetic balance of the swimming problem points at the polymeric part of the stress as the primary Wedecreasing energetic contribution, while the contributions of the work done by the swimmer from the solvent remain essentially Weindependent. In addition, we observe negative values of the polymeric power density in some flow regions, indicating positive elastic work by the polymers on the fluid. The hydrodynamic efficiency, defined as the ratio of the useful to total rate of work, is always above the Newtonian case, with a maximum relative value obtained at intermediate Weissenberg numbers. Finally, the presence of polymeric stresses leads to an increase of the rate of decay of the flow velocity in the fluid, and a decrease of the magnitude of the stresslet governing the magnitude of the effective bulk stress in the fluid.

Red blood cell clustering in Poiseuille microcapillary flow
View Description Hide DescriptionRed blood cells (RBC) flowing in microcapillaries tend to associate into clusters, i.e., small trains of cells separated from each other by a distance comparable to cell size. This process is usually attributed to slower RBCs acting to create a sequence of trailing cells. Here, based on the first systematic investigation of collective RBC flow behavior in microcapillaries in vitro by highspeed video microscopy and numerical simulations, we show that RBC size polydispersity within the physiological range does not affect cluster stability. Lower applied pressure drops and longer residence times favor larger RBC clusters. A limiting cluster length, depending on the number of cells in a cluster, is found by increasing the applied pressure drop. The insight on the mechanism of RBC clustering provided by this work can be applied to further our understanding of RBC aggregability, which is a key parameter implicated in clotting and thrombus formation.

Transport of particles by magnetic forces and cellular blood flow in a model microvessel
View Description Hide DescriptionThe transport of particles (diameter 0.56 μm) by magnetic forces in a small blood vessel (diameter D = 16.9 μm, mean velocity U = 2.89 mm/s, red cell volume fraction H _{ c } = 0.22) is studied using a simulation model that explicitly includes hydrodynamic interactions with realistically deformable red blood cells. A biomedical application of such a system is targeted drug or hyperthermia delivery, for which transport to the vessel wall is essential for localizing therapy. In the absence of magnetic forces, it is seen that interactions with the unsteadily flowing red cells cause lateral particle velocity fluctuations with an approximately normal distribution with variance σ = 140 μm/s. The resulting dispersion is over 100 times faster than expected for Brownian diffusion, which we neglect. Magnetic forces relative to the drag force on a hypothetically fixed particle at the vessel center are selected to range from Ψ = 0.006 to 0.204. The stronger forces quickly drive the magnetic particles to the vessel wall, though in this case the red cells impede margination; for weaker forces, many of the particles are marginated more quickly than might be predicted for a homogeneous fluid by the apparently chaotic stirring induced by the motions of the red cells. A corresponding nondimensional parameter Ψ^{′}, which is based on the characteristic fluctuation velocity σ rather than the centerline velocity, explains the switchover between these behaviors. Forces that are applied parallel to the vessel are seen to have a surprisingly strong effect due to the streamwiseasymmetric orientation of the flowing blood cells. In essence, the cells act as lowReynolds number analogs of turning vanes, causing streamwise accelerated particles to be directed toward the vessel center and streamwise decelerated particles to be directed toward the vessel wall.

Interfacial Flows

New exact solutions for HeleShaw flow in doubly connected regions
View Description Hide DescriptionRadial Hele–Shaw flows are treated analytically using conformal mapping techniques. The geometry of interest has a doubly connected annular region of viscous fluid surrounding an inviscid bubble that is either expanding or contracting due to a pressure difference caused by injection or suction of the inviscid fluid. The zerosurfacetension problem is illposed for both bubble expansion and contraction, as both scenarios involve viscous fluid displacing inviscid fluid. Exact solutions are derived by tracking the location of singularities and critical points in the analytic continuation of the mapping function. We show that by treating the critical points, it is easy to observe finitetime blowup, and the evolution equations may be written in exact form using complex residues. We present solutions that start with cusps on one interface and end with cusps on the other, as well as solutions that have the bubble contracting to a point. For the latter solutions, the bubble approaches an ellipse in shape at extinction.

