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Volume 17, Issue 10, October 2005
 ARTICLES

 Interfacial Flows

Nonlinear oscillations and collapse of elongated bubbles subject to weak viscous effects
View Description Hide DescriptionThe weak viscous oscillations of a bubble are examined, in response to an elongation that perturbs the initial spherical shape at equilibrium. The flow field in the surrounding liquid is split in a rotational and an irrotational part. The latter satisfies the Laplacian and can be obtained via an integral equation. A hybrid boundaryfinite element method is used in order to solve for the velocity potential and shape deformation of axisymmetric bubbles. Weak viscous effects are included in the computations by retaining firstorder viscous terms in the normal stress boundary condition and satisfying the tangential stress balance. An extensive set of simulations was carried out until the bubble either returned to its initial spherical shape, or broke up. For a relatively small initial elongation the bubble returned to its initial spherical state regardless of the size of the Ohnesorge number; . For larger initial elongations there is a threshold value in above which the bubble eventually breaks up giving rise to a “donut” shaped larger bubble and a tiny satellite bubble occupying the region near the center of the original bubble. The latter is formed as the round ends of the liquid jets that approach each other from opposite sides along the axis of symmetry, coalesce. The size of the satellite bubble decreased as the initial elongation or increased. This pattern persisted for a range of large initial deformations with a decreasing threshold value of the as the initial deformation increased. As its equilibrium radius increases the bubble becomes more susceptible to the above collapse mode. The effect of initial bubble overpressure was also examined and it was seen that small initial overpressures, for the range of initial bubble deformations that was investigated, translate the threshold of to larger values while at the same time increasing the size of the satellite bubble.

Thermocapillary instability of coreannular flows
View Description Hide DescriptionThermocapillary instability of a coreannular flow is asymptotically examined in the thin annulus limit. Two sets of scalings are established to study the interplays between base flows,interfacial tension, and thermocapillary effects. For each scaling case, an interfacial evolution equation is derived for describing the leading order stability of the system. Both linear and weakly nonlinear stabilities are examined. When the core fluid is warmer (cooler) than the wall, thermocapillarity linearly stabilizes (destabilizes) the system, and hence suppresses (promotes) the capillary instability. For a moderate thermocapillary force and a strong capillary force, the linear instability can be arrested within the weakly nonlinear regime. For a weak thermocapillary force and a moderately strong interfacial tension, the weakly nonlinear evolution is governed by a modified KuramotoSivashinsky equation. The influence of thermocapillarity on the route to chaos is discussed.

Dynamics and stability of a thin liquid film on a heated rotating disk film with variable viscosity
View Description Hide DescriptionA theoretical analysis of the thermal effects on the dynamics of a thin nonuniform film of a nonvolatile incompressible viscous fluid on a heated rotating disk has been considered and the effects of temperaturedependent viscosity and surface tension have been analyzed. A nonlinear evolution equation describing the shape of the film interface has been derived as a function of space and time and its stability characteristics have been examined using linear theory. It has been observed that the infinitesimal disturbances decay for small wave numbers and are transiently stable for large wave numbers, for both zero and nonzero values of Biot number.

Steady and transient thinjet flow
View Description Hide DescriptionThe interplay between inertia and gravity is examined in this theoretical study for the steady and transient twodimensional thin jet flow free of surface tension. The fluid emerges from a channel and is driven by both a pressure gradient maintained inside the channel and/or gravity. The flow is dictated by the thinfilmequations of the boundary layer type, which are solved by expanding the flow field in terms of orthonormal modes depthwise, and using the Galerkin projection. The strength of inertia relative to gravity is found to be of crucial significance on the film flow. Transient behavior of the film is closely examined for various flow parameters, initial and exit conditions. It is shown that under a wide range of flow parameters, the steady state cannot be achieved.

Dip coating in the presence of a substrateliquid interaction potential
View Description Hide DescriptionIn this work we investigate theoretically the LandauLevich problem of dip coating in the presence of a strong interaction potential normal to the substrate. This study is motivated by dip coating at very low capillary numbers when the deposited film thickness is less than and such interaction forces become important. The objective of this work is to demonstrate that in the presence of an extra body force the solution procedure differs significantly from the classical one and leads to substantial deviations from the LandauLevich law for the entrained film thickness. In particular, attractive potentials produce film thickening and the resulting film thickness is independent of speed to lowest order. Repulsive potentials bring about more complicated behavior and lead either to films whose thickness is also independent of speed, or to a modification of the leading order constant in the classical law. Demonstration of these effects is given for a model potential. The analysis is generally applicable to many physical situations when there is an interaction between a coating liquid and a substrate, e.g., dip coating of ferromagnetic liquids on magnetic substrates, or dip coating of liquids carrying charges.

