Volume 18, Issue 10, October 2006
 LETTERS


Surface switching of rotating fluid in a cylinder
View Description Hide DescriptionWe study the surface shape of water in an open cylinder driven by constant rotation of the bottom. Around the critical Reynolds number for the laminarturbulent transition, the surface deformation, which is of the order of the container size, shows an aperiodic switching phenomenon between an axisymmetric shape and a nonaxisymmetric shape. The axisymmetric shape is observed as a steady state when the Reynolds number is smaller than that in the switching region, while the nonaxisymmetric shape is observed as a (quasi) periodic state in which the surface rotates at almost constant angular velocity when the Reynolds number is larger than that in the switching region. A detailed analysis for the surface shape suggests that the flow with the nonaxisymmetric shape is turbulent.

Angularmomentum conservative smoothed particle dynamics for incompressible viscous flows
View Description Hide DescriptionCurrent smoothed particle dynamics discretizations for macroscopic and mesoscopic viscous flows usually do not conserveangular momentum. Angularmomentum conservation, however, potentially stabilizes the solution for longtime simulations. We show that a simple angularmomentum conservative formulation of the viscous force, which was proposed previously based on empirical findings, can be derived theoretically under the condition of incompressible flow. The properties of this formulation are asserted by numerical simulations of twodimensional TaylorGreen flow.
 Top

 ARTICLES

 Interfacial Flows

Flow of a thin liquid film on an unsteady stretching sheet
View Description Hide DescriptionThe stretching surface is assumed to be stretched impulsively from rest and the effect of inertia of the liquid is considered. Equations describing the laminar flow on the stretching surface are solved analytically by using the singular perturbation technique and the method of characteristics is used to obtain an analytic expression for film thickness. The results show that the final film thickness is independent of the amount of liquid distributed initially and on the initial film thickness be it uniform or nonuniform. It is also shown that the forceful stretching produces quicker thinning of the film on the stretching surface.

A computational study of the coalescence between a drop and an interface in Newtonian and viscoelastic fluids
View Description Hide DescriptionA drop falling onto a fluidfluidinterface may not merge with it at once but may undergo a socalled partial coalescence cascade. Experimental observations of this phenomenon have revealed fascinating features of the process for Newtonian as well as polymeric fluids. In this paper, we describe numerical simulations of partial coalescence based on a phasefield method. Results show that partial coalescence occurs for an intermediate range of drop sizes, and proceeds in two stages: capillary waves propagating along the drop and transforming it into a fluid column, and neck formation on the column and pinchoff of the secondary drop. In the first stage, interfacial energy turns into kinetic energy following film rupture, while in the second, the kinetic energy overcomes an energy barrier due to the initial increase in interfacial area during neck formation. A parametric study establishes a criterion for partial coalescence in terms of a maximum Ohnesorge number that applies to a wide range of fluid densities and viscosities as long as the Bond number is small. Viscoelasticity in either the drop or the matrix tends to delay the pinchoff of the secondary drop, and may even suppress partial coalescence altogether. The underlying mechanism is large tensile polymer stresses resisting the stretching and thinning of the fluid neck. The numerical results are in qualitative, and in some cases quantitative, agreement with prior experiments.

Direct numerical study of a liquid droplet impulsively accelerated by gaseous flow
View Description Hide DescriptionA liquid spherical droplet impulsively accelerated by a gaseous flow is simulated in order to investigate the drag force and the deformation. The dynamics of the droplet immersed in a gaseous flow are investigated by solving the incompressible NavierStokes equations using a finite volume staggered mesh method coupled with a moving mesh interface tracking scheme. The benefit of the current scheme is that the interface conditions are implemented directly on an explicitly located interface with zero thickness. The droplet shape changes as it is accelerated, and the deformation factor of the droplet is as small as 0.2, so mesh adaptation methods are employed to achieve good mesh quality and to capture the interface curvature. The total drag coefficients are found to be larger than typical steadystate drag coefficients of solid spheres at the same Reynolds numbers. This agrees with the observation of Temkin et al. [J. Fluid Mech.96, 133 (1980)] that the unsteady drag of decelerating relative flows was always larger than the steady drag. The large recirculation region behind the deformed droplet may explain this greater drag force. The effects of the viscosity ratio, density ratio, and initial Weber number on the droplet dynamics are also studied. It is found that the initial Weber number and the viscosity ratio have significant effects on the droplet dynamics, while the density ratio does not.