Experimental investigation of convective structure evolution and heat transfer in quasisteady evaporating liquid films
View Description Hide DescriptionThe stability, convective structure, and heat transfer characteristics of upwardfacing, evaporating, thin liquid films were studied experimentally. Dichloromethane, chloroform, methanol, and acetone films with initial thicknesses of 2–5 mm were subjected to constant levels of superheating until film rupture occurred (typically at a thickness of around 50 μm). The films resided on a temperature controlled, polished copper plate incorporated into a closed pressure chamber free of noncondensable gasses. The dynamic film thickness was measured at multiple points using a nonintrusive ultrasound ranging system. Instability wavelength and convective structure information was obtained using doublepass schlieren imaging. The sequence of the convective structures as the film thins due to evaporation is observed to be as follows: (1) large, highly variable cells, (2) concentric rings and spirals, and (3) apparent end of convection. The transition from large, variable cells to concentric rings and spirals occurs at a Rayleigh number of 4800 ± 960. The apparent end of convection occurs at a Rayleigh number of 1580 ± 180. At the cessation of convection, the Nusselt number is nearly unity, indicating that there is little heat transfer in the film due to convection. In films where the Rayleigh number is above this transitional value, the Nusselt number increases with increasing Rayleigh number. The current results suggest that the equilibrium condition at the evaporating surface suppresses surface temperature variation, effectively eliminating thermocapillarydriven instability.

LandauLevich flow visualization: Revealing the flow topology responsible for the film thickening phenomena
View Description Hide DescriptionAn extensive body of experimental work has proven the validity of the analysis of Landau and Levich, who were the first to determine theoretically the thickness of the film deposited by the withdrawal of a flat substrate from a bath of liquid with a clean interface. However, there are a number of experimental investigations that have shown that surfactants in the liquid may result in a thickening of the deposited film. Marangoni phenomena have usually been considered responsible for this effect. However, some careful experiments and numerical simulations reported in the literature seemed to rule out this view as the cause of the observed behavior. Despite all these studies and the number of reports of film thickening, an experimental study of the flow field close to the coated substrate in the presence of surfactants has never been undertaken. In this paper we will present a set of flow visualization experiments on coating of a planar substrate in the range of capillary numbers 10^{−4} ≲ Ca ≲ 10^{−3} for sodium dodecyl sulfate solutions with bulk concentrations of 0.25 CMC ⩽ C ⩽ 5.0 CMC (critical micelle concentration). It was evident during experiments that the flow field near the meniscus region exhibits patterns that can only be explained with a stagnation point residing in the bulk and not at the interface. As opposed to patterns with an interfacial stagnation point, the observed flow fields allow for the increase in film thickness due to the presence of surfactants compared to the clean interface case.

Numerical investigation of collapsing cavity arrays
View Description Hide DescriptionIn most technical applications involving cavitation, vapor bubbles occur in clouds, and their collapse is affected by the interaction with neighboring bubbles. One approach to study the influence of these interactions is the investigation of the collapse of cavity arrays in water under shock wave loading. We describe in detail the collapse mechanisms during the collapse of a horizontal cavity array, with particular consideration of maximum pressures. As general trend, we find a pressure amplification in consecutive cavity collapses. However, by increasing the number of cavities, we are able to demonstrate that the amplification is not monotonic. A parameter study of the bubble separation distance in horizontal arrays shows that a smaller distance generally, but not necessarily, results in larger collapse pressure. Exceptions from the general trend are due to the very complex shock and expansionwave interactions and demonstrate the importance of using stateoftheart numerical methods. By varying boundary conditions, we illustrate the significance of large test sections in experimental investigations, as the expansion wave emitted at a free surface has a large effect on the collapse dynamics.

Acoustically enhanced boiling heat transfer
View Description Hide DescriptionAn acoustic field generated by a lightweight, lowpower acoustic driver is shown to increase the critical heat flux during pool boiling by about 17%. It does this by facilitating the removal of vapor bubbles from the heated surface and suppressing the instability that leads to the transition to film boiling at the critical heat flux. Bubble removal is enhanced because the acoustic field induces capillary waves on the surface of a vapor bubble that interact with the bubble contact line on the heated surface causing the contact line to contract and detach the bubble from the surface. The acoustic field also produces a radiation pressure that helps to facilitate the bubble detachment process and also suppresses the transition to film boiling. The mechanisms associated with these interactions are explored using three different experimental setups with acoustic forcing: an air bubble on the underside of a horizontal surface, a single vapor bubble on the top side of a horizontal heated surface, and pool boiling from a horizontal heated surface. Measurements of the capillary waves induced on the bubbles,bubble motion, and heat transfer from the heated surface were performed to isolate and identify the dominant forces involved in these acoustically forced motions.

