Volume 19, Issue 11, November 2007

Vertical rotating viscous liquid jet experiments show a clear preference for helical instabilities that evolve from initially planar disturbances at large rotation rates for fixed fluid properties. The laboratory setup for the experiments described herein was chosen as the nearest earthbased equivalent to a uniformly rotating viscous liquid column in the absence of gravity. In the ideal situation with stressfree boundaries, the preferred modes of linear temporal instability are theoretically known over the entire physical domain spanned by the Hocking parameter and the rotational Reynolds number, where is the column radius, is its uniform angular velocity, and , , and are, respectively, the fluid density, kinematic viscosity, and surface tension. The theoretical results show that instability in parameter space is dominated by three mode types: The axisymmetric mode, the planar modes, and the first spiral mode. Experiments reveal that, in the region for which the uniformly rotating liquid column is dominated by planar modes of instability, the rotating liquid jet spontaneously gives rise to planar disturbances of mode that rapidly evolve into helical instabilities. However, these observed instabilities are not the spiral normal modes that exist for as posited in linear stability theory. In spite of obvious fundamental differences between the rotating liquid jet and the uniformly rotating liquid column, some remarkable similarities associated with initial growth rates, angular frequencies, and mode transitions between the two systems are found.
 LETTERS


Reducedorder modeling for unsteady transonic flows around an airfoil
View Description Hide DescriptionHightransonic unsteady flows around an airfoil at zero angle of incidence and moderate Reynolds numbers are characterized by an unsteadiness induced by the von Kármán instability and buffet phenomenon interaction. These flows are investigated by means of lowdimensional modeling approaches. Reducedorder dynamical systems based on proper orthogonal decomposition are derived from a Galerkin projection of twodimensional compressible NavierStokes equations. A specific formulation concerning density and pressure is considered. Reducedorder modeling accurately predicts unsteady transonic phenomena.

Nonhomogeneous shear flow in concentrated liquidcrystalline solutions
View Description Hide DescriptionThe dynamics of concentrated solutions of rodlike molecules in nonhomogeneous shear flow are explored using a consistent numerical simulation of the Doi diffusion equation and the nonhomogeneous Onsager model of excludedvolume rod interactions. Simulations of planar, walldriven shear flow show that outofplane structureinstabilities occur when nematic anchoring constraints at the boundaries are removed. A new composite state with misaligned logrolling and flowaligning domains is observed for pressuredriven flow in a planar channel. These results mark the first use of the Doi diffusion equation to show how a nonhomogeneous flow field generates sharp interdomain interfaces analogous to those observed in rheological experiments.

Optimal amplification of the Crow instability
View Description Hide DescriptionA mechanism for promoting the Crow instability in a counterrotating vortex pair is presented within the framework of linear dynamics. It consists of (i) the creation of a periodic array of vortex rings along the length of the vortices by stretching of vorticity at the leading hyperbolic point of the dipole, and (ii) the deformation of the vortices by the vortex rings leading to the Crow instability. A reduction of the characteristic time of the Crow instability by a factor of roughly 2 can be obtained by this mechanism.
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 ARTICLES

 Interfacial Flows

The effect of jet velocity profile on the characteristics of thickness and velocity of the liquid sheet formed by two impinging jets
View Description Hide DescriptionIn this study, the effect of jet velocity profile on the thickness and velocity of the liquidsheet formed by two lowspeed impinging jets was investigated. In addition to the constant jet velocity and the Poiseuille parabolic profile, the jet velocity profile measured experimentally was considered to account for the real nonuniform jet velocity profile. For three jet velocity profiles, the distributions of the thickness and velocity of the liquidsheet were analytically predicted by solving conservation equations for mass, momentum, and energy. The predicted results were compared with the previous experimental results. The jet velocity profile affected the resulting thickness and velocity characteristics of the liquidsheet. The distributions of the thickness and velocity of the liquidsheet predicted by using the measured jet velocity profile produced more acceptable results, which agreed better with the experimental observations, than those obtained by using the constant jet velocity, which has been commonly used in previous theoretical works.

Dynamics of a horizontal thin liquid film in the presence of reactive surfactants
View Description Hide DescriptionWe investigate the interplay between a stable horizontal thin liquid film on a solid substrate and an excitable or bistable reactive mixture on its free surface. Their coupling is twofold. On the one hand, flow in the film transports the reacting surfactants convectively. On the other hand, gradients in the surfactant concentration exert Marangoni stresses on the free surface of the film. A reduced model is derived based on the longwave approximation. We analyze the linear stability of the coupled system as well as the nonlinear behavior, including the propagation of solitary waves, fronts, and pulses. We show, for instance, that the coupling of thin filmhydrodynamics and surfactant chemistry can either stabilize instabilities occurring in the pure chemical system, or in a regime where the pure hydrodynamic and chemical subsystems are both stable, the coupling can induce instabilities.

