Volume 22, Issue 10, October 2010

We study, experimentally and theoretically, the mixing of monodisperse colored beads in rotating cylinders with different crosssectional shapes (square, star, and a circle with two and four triangular wedges), operating in the continuous flow regime, to understand the role of crosssectional shape on the mixing process. The evolution of the mixed state is quantified using two measures: the intensity of segregation and the centroid distance. The mixing rate index, defined as the specific rate of change of the intensity of segregation with time, is initially lower in all the noncircular crosssections compared to a circle. It increases monotonically in noncircular crosssections, whereas it is nearly constant in a circle. This corresponds to a faster than exponential decay of the intensity of segregation for noncircular crosssections as compared to an exponential decay for a circular crosssection. The decay of centroid distance is oscillatory in a circle, whereas it is monotonic in noncircular crosssections. A significant improvement in the mixing rate is obtained for noncircular crosssections relative to circular crosssections. The mixing rate is highest for the circle with four wedges; the mixing rates for square and star crosssections are slightly lower. The circle with two wedges has the lowest mixing rate among the noncircular crosssections. Experiments with smaller glass beads, in which the particle diffusivity is significantly smaller, yield a mixing rate very close to that for the larger particles. Mixing patterns obtained experimentally at short times are well predicted by a convective diffusion model that includes a timeperiodic velocity field for the noncircular crosssections. A quantitative comparison of model predictions with experiments in terms of the intensity of segregation and the centroid distance shows a reasonably good agreement for the different shapes and for the two particle sizes. A scaling analysis is presented to explain the insensitivity of the mixing rate to particle size despite a significant variation in diffusivity. Mixing patterns for a tracer blob and the mixing rates obtained for the different mixers are found to be related to the regions of the chaotic advection in computed Poincaré maps. For the systems studied, the mixing rate increases with an increase in the number of corners in the geometry, but is relatively unaffected by the ratio of the maximum to minimum diameter of the crosssection.
 LETTERS


Transitional flow of a nonNewtonian fluid in a pipe: Experimental evidence of weak turbulence induced by shearthinning behavior
View Description Hide DescriptionThe present letter is a thorough study of the flow regime where an asymmetry of the mean axial velocity profiles is observed for shearthinning fluids flow in a pipe. This study is based on a statistical analysis of the axial velocityfluctuations. It is shown that this flow regime exhibits features of a weak turbulence: chaotic in time and regular in space. More precisely, (i) power spectra of axial velocityfluctuations decay following a power law with an exponent very close to −3, (ii) largescale coherent structures are generated, and (iii) there is essentially no intermittency in this flow regime.

Lateral drift and concentration instability in a suspension of bubbles induced by Marangoni stresses at zero Reynolds number
View Description Hide DescriptionWe report a concentration instability at zero Reynolds number created by hydrodynamically interacting bubbles with surfactant. This instability is driven by Marangoni stresses that force bubbles to migrate in directions perpendicular to gravity. We characterize the lateral motion of a single buoyant bubble when it is subject to a weak, low wavenumber disturbance velocity. We use this result to determine which mean flow wavevectors amplify concentration fluctuations in a dilute suspension. The suspension is linearly unstable at small horizontal wavenumbers by a mechanism similar to the concentration instabilities demonstrated in suspensions of sedimenting nonspherical or deformable particles.
 Top

 ARTICLES

 Micro and Nanofluid Mechanics

Diffusion model for Knudsentype compressor composed of periodic arrays of circular cylinders
View Description Hide DescriptionA rarefied gasflow in a long porous channel having a periodic structure that is consisting of alternately arranged porous media and gaps, the former of which contains a periodic array of parallel circular cylinders, is considered for the case in which the channel is infinitely wide. The cylinder arrays have a periodic temperature distribution with the same period as the structure. Under the assumption that the length of each cylinder array and that of each gap are much larger than the period of the cylinders in the array, a fluiddynamic system describing the overall behavior of the gas in the channel is derived from the kinetic system composed of the Bhatnagar–Gross–Krook equation and the diffuse reflection boundary condition. The derived system is composed of a diffusionmodel for each cylinder array, whose isothermal version has been reported previously [S. Taguchi and P. Charrier, Phys. Fluids20, 067103 (2008)], a set of fluiddynamic equations for each gap, and the macroscopic connection conditions at each junction between an array and a gap. Then, the fluiddynamic system is applied to a long channel consisting of many cylinder arrays and gaps. Some numerical results demonstrating the pumping effect of the flow are presented.