Laserinduced motion in nanoparticle suspension droplets on a surface
View Description Hide DescriptionThe fluid and particle motion in a volatile colloidal nanoparticle suspension droplet (“nanoink”) spreading on a flat surface upon local heating through a laser beam is investigated numerically. The laser diameter, laser intensity, and the absorption coefficient of the nanoink as well as the substrate thermal diffusivity were varied. The simulations are conducted with a finiteelement method discretization of the extended axisymmetric NavierStokes equations in Lagrangian coordinates, accounting for evaporation, thermocapillarity, and Youngforcedriven wetting for the fluid phase as well as for inertiacontrolled particle motion for the solid phase. An additional continuous particle coagulation model with a locally monodispersed particle distribution is solved on the locations of the discrete computational particles for example cases. The localized heating leads, through the action of thermocapillarity, to a displacement of the liquid in the radial (outward) direction. A dimple in the droplet center region is formed as a consequence, which becomes flattened for larger laser beam diameters due to a significant enlargement in spreading. Substrates with high thermal diffusivity or large thermal contact resistance to the liquid can inhibit the Marangoniinduced enlargement of the droplet footprint. The coagulation model predicts for large absorption coefficients particle clustering primarily at the free surface, which prevents the formation of structures (built by the coagulated nanoparticles) with a uniform thickness.

Interfacial waves due to a singularity in a system of two semiinfinite fluids
View Description Hide DescriptionThe threedimensional interfacial waves due to a fundamental singularity steadily moving in a system of two semiinfinite immiscible fluids of different densities are investigated analytically. The two fluids are assumed to be incompressible and homogenous. There are three systems to be considered: one with two inviscid fluids, one with an upper viscous and a lower inviscid fluid, and one with an upper inviscid and a lower viscous fluid. The Laplace equation is taken as the governing equation for inviscid flows while the steady Oseen equations are taken for viscous flows. The kinematic and dynamic conditions on the interface are linearized for smallamplitude waves. The singularity immersed above or beneath the interface is modeled as a simple source in the inviscid fluid while as an Oseenlet in the viscous fluid. Based on the integral solutions for the interfacial waves, the asymptotic representations of wave profiles in the far field are explicitly derived by means of Lighthill’s twostage scheme. An analytical solution is presented for the density ratio at which the maximum wave amplitude occurs. The effects of density ratio, immersion depth, and viscosity on wave patterns are analytically expressed. It is found that the wavelength of interfacial waves is elongated in comparison with that of freesurface waves in a single fluid.

Experimental study of substrate roughness and surfactant effects on the LandauLevich law
View Description Hide DescriptionIn this work we present an experimental study of deviations from the classical LandauLevich law in the problem of dip coating. Among the examined causes leading to deviations are the nature of the liquidgas and liquidsolid interfaces. The thickness of the coating film created by withdrawal of a plate from a bath was measured gravimetrically over a wide range of capillary numbers for both smooth and wellcharacterized rough substrates, and for clean and surfactantinterface cases. In view of the dependence of the lifetime of a film on the type of liquid and substrate, and liquidgas and liquidsolid interfaces, we characterized the range of measurability of the film thickness in the parameter space defined by the withdrawal capillary number, the surfactant concentration, and substrate roughness size. We then study experimentally the effect of a film thickening due to the presence of surfactants. Our recent theory based on a purely hydrodynamic role of the surface active substance suggests that there is a sorptioncontrolled coating regime in which Marangoni effects should lead to film thinning. However, our experiments conducted in this regime demonstrate film thickening, calling into question the conventional wisdom, which is that Marangoni stresses (as accounted by the conventional interfacial boundary conditions) lead to film thickening. Next we examine the effect of wellcharacterized substrate roughness on the coated film thickness, which also reveals its influence on wettingrelated processes and an effective boundary condition at the wall. In particular, it is found that roughness results in a significant thickening of the film relative to that on a smooth substrate and a different power of capillary number than the classical LandauLevich law.
 Viscous and NonNewtonian Flows

Validity of the Taylor hypothesis in a random spatially smooth flow
View Description Hide DescriptionThe validity of the Taylor frozen flow hypothesis in a chaotic flow of a dilute polymer solution in a regime of elasticturbulence is investigated experimentally. By accurate timedependent measurements of the flow field we study the velocity coherence between pairs of points displaced both in time and space and quantify the degree of applicability of the Taylor hypothesis. Alternatively, the frozen flow assumption is assessed by comparison of the measuredvelocity structure functions with the ones derived by a frozen flow assumption. The breakdown of the Taylor hypothesis is further discussed in both the context of strong velocityfluctuations and longrange spatial correlations, which are the result of the flow smoothness and lack of scale separation. Different choices of the advection velocity are tested and discussed.