The effects of a diffusion controlled surfactant on a viscous drop injected into a viscous medium
View Description Hide DescriptionThe effects of a diffusion controlled surfactant on the evolution of a buoyant viscous drop injected into a viscous fluid are studied numerically for the case of finite bulk convection to resolve neck dynamics and detaching drop volumes. When the drop is formed, its interface initially expands. The surfactant adsorbs and depletes a region around the drop. When the drop is sufficiently elongated, a neck begins to form. The surface contracts rapidly above the neck, driving the surface concentration above its equilibrium value. The surfactant subsequently desorbs into the region adjacent to the interface that had previously been depleted of the surfactant. This creates diffusion fluxes away from the neck that are larger than suggested by an a priori scaling of the governing equations. The rapid flux removes the surfactant effectively from the contracting neck, preventing the occurrence of strong local reductions in the surface tension. Through this mechanism, neck dynamics are altered only weakly for surface coverages less than or equal to 0.9. For surface coverages close enough to maximum packing and for diffusion fluxes that are sufficiently slow, surfactant accumulation can reduce the local surface tension sufficiently to prevent drops from detaching. A phase diagram summarizing neck shapes and regimes where drops fail to detach is presented as a function of , the ratio of surfactantdiffusion rate (between the interface and the bulk) to the rate of surface contraction.

Wetting kinetics of a thin film evaporating in air
View Description Hide DescriptionThe conservation equation and the equations of motion are solved for a case where a thin liquid film moves out of a slot onto a horizontal surface. The liquid is allowed to evaporate into air. The evaporation process is taken to be isothermal. Lubrication theory approximation is used where only the tangential velocity and its dependence only in the normal direction are considered. The dynamics of thin films includes the use of disjoining pressure for a pure liquid and where there is a dissolved polymer. The results show that evaporation is quicker than film thinning such that a spreading regime dominated by the effects of disjoining pressure is never achieved. However, unlike the cases of pinning studied so far, there is no singularity in the evaporative flux near the contact line because of the use of disjoining pressure on evaporation. It is also observed that a balance between the rate of viscous dissipation and surface work is able to quantify the steady state contact angle. Consequently, a more macroscopic (and quantitative) description of contact line can be found that avoids the singularities discussed earlier and also the detailed calculations shown here. However, the detailed calculations are necessary to make the above point.
 Viscous and NonNewtonian Flows

Kinetic theory of a confined polymer driven by an external force and pressuredriven flow
View Description Hide DescriptionKinetic theory is used to investigate the mechanisms causing crossstream migration of confined polymers and polyelectrolytes under the influence of external forces and flow fields. Numerical simulations and experiments have demonstrated that confined polymers migrate towards the center of the channel in response to both external forces and uniaxial flows. Yet, migration towards the walls has been observed with combinations of external force and flow. In this paper, the kinetic theory for an elastic dumbbell developed by Ma and Graham [Phys. Fluids17, 083103 (2005)] has been extended to account for the effects of an external force. Further modifications account for counterion screening within a DebyeHückel approximation. This enables qualitative comparison with experimental results [Zheng and Yeung, Anal. Chem.75, 3675 (2003)] on DNA migration under combined electric and pressuredriven flow fields. The comparison supports the contention [Long et al., Phys. Rev. Lett.76, 3858 (1996)] that the hydrodynamic interactions in polyelectrolytes decay algebraically, as , rather than exponentially. The theory qualitatively reproduces results of both simulations and experiments for the migration of neutral polymers and polyelectrolytes. Concentration profiles similar to those found in numerical simulations are observed, but the Peclet numbers differ by factors of 2–3.

Twodimensional Stokes flow driven by elliptical paddles
View Description Hide DescriptionA fast and accurate numerical technique is developed for solving the biharmonic equation in a multiply connected domain, in two dimensions. We apply the technique to the computation of slow viscousflow(Stokes flow) driven by multiple stirring rods. Previously, the technique has been restricted to stirring rods of circular cross section; we show here how the prior method fails for noncircular rods and how it may be adapted to accommodate general rod cross sections, provided only that for each there exists a conformal mapping to a circle. Corresponding simulations of the flow are described, and their stirring properties and energy requirements are discussed briefly. In particular the method allows an accurate calculation of the flow when flat paddles are used to stir a fluid chaotically.