Spray impact: Rim transverse instability initiating fingering and splash, and description of a secondary spray
View Description Hide DescriptionIn this paper, normal spray impact onto a rigid wall, leading to the formation of secondary spray, is considered. The mechanism of splash is explained by the bending instability of a rim bounding a free liquid sheet. The linear stability analysis of the rim is performed in the framework of the longwave, quasionedimensional approach. The rim instability is caused by the moment of forces associated with the inertia of the liquid entering the rim. Next, two components of the drop velocity and their diameter, as well as various flux density vectors (number, volume, mass fluxes) and tensors (momentum flux), are measured using a phase Doppler instrument. It is shown that the viscous length scale of drop impact can be used in describing the splash threshold, diameter of secondary droplets, and their velocity. Consequently, a closed semiempirical model for the secondary spray has been proposed and validated using a numerical simulation of spray transport based on an EulerLagrange approach.
 Viscous and NonNewtonian Flows

Flow induced by a sphere settling in an aging yieldstress fluid
View Description Hide DescriptionWe have studied the flow induced by a macroscopic spherical particle settling in a Laponite suspension that exhibits a yield stress, thixotropy, and shear thinning. We show that the fluid thixotropy (or aging) induces an increase with time of both the apparent yield stress and shearthinning properties but also a breaking of the flow foreaft symmetry predicted in HershelBulkley fluids (yieldstress, shearthinning fluids with no thixotropy). We have also varied the stress exerted by the particles on the fluid by using particles of different densities. Although the stresses exerted by the particles are of the same order of magnitude, the velocity field presents utterly different features: whereas the flow around the lighter particle shows a confinement similar to the one observed in shearthinning fluids, the wake of the heavier particle is characterized by an upward motion of the fluid (“negative wake”), whatever the fluid’s age. We compare the features of this negative wake to the one observed in viscoelastic shearthinning fluids (polymeric or micelle solutions). Although the flows around the two particles strongly differ, their settling behaviors display no apparent difference which constitutes an intriguing result and evidences the complexity of the dependence of the drag factor on flow field.

Monodomain dynamics for rigid rod and platelet suspensions in strongly coupled coplanar linear flow and magnetic fields. II. Kinetic theory
View Description Hide DescriptionWe establish reciprocity relations of the DoiHess kinetic theory for rigid rod macromolecular suspensions governed by the strong coupling among an excluded volume potential, linear flow, and a magnetic field. The relation provides a reduction of the flow and field driven Smoluchowski equation: from five parameters for coplanar linear flows and magnetic field, to two field parameters. The reduced model distinguishes flows with a rotational component, which map to simple shear (with rate parameter) subject to a transverse magnetic field (with strength parameter), and irrotational flows, for which the reduced model consists of a triaxial extensional flow (with two extensional rate parameters). We solve the Smoluchowski equation of the reduced model to explore: (i) the effect of introducing a coplanar magnetic field on each sheared monodomain attractor of the DoiHess kinetic theory and (ii) the coupling of coplanar extensional flow and magnetic fields. For (i), we show each sheared attractor (steady and unsteady, with peak axis in and out of the shearing plane, periodic and chaotic orbits) undergoes its own transition sequence versus magnetic field strength. Nonetheless, robust predictions emerge: outofplane degrees of freedom are arrested with increasing field strength, and a unique flowaligning or tumbling/wagging limit cycle emerges above a threshold magnetic field strength or modified geometry parameter value. For (ii), irrotational flows coupled with a coplanar magnetic field yield only steady states. We characterize all (generically biaxial) equilibria in terms of an explicit Boltzmann distribution, providing a natural generalization of analytical results on pure nematic equilibria [P. Constantin, I. Kevrekidis, and E. S. Titi, Arch. Rat. Mech. Anal.174, 365 (2004); P. Constantin, I. Kevrekidis, and E. S. Titi, Discrete and Continuous Dynamical Systems11, 101 (2004); P. Constantin and J. Vukadinovic, Nonlinearity18, 441 (2005); H. Liu, H. Zhang, and P. Zhang, Comm. Math. Sci.3, 201 (2005); C. Luo, H. Zhang, and P. Zhang, Nonlinearity18, 379 (2005); I. Fatkullin and V. Slastikov, Nonlinearity18, 2565 (2005); H. Zhou, H. Wang, Q. Wang, and M. G. Forest, Nonlinearity18, 2815 (2005)] and extensional flowinduced equilibria [Q. Wang, S. Sircar, and H. Zhou, Comm. Math. Sci.4, 605 (2005)]. We predict large parameter regions of bistable equilibria; the lowest energy state always has principal axis aligned in the flow plane, while another minimum energy state often exists, with primary alignment transverse to the coplanar field.