On the effects of liquidgas interfacial shear on slip flow through a parallelplate channel with superhydrophobic grooved walls
View Description Hide DescriptionComparisons between slip lengths predicted by a liquidgas coupled model and that by an idealized zerogasshear model are presented in this paper. The problem under consideration is pressuredriven flow of a liquid through a plane channel bounded by two superhydrophobic walls which are patterned with longitudinal or transverse gasfilled grooves. Effective slip arises from lubrication on the liquidgas interface and intrinsic slippage on the solid phase of the wall. In the mathematical models, the velocities are analytically expressed in terms of eigenfunction series expansions, where the unknown coefficients are determined by the matching of velocities and shear stresses on the liquidgas interface. Results are generated to show the effects due to small but finite gas viscosity on the effective slip lengths as functions of the channel height, the depth of grooves, the gas area fraction of the wall, and intrinsic slippage of the solid phase. Conditions under which even a gas/liquid viscosity ratio as small as 0.01 may have appreciable effects on the slip lengths are discussed.
 Interfacial Flows

The height of a static liquid column pulled out of an infinite pool
View Description Hide DescriptionWe consider a solid cone whose vertex points down and dips in an infinite pool of liquid. If the cone is slowly lifted, a liquid column with its top attached to the cone is pulled out of the pool. In this paper, we compute the maximum height of the cone before the column ruptures. Two reasons for rupturing are identified. In some cases, no solution for a higher position of the cone exists. In other cases, a solution does exist, but is unstable.

Nonlinear development of twolayer Couette–Poiseuille flow in the presence of surfactant
View Description Hide DescriptionThe twodimensional nonlinear evolution of the interface between two superposed layers of viscous fluid moving in a channel in the presence of an insoluble surfactant is examined. A pair of coupled weakly nonlinear equations is derived describing the interfacial and surfactantdynamics when one of the two fluid layers is very thin in comparison to the other. In contrast to previous work, the dynamics in the thin film are coupled to the dynamics in the thicker layer through a nonlocal integral term. For asymptotically small Reynolds number, the flow in the thicker layer is governed by the Stokes equations. A linearized analysis confirms the linear instability identified by previous workers and it is proven that the film flow is linearly unstable if the undisturbed surfactant concentration exceeds a threshold value. Numerical simulations of the weakly nonlinear equations reveal the existence of finite amplitude travelingwave solutions. For order one Reynolds number, the flow in the thicker layer is governed by the linearized Navier–Stokes equations. In this case the weakly nonlinear film dynamics are more complex and include the possibility of periodic travelingwaves and chaotic flow.

Continuum models for the contact line problem
View Description Hide DescriptionContinuum models are derived for the moving contact line problem through a combination of macroscopic and microscopic considerations. Macroscopic thermodynamic argument is used to place constraints on the form of the boundary conditions at the solid surface and the contact line. This information is then used to set up molecular dynamics to measure the detailed functional dependence of the boundary conditions. Long range molecular forces are taken into account in the form of a surface potential. This allows us to handle the case of complete wetting as well as the case of partial wetting. In particular, we obtain a new continuum model for both cases in a unified form. Two main parameters and different spreading regimes are identified from the analysis of the energy dissipations for the continuum model. Scaling laws in these different regimes are derived. The new continuum model also allows us to derive boundary conditions for the lubrication approximation. Numerical results are presented for the thin film model and the effect of the boundary condition is investigated.
 Viscous and NonNewtonian Flows

Noncontinuum drag force on a nanowire vibrating normal to a wall: Simulations and theory
View Description Hide DescriptionNanoelectromechanical oscillators are very attractive as sensing devices because of their low power requirements and high resolution, especially at low pressures. While many experimental studies of such systems are available in the literature, a fundamental theoretical understanding over the entire range of operating conditions is lacking. In this article, we use our newly developed Bhatnagar–Gross–Krook based low Mach number direct simulation Monte Carlo method to study the noncontinuum drag force acting on a cylinder oscillating normal to a wall. We explore quasisteady flows in which as well as unsteady flows for which . Here is the oscillation frequency and is the characteristic time for the development of the gas flow. The drag force per unit length acting on a long cylindrical wire is studied as a function of the Knudsen number, defined in terms of the mean free path and the radius of the cylinder as . For quasisteady flows, we also present theoretical calculations for the slip regime, , and the free molecular flow regime, . Simulations of unsteady gas flow around a sinusoidally oscillating cylinder near a wall indicate that the drag force per unit length nondimensionalized by approaches constant values for (quasisteady flow) and for . Here is the gas viscosity and is the maximum value of the nanowirevelocity. The simulation results are compared with experimental measurements in the quasisteady regime.