An experimental study of small Atwood number RayleighTaylor instability using the magnetic levitation of paramagnetic fluids
View Description Hide DescriptionExperiments that take advantage of the properties of paramagnetic liquids are used to study RayleighTaylor (RT) instability. A gravitationally unstable, miscible combination of a paramagnetic salt solution and a nonmagnetic solution is initially stabilized by a magnetic field gradient that is produced by the contoured polecaps of a large electromagnet. RayleighTaylor instability originates from infinitesimal random background noise with the rapid removal of current from the electromagnet, which results in the heavy liquid falling into the light liquid due to gravity and, thus, mixing with it. The mixing zone is visualized by backlit photography and is recorded with a digital video camera. Several miscible, small Atwood number (A ⩽ 0.1) combinations of paramagnetic and nonmagnetic solutions are used. It is found that the RT flow is insensitive to the viscosities of the fluids composing the twofluid system, and that the growth parameter α also does not show dependence on the Atwood number when the experiments are initialized under the same conditions. It is also observed that the turbulent mixing zone grows linearly with time following a period of selfsimilar quadratic growth. When the width of the mixing zone becomes comparable with the crosssectional length scale of the experimental container, the bubble front characteristic velocity approaches a constant value, similar to that observed with a single bubble rising in the confined volume, with Froude number measured in the range Fr = 0.38÷0.45. However, flow visualization does not reveal any persistent largescale perturbations, which would dominate the flow during this stage. We believe that this phenomenon is not an attribute of the given magnetic experiments and has been observed in many other experimental studies, which involve RT instability evolving in confined volumes.

Viscous and NonNewtonian Flows

Internal dynamics of Newtonian and viscoplastic fluid avalanches down a sloping bed
View Description Hide DescriptionWe experimentally investigated the spreading of fluid avalanches (i.e., fixed volumes of fluid) down an inclined flume. Emphasis was given to the velocity field within the head. Using specific imaging techniques, we were able to measurevelocity profiles within the flowing fluid far from the sidewalls. We studied the behavior of Newtonian and viscoplastic fluids for various flume inclinations and initial masses. For the Newtonian fluids tested (glycerol and Triton X100), we compared the measuredvelocity field with that predicted by lubricationtheory. Provided that the flow Reynolds numberRe was sufficiently low (typically Re < 1), there was excellent agreement between theory and experiment except for the very thin region just behind the contact line. For higher Reynolds numbers (typically Re ∼ 10), the discrepancy between theory and experiment was more marked (relative errors up to 17% for the body). As viscoplastic materials, we used Carbopol ultrez 10. For the body, agreement between theoretical and measuredvelocity profiles was fairly satisfactory whereas it was very poor for the tip region as the curvature of the free surface became more pronounced: the velocities were not only much lower than those predicted by lubricationtheory, but there was also evidence of slipping in the flow part adjacent to the contact line.

Visualization of the flow profile inside a thinning filament during capillary breakup of a polymer solution via particle image velocimetry and particle tracking velocimetry
View Description Hide DescriptionWe investigated the flow profile of a polymer solution in a thinning capillary bridge. Fluorescent tracer particles with a diameter of 3 μm were used to visualize the flow. The cylindrical shape of the filament introduced strong optical abberations that could be corrected for, and we were able to characterize the flow in filaments with a thickness ranging from 150 to 30 μm. In the first regime when the filament was still sufficiently large, we used a PIV algorithm to deduce the flow field. At later stages when the number of particles in the observation plane decreased a PTV algorithm was used. The main two results of our measurements are as follows. First, the flow profile at the formation of the cylindrical filament is highly inhomogeneous and there is only flow in the outer parts of the filament. Second, we find that in most parts of the regime, where the temporal radius of the thinning filament can be fitted with an exponential law the flow indeed is purely extensional.

On the effects of mass and momentum transfer from droplets impacting on steady twodimensional rimming flow in a horizontal cylinder
View Description Hide DescriptionMotivated by applications in aeroengines, steady twodimensional thinfilmflow on the inside of a circular cylinder is studied when the filmsurface is subject to mass and momentum transfer from impacting droplets. Asymptotic analysis is used systematically to identify distinguished limits that incorporate these transfer effects at leading order and to provide a new mathematical model. Applying both analytical and numerical approaches to the model, a set of stable steady, twodimensional solutions that fit within the rational framework is determined. A number of these solutions feature steep fronts and associated recirculating pools, which are undesirable in an aeroengine since oil may be stripped away from the steep fronts when there is a core flow external to the film, and recirculation may lead to oil degradation. The model, however, provides a means of investigating whether the formation of the steep fronts on the filmsurface and of internal recirculation pools can be delayed, or inhibited altogether, by designing jets to deliver prescribed distributions of oil droplets or by the judicious siting of oil sinks. Moreover, by studying pathlines, oilresidence times can be predicted and systems optimized.