Topological mixing study of nonNewtonian duct flows
View Description Hide DescriptionTracer advection of nonNewtonian fluids in reoriented duct flows is investigated in terms of coherent structures in the web of tracer paths that determine transport properties geometrically. Reoriented duct flows are an idealization of inline mixers, encompassing many micro and industrial continuous mixers. The topology of the tracer dynamics of reoriented duct flows is Hamiltonian. As the stretching per reorientation increases from zero, we show that the qualitative route from the integrable state to global chaos and good mixing does not depend on fluid rheology. This is due to a universal symmetry of reoriented duct flows, which we derive, controlling the topology of the tracer web. Symmetry determines where in parameter space global chaos first occurs, while increasing nonNewtonian effects delays the quantitative value of onset. Theory is demonstrated computationally for a representative duct flow, the rotated arc mixing flow.
 Particulate, Multiphase, and Granular Flows

Transient development of instabilities in a uniformly driven layer
View Description Hide DescriptionWe present a nonlinear stability analysis on a uniformly driven granular layer to investigate the transient development of the instabilities induced by small perturbations. A continuum model based on the grain kinetic theory was adopted to trace the pattern evolution from the original unstable base state to a new base state and was numerically solved using a finiteelement method. In the stability diagram characterized by two operating parameters, dimensionless mass holdup , and energy input , one point (, ) in the stationary mode is selected to examine the fate of the twodimensional density wave found from the linear stability analysis. Upon perturbing the base state, the bed height, solid fraction, particle velocities, and granular temperature profiles move away from their original to new, steady values after undergoing some oscillations. The stripes of solid fraction and granular temperature profiles finally change into a layerlike pattern, while the top surface is not uniform anymore but appears as periodic peaks and valleys due to the existence of uneven vertical velocities during the evolution. It is found that the finite amplitude of the initial perturbations is important for the pattern transition, and the evolution time increases with the increasing energy input and the increasing collision restitution coefficient. With the top surface of the granular layer being fixed at its original position, the oscillatory feature in the evolution curves of mass and temperature profiles disappears, and a lower solid fraction is obtained as compared to that observed in the presence of a moving upper boundary.

Numerical study of turbulent bubbly downflows in a vertical channel
View Description Hide DescriptionDirect numerical simulations are used to study turbulent bubbly downflows in a vertical channel. All flow scales, including the bubbles and the flow around them, are fully resolved using a fronttracking/finitevolume method. The turbulent bubbly channel flow is driven downward by an imposed constant pressure gradient, and the frictionReynolds number of the flow, based on the friction velocity and halfwidth of the channel, is 127.3, corresponding to a bulk Reynolds number of 3786 for a flow without bubbles. Three cases with several nearly spherical bubbles are examined. The bubble diameter is 31.8 wall units for all cases but the number of bubbles is varied, giving average void fractions of 1.5%, 3%, and 6%. The lift force on the bubbles drives them away from the walls until the mixture in the center of the channel is in hydrostatic equilibrium. Thus, the flow consists of a core region where the average void fraction and the mean vertical velocity are approximately constant and a bubblefree wall layer. The vertical velocity fluctuations in the wall layer decrease as the void fraction increases and the width of the wall layer decreases, but in the bubblerich core the velocity fluctuations are higher than for a corresponding singlephase turbulent flow.

Computing stationary freesurface shapes in microfluidics
View Description Hide DescriptionA finiteelement algorithm for computing freesurface flows driven by arbitrary body forces is presented. The algorithm is primarily designed for the microfluidic parameter range where (i) the Reynolds number is small and (ii) forcedriven pressure and flow fields compete with the surface tension for the shape of a stationary free surface. The free surface shape is represented by the boundaries of finite elements that move according to the stress applied by the adjacent fluid. Additionally, the surface tends to minimize its free energy and by that adapts its curvature to balance the normal stress at the surface. The numerical approach consists of the iteration of two alternating steps: The solution of a fluidic problem in a prescribed domain with slip boundary conditions at the free surface and a consecutive update of the domain driven by the previously determined pressure and velocity fields. For a Stokes problem the first step is linear, whereas the second step involves the nonlinear freesurface boundary condition. This algorithm is justified both by physical and mathematical arguments. It is tested in two dimensions for two cases that can be solved analytically. The magnitude of the errors is discussed in dependence on the approximation order of the finite elements and on a stepwidth parameter of the algorithm. Moreover, the algorithm is shown to be robust in the sense that convergence is reached also from initial forms that strongly deviate from the final shape. The presented algorithm does not require a remeshing of the used grid at the boundary. This advantage is achieved by a builtin mechanism that causes a smooth change from the behavior of a free surface to that of a rubber blanket if the boundary mesh becomes irregular. As a side effect, the element sides building up the free surface in two dimensions all approach equal lengths. The presented variational derivation of the boundary condition corroborates the numerical finding that a secondorder approximation of the velocity also necessitates a secondorder approximation for the free surface discretization.