Brownian motion near a partialslip boundary: A local probe of the noslip condition
View Description Hide DescriptionMotivated by experimental evidence of violations of the noslip boundary condition for liquidflow in micrometerscale geometries, we propose a simple, complementary experimental technique that has certain advantages over previous studies. Instead of relying on externally induced flow or probe motion, we suggest that colloidal diffusivity near solid surfaces contains signatures of the degree of fluid slip exhibited on those surfaces. To investigate, we calculate the imagesystem for point forces (Stokeslets) oriented perpendicular and parallel to a surface with a finite slip length, analogous to Blake’s solution for a Stokeslet near a noslip wall. Notably, the imagesystem for the point source and perpendicular Stokeslet contain the same singularities as Blake’s solution; however, each is distributed along a line with a magnitude that decays exponentially over the slip length. The imagesystem for the parallel Stokeslet involves a larger set of fundamental singularities, whose magnitude does not decay exponentially from the surface. Using these imagesystems, we determine the wallinduced correction to the diffusivity of a small spherical particle located “far” from the wall. We also calculate the coupled diffusivities between multiple particles near a partially slipping wall. Because, in general, the diffusivity depends on “local” wall conditions, patternedsurfaces would allow differential measurements to be obtained within a single experimental cell, eliminating potential celltocell variability encountered in previous experiments. In addition to motivating the proposed experiments, our solutions for point forces and sources near a partialslip wall will be useful for boundary integral calculations in slip systems.

Deformation of a surfactantcovered drop in a linear flow
View Description Hide DescriptionWe study the effect of adsorbed surfactant on dropdeformation in linear flows by means of analytical solutions for small perturbations of the drop shape and surfactant distribution, and by numerical simulations for large distortions. We consider a drop with the same viscosity as the suspending fluid. Under these conditions, the problem simplifies because the disturbance flow field results solely from the interfacial stresses that oppose the distortion of shape and surfactant distribution induced by the incident flow. A general form of perturbation analysis valid for any flow is presented. The analysis can be carried out to arbitrary order given its recursive structure; a thirdorder perturbation solution is explicitly presented. The expansions are compared to results from boundary integral simulations for drops in axisymmetric extensional and simple shear flows. Our results indicate that under weakflow conditions, deformation is enhanced by the presence of surfactant, but the leadingorder perturbation of the drop shape is independent of the (nonzero) surfactantelasticity. In strong flows,dropdeformation depends nonmonotonically on surfactantelasticity. The nonNewtonian rheology in a dilute emulsion that results from dropdeformation and surfactant redistribution is predicted. Shear thinning is most pronounced for low values of the surfactantelasticity. In the weakflow limit with finite surfactantelasticity, the emulsion behaves as a suspension of rigid spheres. In strong flows, the stresses can approach the behavior for surfactantfree drops.
 Particulate, Multiphase, and Granular Flows

Submarine granular flows down inclined planes
View Description Hide DescriptionSubmarine flows of granular material down a rough inclined plane are experimentally investigated. We focus on the dense flow regime when the whole sediment layer is flowing down the slope and when no deposition nor entrainment occurs. In this regime, steady uniform flows are observed for which we systematically measure the depthaveraged velocity, the thickness, and the excess pore pressure for different inclinations and different flow rates. The experimental measurements are analyzed within a theoretical approach inspired by recent results obtained for dry granular flows. The basic assumption of the model is that the constitutive law obtained in the dry case still holds for submarine flow, if one substitutes the inertial time scale coming into play in the rheology by a viscous time scale. The agreement between the measurements and the theory supports this assumption.