Particulate, Multiphase, and Granular Flows

Flow properties of particles in a model annular shear cell
View Description Hide DescriptionIn order to quantitatively investigate the mechanical and rheological properties of solid flow in a shear cell under conditions relevant to those in an annular cell, we performed a series of discrete particle simulations of slightly polydispersed spheres from quasistatic to intermediate flow regimes. It is shown that the average values of stress tensor components are uniformly distributed in the cell space away from the stationary walls; however, some degree of inhomogeneity in their spatial distributions does exist. A linear relationship between the (internal/external) shear and normal stresses prevails in the shear cell and the internal and external friction coefficients can compare well with each other. It is confirmed that annular shear cells are reasonably effective as a method of measuring particle flow properties. The socalled Irheology proposed by Jop et al. [Nature (London)441, 727 (2006)] is rigorously tested in this cell system. The results unambiguously display that the Irheology can effectively describe the intermediate flow regime with a high correlation coefficient. However, significant deviations take place when it is applied to the quasistatic regime, which corresponds to very small values of inertial number.

Transition due to base roughness in a dense granular flow down an inclined plane
View Description Hide DescriptionParticle simulations based on the discrete element method are used to examine the effect of base roughness on the granular flow down an inclined plane. The base is composed of a random configuration of fixed particles, and the base roughness is decreased by decreasing the ratio of diameters of the base and moving particles. A discontinuous transition from a disordered to an ordered flow state is observed when the ratio of diameters of base and moving particles is decreased below a critical value. The ordered flowing state consists of hexagonally close packed layers of particles sliding over each other. The ordered state is denser (higher volume fraction) and has a lower coordination number than the disordered state, and there are discontinuous changes in both the volume fraction and the coordination number at transition. The Bagnold law, which states that the stress is proportional to the square of the strain rate, is valid in both states. However, the Bagnold coefficients in the ordered flowing state are lower, by more than two orders of magnitude, in comparison to those of the disordered state. The critical ratio of base and moving particle diameters is independent of the angle of inclination, and varies very little when the height of the flowing layer is doubled from about 35 to about 70 particle diameters. While flow in the disordered state ceases when the angle of inclination decreases below 20°, there is flow in the ordered state at lower angles of inclination upto 14°.

Suspension flow and sedimentation in selfaffine fractures
View Description Hide DescriptionThe transport and gravitational sedimentation of a particulate suspension in fracture joints with selfaffinely rough walls is studied by lattice Boltzmann numerical simulations. We consider either homogeneous or bidisperse distributions of nonBrownian spheres in a Newtonian fluid, driven through a fracture by a pressure gradient, and acted upon by gravity. Most results concern the case of open fractures, in which the two walls of the channel do not approach closely enough to block the flow. We present profiles of particle density and profiles of particle and fluid velocities, along with total flow rates and characterizations of the sediment, for three values of particle concentration and a range of buoyancy and Reynolds numbers, principally in the inertial regime. We systematically study the effects of increasing the pressure gradient and the strength of sedimentation and compare the results to those for channel bounded by flat surfaces. We find that both the flow rate and the average particle velocity for flows through an open fracture, when suitably normalized, depend only on the volume fraction of the particles and the buoyancy number in the steady state regardless of the pressure drop, and observe interesting scaling laws in the large buoyancy number limit. We also investigate the possibility for correlations between the surface morphology of the sediment region and the geometry of the underlying fracturesurface in the strong sedimentation limit, but no evidence for correlation is found.

Laminar Flows

Lowdimensional modeling of streaks in a wedge flow boundary layer
View Description Hide DescriptionThis paper is concerned with the low dimensional structure of optimal streaks in a wedge flow boundary layer, which have been recently shown to consist of a unique (up to a constant factor) threedimensional streamwise evolving mode, known as the most unstable streaky mode. Optimal streaks exhibit a still unexplored/unexploited approximate selfsimilarity (not associated with the boundary layer selfsimilarity), namely the streamwise velocity rescaled with their maximum remains almost independent of both the spanwise wavenumber and the streamwise coordinate; the remaining two velocity components instead do not satisfy this property. The approximate selfsimilar behavior is analyzed here and exploited to further simplify the description of optimal streaks. In particular, it is shown that streaks can be approximately described in terms of the streamwise evolution of the scalar amplitudes of just three onedimensional modes, providing the wall normal profiles of the streamwise velocity and two combinations of the cross flow velocity components; the scalar amplitudes obey a singular system of three ordinary differential equations (involving only two degrees of freedom), which approximates well the streamwise evolution of the general streaks.