Phasefield simulations of viscous fingering in shearthinning fluids
View Description Hide DescriptionA phasefield model for the HeleShaw flow of nonNewtonian fluids is developed. It extends a previous model for Newtonian fluids to a wide range of fluids with a sheardependent viscosity. The model is applied to simulate viscous fingering in shearthinning fluids and found to capture the complete crossover from the Newtonian regime at lowshear rate to the strongly shearthinning regime. The width selection of a single steadystate finger is studied in detail for a twoplateau shearthinning law (Carreau’s law) in both its weakly and strongly shearthinning limits, and the results are related to the previous analyses. For powerlaw (Ostwald–de Waele) fluids in the strongly shearthinning regime, good agreement with experimental data from the literature is obtained.
 Particulate, Multiphase, and Granular Flows

Particle motion between parallel walls: Hydrodynamics and simulation
View Description Hide DescriptionThe lowReynoldsnumber motion of a single spherical particle between parallel walls is determined from the exact reflection of the velocity field generated by multipoles of the force density on the particle’s surface. A grand mobility tensor is constructed and couples these force multipoles to moments of the velocity field in the fluid surrounding the particle. Every element of the grand mobility tensor is a finite, ordered sum of inverse powers of the distance between the walls. These new expressions are used in a set of Stokesian dynamics simulations to calculate the translational and rotational velocities of a particle settling between parallel walls and the Brownian drift force on a particle diffusing between the walls. The Einstein correction to the Newtonian viscosity of a dilute suspension that accounts for the change in stress distribution due to the presence of the channel walls is determined. It is proposed how the method and results can be extended to computations involving many particles and periodic simulations of suspensions in confined geometries.

Mixing of granular material in rotating cylinders with noncircular crosssections
View Description Hide DescriptionWe study, experimentally and theoretically, the mixing of monodisperse colored beads in rotating cylinders with different crosssectional shapes (square, star, and a circle with two and four triangular wedges), operating in the continuous flow regime, to understand the role of crosssectional shape on the mixing process. The evolution of the mixed state is quantified using two measures: the intensity of segregation and the centroid distance. The mixing rate index, defined as the specific rate of change of the intensity of segregation with time, is initially lower in all the noncircular crosssections compared to a circle. It increases monotonically in noncircular crosssections, whereas it is nearly constant in a circle. This corresponds to a faster than exponential decay of the intensity of segregation for noncircular crosssections as compared to an exponential decay for a circular crosssection. The decay of centroid distance is oscillatory in a circle, whereas it is monotonic in noncircular crosssections. A significant improvement in the mixing rate is obtained for noncircular crosssections relative to circular crosssections. The mixing rate is highest for the circle with four wedges; the mixing rates for square and star crosssections are slightly lower. The circle with two wedges has the lowest mixing rate among the noncircular crosssections. Experiments with smaller glass beads, in which the particle diffusivity is significantly smaller, yield a mixing rate very close to that for the larger particles. Mixing patterns obtained experimentally at short times are well predicted by a convective diffusion model that includes a timeperiodic velocity field for the noncircular crosssections. A quantitative comparison of model predictions with experiments in terms of the intensity of segregation and the centroid distance shows a reasonably good agreement for the different shapes and for the two particle sizes. A scaling analysis is presented to explain the insensitivity of the mixing rate to particle size despite a significant variation in diffusivity. Mixing patterns for a tracer blob and the mixing rates obtained for the different mixers are found to be related to the regions of the chaotic advection in computed Poincaré maps. For the systems studied, the mixing rate increases with an increase in the number of corners in the geometry, but is relatively unaffected by the ratio of the maximum to minimum diameter of the crosssection.