Granular slumping on a horizontal surface
View Description Hide DescriptionWe report the results of an experimental investigation of the flow induced by the collapse of a column of granular material (glass beads of diameter ) over a horizontal surface. Two different setups are used, namely, a rectangular channel and a semicircular tube, allowing us to compare twodimensional and axisymmetric flows, with particular focus on the internal flow structure. In both geometries the flowdynamics and the deposit morphologies are observed to depend primarily on the initial aspect ratio of the granular column , where is the height of the initial granular column and its length along the flow direction. Two distinct regimes are observed depending on : an avalanche of the column flanks producing truncated deposits for small and a column free fall leading to conical deposits for large . In both geometries the characteristic time scale is the free fall of the granular column . The flow initiated by Coulomblike failure never involves the whole granular heap but remains localized in a surface layer whose size and shape depend on and vary in both space and time. Except in the vicinity of the pile foot where the flow is pluglike, velocity profiles measured at the side wall are identical to those commonly observed in steady granular surface flows: the velocity varies linearly with depth in the flowing layer and decreases exponentially with depth in the static layer. Moreover, the shear rate is constant, , independent of the initial aspect ratio, the flow geometry, position along the heap, or time. Despite the rather complex flowdynamics, the scaled deposit height and runout distance both exhibit simple power laws whose exponents depend on and on the flow geometry. We show that the physical origin of these power laws can be understood on the basis of a dynamic balance between acceleration, pressure gradient, and friction forces at the foot of the granular pile. Two asymptotic behaviors can be distinguished: the flow is dominated by friction forces at small and by pressure forces at large . The effect of the flow geometry is determined primarily by mass conservation and becomes important only for large .

Numerical simulation of twodimensional steady granular flows in rotating drum: On surface flow rheology
View Description Hide DescriptionThe rheology of twodimensional steady surfaceflow of cohesionless cylinders in a rotating drum is investigated through nonsmooth contact dynamics simulations. Profiles of volume fraction, translational and angular velocity, rms velocity, strain rate, and stress tensor are measured at the midpoint along the length of the surfaceflowing layer, where the flow is generally considered as steady and homogeneous. Analysis of these data and their interrelations suggest the local inertial number—defined as the ratio between local inertial forces and local confinement forces—to be the relevant dimensionless parameter to describe the transition from the quasistatic part of the packing to the flowing part at the surface of the heap. Variations of the components of the stress tensor as well as the ones of rms velocity as a function of the inertial number are analyzed within both the quasistatic and the flowing phases. Their implications are discussed.

Onset of erosion and avalanche for an inclined granular bed sheared by a continuous laminar flow
View Description Hide DescriptionThe transport inception of immersed grains is studied experimentally with laminar flow conditions in a HeleShaw cell when varying the tilt angle of the cell and the water flow rate. Varying these two parameters, grains are either motionless, rolling on the bed surface, or avalanching downwards. This paper focuses on the determination of the onset for grain motion either by erosion or by avalanche. For a horizontal interface, onset for erosion corresponds to a constant critical Shields number at small particle Reynolds number but decreases as at larger particle Reynolds number. For tilted bed, the onset of erosion increases when the flow is opposed to gravity. Both results are compared to a standard model based on a balance of the forces acting on a single grain lying on a tilted plane. When tilt angles are large, avalanches occur. The maximum angle of stability is modified by the flow and increases slightly when the flow acts against gravity. This behavior is compared to a continuous model where a few layers of grains are about to slide.
 Laminar Flows

On molecular transport effects in real gas laminar diffusion flames at large pressure
View Description Hide DescriptionDirect numerical simulations are conducted of unsteady, exothermic and onedimensional laminar diffusionflames at large pressures. The simulations are used to assess the impact of molecular diffusion and real gas effects under high pressure conditions with simplified chemical kinetics. The formulation includes the fully compressible form of the governing equations, real gas effects modeled by the cubic Peng–Robinson equation of state, and a generalized form of the Soret and Dufour mass and heat diffusion vectors derived from nonequilibrium thermodynamics and fluctuation theory. The cross diffusion fluxes are derived for a ternary species system and include the effects of both heat and mass diffusion in the presence of temperature, concentration and pressure gradients (i.e., Soret and Dufour diffusion). The ternary species formulation is applied to a simplified single step reaction elucidating molecular and thermodynamic effects apparent in general combustion. Realistic models for pressure, temperature and species dependent heat capacities, viscosities, thermal conductivities and mass diffusivities are also included. Three different model reactions are simulated both including and neglecting Soret and Dufour cross diffusion. The simulation results show that Soret and Dufour effects are negligible for reactions comprised of species with equal or near equal molecular weights. However, Soret diffusion effects are apparent when species with nonequal molecular weights are involved in the reaction and result in reductions of the peak flame temperature. In addition, it is shown that neglect of cross diffusion leads to deviations in the predicted flame thicknesses, with under predictions for a hydrogenoxygen system and over predictions for a heavy hydrocarbon reaction. These effects are explained in detail through examinations of the individual heat and mass flux vectors as well as through associated thermodynamic properties. A parametric study addresses the effects of the ambient pressure, the initial “flame Reynolds number,” the Damkohler number and the heat release parameter.