Drift in supported membranes
View Description Hide DescriptionAn object moving in a fluid transports the fluid along the direction of its motion. Using the concept of drift, i.e., the net motion of a small volume of fluid or a tracer particle due to a moving body, we quantify this entrainment for an inclusion in a supported lipid bilayer membrane. Our analysis demonstrates that a moving object in a supported membrane transports a small volume of fluid by a significant distance only when the initial position of the fluid volume in question is within a distance from the line of motion, where is the screening length of the membrane. The total area swept out by a line of such fluid volume elements, initially at rest and oriented perpendicular to the direction of motion, is the drift area. We show that the drift area is related quadratically to the screening length. These calculations suggest that dynamic domains of entrained lipids of size form spontaneously around moving objects in supported membranes due to hydrodynamicinteractions. This effect is potentially important for transport processes in biological and artificial membranes.

Hydrodynamic studies on two traveling wavy foils in tandem arrangement
View Description Hide DescriptionIn this study, the hydrodynamic interactions between two tandem foils undergoing fishlike swimming motion are investigated numerically by solving the Navier–Stokes equations with the immersedboundary method. The two foils represent two tandem propellers attached on a concept ship. The thrusts and efficiencies at three typical Strouhal numbers, i.e., , 0.6, and 0.8, are investigated. The results show that a fish situated directly behind another one does not always undergo a lower thrust. Whether it experiences a thrust enhancement or reduction depends on the Strouhal number. At a relatively low Strouhal number (e.g., ), the usual wake dragreduction effect predominates over the dragenhancement effect caused by the reverse von Kármán vortices, resulting in a thrust enhancement. The opposite happens at a relatively high Strouhal number (e.g., ). The downstream fish can benefit from the upstream one by slalom between the vortices rather than through them. For the upstream fish, the thrusts and efficiencies for all Strouhal numbers studied are higher than those for a single fish when the two fish are closely spaced, and approach those for a single fish as the spacing is increased.
 Particulate, Multiphase, and Granular Flows

Transient deformation and drag of decelerating drops in axisymmetric flows
View Description Hide DescriptionTransient deformation and drag coefficients of decelerating drops in axisymmetric flows are numerically computed. The drag coefficients are compared with those of solid spheres. In the case of drops, the behavior of the drag coefficient is dependent on the deformation and internal circulation of the drops in addition to the factors which are important for solid spheres. These, in turn, are dependent on the gasbased Weber number and the Ohnesorge number . At the relatively low of 1, when the deformation is small, the drag coefficients are about the same for the solid sphere and drop. When is increased, the deformation increases and the difference increases. At the highest of 100, the drop reaches a point of secondary breakup. In general, oblate shapes result in greater drag and prolate shapes in lower drag relative to the solid sphere. Increasing , which implies increasing viscous forces in the liquid relative to surface tension forces, leads to less deformation and hence lesser differences between solid and drop drag coefficients for a given .

Evolution of wake structure and wakeinduced loads along the path of freely rising axisymmetric bodies
View Description Hide DescriptionThis paper reports on the zigzagging motion of disks of various thicknesses rising freely under the effect of buoyancy in a liquid otherwise at rest. Time sequences of the velocity fields around the moving body were obtained using particle imagevelocimetry and are presented for two bodies with contrasted diametertothickness ratios. The differences observed between the two cases are discussed in relation with the evolution of the loads acting on the body and of the displacement and rotation of the body during a period of the zigzag. The crucial influence of the phase difference between the vortical force and torque on the path is underlined.

Unmixing in random flows
View Description Hide DescriptionWe consider particles suspended in a randomly stirred or turbulent fluid. When effects of the inertia of the particles are significant, an initially uniform scatter of particles can cluster together. We analyze this “unmixing” effect by calculating the Lyapunov exponents for dense particles suspended in such a random threedimensional flow, concentrating on the limit where the viscous damping rate is small compared to the inverse correlation time of the random flow (that is, the regime of large Stokes number). In this limit Lyapunov exponents are obtained as a power series in a parameter which is a dimensionless measure of the inertia. We report results for the first seven orders. The perturbation series is divergent, but we obtain accurate results from a PadéBorel summation. We deduce that particles can cluster onto a fractal set and show that its dimension is in satisfactory agreement with previously reported simulations of turbulent NavierStokes flows. We also investigate the rate of formation of caustics in the particle flow.

Velocity fluctuations of initially stratified sedimenting spheres
View Description Hide DescriptionThe study of velocityfluctuations in the sedimentation of spheres is complicated by the time evolution of the underlying particle distribution, both at the microscale and in the bulk. We perform a series of experiments and simulations to isolate the effect of an initial, stable stratification in the particle concentration. The directly observed dependence of velocityfluctuations on stratification agrees with a previously obtained scaling theory.