The effects of turbulence on nanoparticle growth in turbulent reacting jets
View Description Hide DescriptionThe effects of turbulence on nanoparticle growth in turbulent reacting flows are studied via a priori analysis of direct numerical simulation data. The formation and growth of titanium dioxide nanoparticles in incompressible planar jets are simulated via gasphase hydrolysis of titanium tetrachloride. The particle field is captured by utilizing a nodal approach which accounts for nucleation,condensation, and Brownian coagulation. Simulations are performed at a single Reynolds number and two different precursor concentration levels. Instantaneous, filtered, and averaged data are presented to convey the nature of turbulent or unresolved contributions to the growth of nanoparticles. The effects of turbulence on particle dynamics, in the context of both Reynoldsaveraged Navier–Stokes simulation and largeeddy simulation, are assessed by comparing the exact, turbulent, and subgridscale growth rates. The results show that large particles are produced in the regions away from the jet core, and an increase in the precursor concentration level increases the particle mean diameter. Particles grow faster when the precursor concentration is increased. It is further observed that the growth rate of the particles is higher inside the eddies and it increases as the jet grows. Additionally, the results show that the unresolved smallscale fluctuations can both augment and inhibit particle growth. However the predominant effect is to reduce particle growth. This tendency is increased (in magnitude) as the precursor concentration level is increased.

Preferential concentration of heavy particles: A Voronoï analysis
View Description Hide DescriptionWe present an experimental characterization of preferential concentration and clustering of inertial particles in a turbulent flow obtained from Voronoï diagram analysis. Several results formerly obtained from various data processing techniques are successfully recovered and further analyzed with Voronoï tesselations as the main single tool. We introduce a simple and nonambiguous way to identify particle clusters. We emphasize the maximum preferential concentration for particles with Stokes numbers around unity and the selfsimilar nature of clustering and we report new unpredicted results concerning clusters inner concentration dependence on Stokes number and global seeding density. Some of these experimental observations can be consistently interpreted in the context of the socalled sweepstick mechanism. Finally, we stress the great potential of Voronoï analysis that offers important openings for new investigations of particle laden flows in terms, for instance, of simultaneous Lagrangian statistics of particle dynamics and local concentration field.

Axial pressuredifference between farfields across a sphere in viscous flow bounded by a cylinder
View Description Hide DescriptionThe presence of a particle with specified velocity inside a cylindrical channel affects the pressurefield along the length of the conduit. In this article, we quantify this effect by using a new general method, which describes hydrodynamic interactions between a cylindrical confinement and a spherical particle under creeping flow assumption. The generality of the scheme enables us to consider arbitrary values for systemdefining parameters like cylindertosphere ratio or separation between their centers. As a result, we can obtain accurate results for the parameter values hitherto unexplored by previous studies. Our simulations include three cases. First, we consider a fixed spherical obstacle in a pressuredriven flow through the cylinder and find the additional pressure drop due to the blockage. Then, we compute the pressure created by the pistonlike effect of a translating sphere inside a cylinderbound quiescent fluid. Finally, we analyze the farfield pressure variation due to rotation of an asymmetrically situated sphere in confined quiescent fluid. For limiting cases, our calculations agree with existing results within 0.5% relative error. Moreover, the efficiency of the scheme is exploited in a dynamic simulation where flow dynamics due to a sedimenting sphere under gravity inside a cylinder with different inclination is explored. We determine the particle trajectory as well as the timedependent farfield pressuredifference created due to the sedimentation process. The results agree well with approximate analytical expressions describing the underlying physics.
 Laminar Flows

On the secondary flow through bifurcating pipes
View Description Hide DescriptionThe secondary motion induced by flow through curves and bifurcations has been subject to investigation over long time due to its importance in physiological and technological applications. In contrast to the flow in a straight pipe, curvature leads to the formation of secondary flow which is often unsteady. Streamline curvature occurs also in bifurcating pipes leading to some corresponding secondary, unsteady flow. This paper presents a detailed description of the unsteady flow in the daughter branch after a 90° bifurcation. A range of Reynolds and Womersley numbers are investigated. The results show the presence of Dean vortices and additional vortical patterns not reported in the literature. Both the streamwise (axial) and the secondary velocity components change character at larger Womersley numbers, leading to less complex secondary flow. Also, at larger Reynolds numbers,flow instabilities are observed. The secondary flow may lead to the formation of unsteady separation bubbles. This in turn yields peaks in the wall shear stress components. Such wall shear stress variations have often been related in the literature to the development process of atherosclerosis.
 Instability and Transition

Influence of a low frequency vibration on a longwave Marangoni instability in a binary mixture with the Soret effect
View Description Hide DescriptionWe study the influence of a low frequency vibration on a longwave Marangoni convection in a layer of a binary mixture with the Soret effect. A linear stability analysis is performed numerically by means of the Floquet theory; several limiting cases are treated analytically. Competition of subharmonic, synchronous, and quasiperiodic modes is considered. The vibration is found to destabilize the layer, decreasing the stability threshold. Also, a vibrationinduced mode is detected, which takes place even for zero Marangoni number.