An efficient approach for eigenmode analysis of transient distributive mixing by the mapping method
View Description Hide DescriptionThe mapping method is an efficient tool to investigate distributive mixing induced by periodic flows. Computed only once, the mapping matrix can be applied a number of times to determine the distribution of concentration inside the flow domain. Spectral analysis of the mapping matrix reveals detailed properties of the distributive mixing as all relevant information is stored in its eigenmodes. Any vector that describes a distribution of concentration can be expanded in the complete system of linearly independent eigenvectors of the mapping matrix. The rapid decay of the contribution of each mode in the eigenmode decomposition allows for a truncation of the eigenmode expansion from the whole spectrum to only the dominant eigenmodes (characterized by a decay rate significantly lower than the duration of the mixing process). This truncated decomposition adequately represents the distribution of concentration inside the flow domain already after a low number of periods, because contributions of all nondominant eigenmodes rapidly become insignificant. The truncation is determined independently of the initial distribution of concentration and based on the decay rates of the eigenmodes, which are inversely proportional to the corresponding eigenvalues. Only modes with eigenvalues above a certain threshold are retained. The key advantage of the proposed compact eigenmode representation of the mapping method is that it includes practically relevant transient states and not just the asymptotic one. As such the method enables an eigenmodeanalysis of realistic problems yet with a substantial reduction in computational effort compared to the conventional approach.

Two regimes of selfpropelled motion of a torus rotating about its centerline in a viscous incompressible fluid at intermediate Reynolds numbers
View Description Hide DescriptionIn the present work, the problem of the motion of a selfpropelled torus in a viscous incompressible fluid is investigated numerically. The surface of the torus rotates with constant velocity around its centerline. The rotating boundary of a torus generates inertia in the surrounding fluid. The outer and inner surfaces produce inertia in opposite directions. There are two selfmotion regimes. In one of them, the torus moves in the direction of the inner surfacemotion due to the larger production of inertia by the outer portion of the torus boundary. The direction of propulsion is the same as in the case of a zero Reynolds number. In the other regime the torus moves in opposite direction due to the high momentum flux associated with the jet of fluid expelled from the hole. The drag coefficients and flowpatterns are analyzed at Reynolds numbersRe = 20 − 60, (Reynolds number defined by velocity of a uniform stream and a smaller diameter of torus), the aspect ratios Ar = 2, 3 (aspect ratio defined as ratio of torus diameter to crosssection diameter), and a range of rotational rate −5.6 ⩽ α ⩽ 2.5 (α defined as ratio of tangential tanktreading motion of torus surface to the uniform farfield velocity).

Instability and Transition

Experimental investigation of reshocked spherical gas interfaces
View Description Hide DescriptionThe evolution of a spherical gas interface under reshock conditions is experimentally studied using the highspeed schlieren photography with high time resolutions. A number of experimental sets of helium or SF_{6}bubble surrounded by air for seven different end wall distances have been performed. Distinct flow structures are observed due to the additional vorticity and wave configuration caused by the reshock. In the air/helium case, the deformation of the reshocked bubble is dependent on the development of the penetrating air jet along the symmetry axis of the bubble. In general, two separate vortex rings can be observed, i.e., one develops slowly, and the other approaches and eventually impinges on the shock tube end wall. In the air/SF_{6} case, two SF_{6} jets moving in opposite directions are generated and the oscillation of the interface is observed for small end wall distances, while small scale vortex morphologies on the gas interface are found for large end wall distances. The physical mechanisms of the baroclinic vorticity generation and the pressure perturbation are highlighted in the interface evolution process. Based on the sequence of the schlieren images obtained during a single run for each case, the xt diagrams of the shock and reshock interacting with the helium or SF_{6}bubble are plotted and the velocities estimated in linear stages are compared with those calculated from onedimensional gas dynamics. The changes with time in the characteristic bubble sizes including the interface length, height, and vortex diameter are also measured.