Hydrodynamic interaction of two particles in confined linear shear flow at finite Reynolds number
View Description Hide DescriptionWe discuss the hydrodynamic interactions of two solid bodies placed in linear shear flow between parallel plane walls in a periodic geometry at finite Reynolds number. The computations are based on the lattice Boltzmann method for particulate flow, validated here by comparison to previous results for a single particle. Most of our results pertain to cylinders in two dimensions but some examples are given for spheres in three dimensions. Either one mobile and one fixed particle or else two mobile particles are studied. The motion of a mobile particle is qualitatively similar in both cases at early times, exhibiting either trajectory reversal or bypass, depending upon the initial vector separation of the pair. At longer times, if a mobile particle does not approach a periodic image of the second, its trajectory tends to a stable limit point on the symmetry axis. The effect of interactions with periodic images is to produce nonconstant asymptotic longtime trajectories. For one free particle interacting with a fixed second particle within the unit cell, the free particle may either move to a fixed point or take up a limit cycle. Pairs of mobile particles starting from symmetric initial conditions are shown to asymptotically reach either fixed points, or mirror image limit cycles within the unit cell, or to bypass one another (and periodic images) indefinitely on a streamwise periodic trajectory. The limit cycle possibility requires finite Reynolds number and arises as a consequence of streamwise periodicity when the system length is sufficiently short.

Simulation of hydrodynamically interacting particles near a noslip boundary
View Description Hide DescriptionThe dynamics of spherical particles near a single plane wall are computed using an extension of the Stokesian dynamics method that includes longrange manybody and pairwise lubrication interactions between the spheres and the wall in Stokes flow. Extra care is taken to ensure that the mobility and resistance tensors are symmetric, positive, and definite—something which is ineluctable for particles in lowReynoldsnumber flows. We discuss why two previous simulation methods for particles near a plane wall, one using multipole expansions and the other using the RotnePrager tensor, fail to produce symmetric resistance and mobility tensors. Additionally, we offer some insight on how the Stokesian dynamics paradigm might be extended to study the dynamics of particles in any confining geometry.

Simulation of the motion of flexible fibers in viscous fluid flow
View Description Hide DescriptionA model for flexible fibers in viscous fluid flow is proposed, and its predictions compared with experiments found in the literature. The incompressible threedimensional Navier–Stokes equations are employed to describe the fluid motion, while fibers are modeled as chains of fiber segments, interacting with the fluid through viscous and dynamic drag forces. Fiber segments, from the same or from different fibers, interact with each other through normal, frictional, and lubrication forces. Momentum conservation is enforced on the system to capture the twoway coupling between phases. Quantitative predictions could be made, and showed good agreement with experimental data, for the period of Jeffery orbits in shear flow, as well as for the amount of bending of flexible fibers in shear flow. Simulations, using the proposed model, also successfully reproduced the different regimes of motion for threadlike particles, ranging from rigid fiber motion to complicated orbiting behavior, including coiling and selfentanglement.

Refinement of the probability density function model for preferential concentration of aerosol particles in isotropic turbulence
View Description Hide DescriptionThe purposes of the paper are threefold: (i) to refine the statistical model of preferential particle concentration in isotropic turbulence that was previously proposed by Zaichik and Alipchenkov [Phys. Fluids15, 1776 (2003)], (ii) to investigate the effect of clustering of lowinertia particles using the refined model, and (iii) to advance a simple model for predicting the collision rate of aerosol particles. The model developed is based on a kinetic equation for the twopoint probability density function of the relative velocity distribution of particle pairs. Improvements in predicting the preferential concentration of lowinertia particles are attained due to refining the description of the turbulentvelocity field of the carrier fluid by including a difference between the time scales of the of strain and rotation rate correlations. The refined model results in a better agreement with direct numerical simulations for aerosol particles.

Solution of threedimensional fiber orientation in twodimensional fiber suspension flows
View Description Hide DescriptionThe orientation of fibers in simple twodimensional flows is investigated. According to different ranges of the Péclet number, , defined as the ratio of a characteristic rotational speed of fibers and the orientational diffusivity, three methods are developed: characteristic method for , regular perturbation method for , and spectral method for everything else. All the methods subtly utilize the evolving solution of the rotational dynamics of fibers, which is also given in this paper. Especially, the adoption of spherical harmonics in the spectral method eliminates the singularity of the Fokker–Planck equation in spherical coordinates, and provides high precision and efficiency. The evolving solution of orientation distribution with is obtained through the solution of rotational dynamics. Using a regular perturbation method, the solution of orientation distribution with is extended for the condition of . This paper provides systematical and high efficient techniques to deal with the fiber orientation.