Characteristics of laminar flow past a sphere in uniform shear
View Description Hide DescriptionNumerical simulations are performed to investigate the characteristics of laminar flow past a sphere in uniform shear. The Reynolds numbers considered are , 425, and 480 based on the inlet center velocity and sphere diameter . The nondimensional shear rate of inlet uniform shear is varied from 0 to 0.15, where () and is the shear rate at inlet. For all Reynolds numbers investigated, the head of the hairpin vortex loop is always located on the highvelocity side in uniform shear. The flow maintains planar symmetry at . At and 480, the temporal variation in the azimuthal angle of the hairpin vortex formation appearing in the uniform inlet flow is greatly reduced in uniform shear, but the flows still keep asymmetry for most inlet shear rates. However, in the cases of and 0.1, at , the flows become planar symmetric and their characteristics of formation and evolution of the hairpin vortex loops are different from those of asymmetric flows. In most cases, except the instances showing planar symmetry at , the Strouhal number and timeaveraged drag and lift coefficients increase with increasing inlet shear rate. On the other hand, for and 0.1, showing planar symmetry at , three different vortices are shed in the wake, resulting in three distinct peak frequencies. Finally, a hysteresis phenomenon switching from planar symmetry to asymmetry (or vice versa) depending on the initial condition is observed at and 450, implying that small variations in the flow or initial conditions change the flow field at these Reynolds numbers.

The dragout problem in film coating
View Description Hide DescriptionThe classical coating flow problem of determining the asymptotic film thickness (and hence the load) on a flat plate being withdrawn vertically from an infinitely deep bath is examined via a numerical solution of the steadystate NavierStokes equations. Under creeping flow conditions, the dimensionless load is computed as a function of the capillary number and, for , is found to agree with Wilson’s extension [J. Eng. Math.16, 209 (1982)] of Levich’s wellknown expression. On the other hand, for , asymptotes to 0.582, well below the value of postulated by Deryagin and Levi [Film Coating Theory (Focal, London, 1964)]. For finite Reynolds numbers, where is a dimensionless number involving only the gravitational acceleration and the properties of the fluid, is found to remain essentially independent of at a given , but only up to a critical capillary number , dependent on , beyond which our numerical scheme failed. Analogous results, but only for creeping flows, are presented for the case where the plate is inclined at an angle from the vertical. Here, the corresponding dimensionless flow rate depends on both and , and its maximum is found to increase monotonically with and to become equal to when exceeds a critical angle , where the plate is inclined midway to the horizontal with its coating surface on the topside.

Evolution of zeromean magnetic fields in cellular flows
View Description Hide DescriptionThis paper presents the results of numerical simulations to examine the evolution of a zeromean magnetic field in a steady cellular flow. In the kinematic model the evolution consists of two phases. In general, the time scale for decay in these two phases are those presented by Rhines and Young [“How rapidly is a passive scalar mixed within closed streamlines,” J. Fluid Mech.133, 135 (1983)]. However, we show that there is a case for which the magnetic energy does not decay on these time scales. In the dynamic model, the evolution is dependent on the strength of the imposed field. For weak fields, the evolution can be satisfactorily approximated by that for the kinematic model. However, as the field strength is increased the approximation ceases to be valid. For stronger fields, there are three primary phases to the decay. For such fields there is an additional phase, in between those for the kinematic model, where the field decays at the ohmic rate.

Migration of twisted ribbonlike particles in simple shear flow
View Description Hide DescriptionThe motion of rigid twisted ribbonlike particles in shear flow is studied by computer simulation. It is shown that the twisted ribbon migrates in the vorticity direction in strong shear flow with the sign determined by the chirality: the righthanded particles and their mirrorimaged particles move in opposite directions. This enables the separation of chiral particles from the racemic mixture. The average migration velocity in the vorticity direction is studied as a function of the shear rate, the length, and the pitch of the particle.