Dynamics of a twodimensional upflowing mixing layer seeded with bubbles: Bubble dispersion and effect of twoway coupling
View Description Hide DescriptionThe evolution and structure of a spatially evolving twodimensional mixing layer seeded with small bubbles are numerically investigated. The oneway coupling approach is first employed to show that characteristics of bubble dispersion are dominated by the possibility for sufficiently small bubbles to be captured in the core of the vortices. A stability analysis of the ordinary differential equation system governing bubble trajectories reveals that this entrapment process is governed by the presence of stable fixed points advected by the mean flow. Twoway coupling simulations are then carried out to study how the global features of a twodimensional flow are affected by bubbleinduced disturbances. The local interaction mechanism between the two phases is first analyzed using detailed simulations of a single bubbly vortex. The stability of the corresponding fixed point is found to be altered by the collective motion of bubbles. For trapped bubbles, the interphase momentum transfer yields periodic sequences of entrapment, local reduction of velocity gradients, and eventually escape of bubbles. Similar mechanisms are found to take place in the spatially evolving mixing layer. The presence of bubbles is also found to enhance the destabilization of the inlet velocity profile and to shorten the time required for the rollup phenomenon to occur. The most spectacular effects of small bubbles on the largescale flow are a global tilting of the mixing layer centerline towards the lowvelocity side and a strong increase of its spreading rate. In contrast, no significant modification of the flow is observed when the bubbles are not captured in the largescale vortices, which occurs when the bubble characteristics are such that the drift parameter defined in the text exceeds a critical value. These two contrasted behaviors agree with available experimental results.

A novel gravityinduced flow transition in twophase fluids
View Description Hide DescriptionExperimental results are reported that show a gravityinduced flow transition in wellmixed suspensions and emulsions, even when the buoyancydriven velocity of isolated drops or particles is several orders of magnitude smaller than the imposed velocity. The experiments were conducted with emulsions of isooctane in water and suspensions of polymethylmethacrylate particles in water. Both the drop and particle diameters were approximately , and concentrations of the dispersed phases ranged from dilute (2%) to concentrated (40%). The twophase fluids were confined to a horizontal, concentriccylinder apparatus in which the outer cylinder was rotated, and the velocity profiles were measured by nuclear magnetic resonance imaging. The results show that the flow transition is relatively insensitive to the volume fraction of the dispersed phase. The flow transition occurs because, although the buoyancydriven velocity is relatively small on the length scale of the particle or drop dimension, the flow itself induces a slight variation in the suspension concentration and, hence, density. Although only on the order of , this density difference spans a macroscopic length scale, making the buoyancy effect competitive with the imposed flow. These arguments yield a dimensionless parameter that predicts very closely the nonequilibrium phase diagram generated by the experiments.

Unsteady motion of two solid spheres in Stokes flow
View Description Hide DescriptionThis study is concerned with the unsteady motion of two solid spherical particles in an unbounded incompressible Newtonian flow. The background flow is uniform and can be time dependent. In addition, the particle Reynolds numbers and , based on characteristic particles velocities and , are assumed to remain small throughout the motion. Here, and denote the particle radii and is the kinematicviscosity of the fluid. Two approximate methods are employed in order to calculate the unsteady force exerted on each particle. In the first approach, a simplified method of reflections in combination with the pointforce method is employed. In the second approach, a simplified method of reflections combined with Burger’s unsteady flow solution is considered. The forces due to the background flow and the disturbed flow created by the presence of particles are treated separately. The equation of motion for each particle is derived and some special cases are presented in detail including the motion with constant acceleration and the motion in a gravitational field. The results indicate that using the Basset force corresponding to the motion of two spheres gives rise to a larger drag force as compared to the solution utilizing the solitaryparticle Basset force.

Dense shearing flows of inelastic disks
View Description Hide DescriptionWe introduce a simple phenomenological modification to the hydrodynamic equations for dense flows of identical, frictionless, inelastic disks and show that the resulting theory describes the area fraction dependence of quantities that are measured in numerical simulations of steady, homogeneous shearing flows and steady, fully developed flows down inclines. The modification involves the incorporation of a length scale other than the particle diameter in the expression for the rate of collisional dissipation. The idea is that enduring contacts between grains forced by the shearing reduce the collisional rate of dissipation while continuing to transmit momentum and force. The length and orientation of the chains of particles in contact are determined by a simple algebraic equation. When the resulting expression for the rate of dissipation is incorporated into the theory,numerical solutions of the boundaryvalue problem for steady, fully developed flow of circular disks down a bumpy incline exhibit a core with a uniform area fraction that decreases with increasing angles of inclination. When the height at which an inclined flow stops is assumed to be proportional to this chain length, a scaling between the average velocity, flow height, and stopping height similar to that seen in experiments and numerical simulations is obtained from the balance of fluctuation energy.
 Laminar Flows

Rotational flow in tapered slab rocket motors
View Description Hide DescriptionInternal flow modeling is a requisite for obtaining critical parameters in the design and fabrication of modern solid rocket motors. In this work, the analytical formulation of internal flows particular to motors with tapered sidewalls is pursued. The analysis employs the vorticitystreamfunction approach to treat this problem assuming steady, incompressible, inviscid, and nonreactive flow conditions. The resulting solution is rotational following the analyses presented by Culick for a cylindrical motor. In an extension to Culick’s work, Clayton has recently managed to incorporate the effect of tapered walls. Here, an approach similar to that of Clayton is applied to a slab motor in which the chamber is modeled as a rectangular channel with tapered sidewalls. The solutions are shown to be reducible, at leading order, to Taylor’s inviscid profile in a porous channel. The analysis also captures the generation of vorticity at the surface of the propellant and its transport along the streamlines. It is from the axial pressure gradient that the proper form of the vorticity is ascertained. Regular perturbations are then used to solve the vorticityequation that prescribes the mean flow motion. Subsequently, numerical simulations via a finite volume solver are carried out to gain further confidence in the analytical approximations. In illustrating the effects of the taper on flow conditions, comparisons of total pressure and velocity profiles in tapered and nontapered chambers are entertained. Finally, a comparison with the axisymmetric flow analog is presented.

Convective heat transfer in unsteady laminar parallel flows
View Description Hide DescriptionIn laminar, fully developed pipe and channel flows that undergo transients from a known initial state, exact analytical solutions for the momentary velocity field as functionals of the flow rate can be determined from the NavierStokes equations, for arbitrary flow unsteadiness [Phys. Fluids12, 518 (2000)]. For laminar, fully developed duct flows with uniform wall heating that undergo large flow transients, the companion thermal energy equation can be approximated in a form that may also be solved analytically, yielding solutions for the instantaneous temperature field for arbitrary time unsteadiness in both the flow and the wall heat flux. Expressions for Nusselt numbers in convective heat transfer in duct flows with arbitrary temporal flow and heat flux unsteadiness can then be found, which illustrate how the flow and heat flux transient histories determine whether the unsteadiness enhances or reduces the overall heattransfer effectiveness. These expressions are used to show how significant enhancements or reductions in the average Nusselt number can be achieved in duct flow by applying appropriate temporal flow control.

An experimental investigation of hydrodynamic cavitation in microVenturis
View Description Hide DescriptionThe existence of hydrodynamiccavitation in the flow of deionized water through microVenturis has been witnessed in the form of traveling bubble cavitation and fully developed streamer bubble/supercavitation, and their mechanisms have been discussed. Highspeed photography and flow visualization disclose inchoate cavitation bubbles emerging downstream from the microVenturi throat and the presence of a single streamer bubble/supercavity, which is equidistant from the micro device walls. The supercavity initiates inside the diffuser section and extends until the microchannel exit and proceeds to bifurcate the incoming flow. This article strives to provide numerical data and experimental details of hydrodynamiccavitation taking place within microVenturis.

Threedimensional simulation of gaseous slip flow in different aspect ratio microducts
View Description Hide DescriptionThreedimensional lattice Boltzmann method based simulations of a microduct have been undertaken in this paper. The objectives are to understand the different physical phenomena occurring at these small scales and to investigate when the flow can be treated as two dimensional. Toward this end, the Knudsen number and aspect ratio (depth to width ratio) are varied for a fixed pressure ratio. The pressure in the microduct is nonlinear with the nonlinearity in pressure reducing with an increase in the Knudsen number. The pressure behaves somewhat similar to twodimensional microchannels, even when the aspect ratio is unity. The slip velocity at the impenetrable wall has two components: along and perpendicular to the primary flow direction. Our results show that the streamwise velocity near the centerline is relatively invariant along the depth for an aspect ratio of more than three, suggesting that the microduct can be modeled as a twodimensional microchannel. On the other hand, the velocity component along the depth is never identically zero, implying that the flow is not truly two dimensional, although for practical purposes a twodimensional treatment might suffice. A curious change in the vector direction in a plane normal to the flow direction is observed around an aspect ratio of four. These threedimensional results are significant because they will help in theoretical development and flow modeling at microscales.