Thin film lubrication dynamics of a binary mixture: Example of an oscillatory instability
View Description Hide DescriptionWe study thin film instabilities in liquid films with deformable surface using the lubrication theory. An externally applied vertical temperature gradient may give cause to an instability (Marangoni instability) of the flat motionless film. Contrary to the earlier work where mostly pure fluids were discussed, the focus of the present paper lays on instabilities in mixtures of two completely miscibleliquids. We show that the normally found monotonic longwave instability may turn into an oscillatory one if the two components have a different surface tension and if the Soret coefficient establishes a stabilizing vertical concentration gradient. A systematic derivation of the basic equations in longwave approximation is given. The character of instabilities is studied using linear stability analysis. Finally, a real system consisting of a waterisopropanol mixture is discussed in some detail.

On Richtmyer–Meshkov instability in dilute gasparticle mixtures
View Description Hide DescriptionRichtmyer–Meshkov instability(RMI) in gasparticle mixtures is investigated both numerically and analytically. The linear amplitude growth rate for a RMI in a twophase mixture is derived by using a dusty gas formulation for small Stokes number , and it is shown that the problem can be characterized by mass loading and St. The model predictions are compared with numerical results under two conditions, i.e., a shock wave hitting (1) a perturbed species interface of air and surrounded by uniformly distributed particles, and (2) a perturbed shape particle cloud in uniform air. In the first case, the interaction between the instability of the species perturbation and the particles is investigated. The multiphase growth model accurately predicts the growth rates when , and the amplitude growth normalized by the twophase RMI velocity shows good agreement with the singlephase RMI growth rate as well. It is also shown that the twophase model results are in accordance with the growth rates obtained from the simulations even for cases corresponding to . However, for , particles do not follow the RMImotion, and the RMI growth rate agrees with the original Richtmyer’s model [R. D. Richtmyer, “Taylor instability in shock acceleration of compressible fluids,” Commun. Pure Appl. Math.13, 297 (1960)]. Preferential concentration of particles are observed around the RMI rollups at late times when St is of order unity, whereas when , the particles respond rapidly to the flow, causing them to distribute within the rollups. In the second problem, the twophase RMI growth model is extended to study whether a perturbed dusty gas front shows RMIlike growth due to the impact of a shock wave. When , good agreement with the multiphase model is again seen. Moreover, the normalized growth rates are very close to the singlephase RMI growth rates even at late times, which suggest that the twophase growth model is applicable to this type of perturbed shape particle clouds as well. However, when St is close to unity or larger , the particles do not experience impulsive acceleration but rather a continuous one, which results in exponential growth rates as seen in a Rayleigh–Taylor instability.
 Turbulent Flows

Experimental study of highly turbulent isothermal opposedjet flows
View Description Hide DescriptionOpposedjet flows have been shown to provide a valuable means to study a variety of combustion problems, but have been limited to either laminar or modestly turbulent conditions. With the ultimate goal of developing a burner for laboratory flames reaching turbulence regimes of relevance to practical systems, we characterized highly turbulent, strained, isothermal, opposedjet flows using particle image velocimetry (PIV). The bulk strain rate was kept at and specially designed and properly positioned turbulence generation plates in the incoming streams boosted the turbulence intensity to well above 20%, under conditions that are amenable to flame stabilization. The data were analyzed with proper orthogonal decomposition (POD) and a novel statistical analysis conditioned to the instantaneous position of the stagnation surface. Both POD and the conditional analysis were found to be valuable tools allowing for the separation of the truly turbulent fluctuations from potential artifacts introduced by relatively lowfrequency, largescale instabilities that would otherwise partly mask the turbulence. These instabilities cause the stagnation surface to wobble with both an axial oscillation and a precession motion about the system axis of symmetry. Once these artifacts are removed, the longitudinal integral length scales are found to decrease as one approaches the stagnation line, as a consequence of the strained flow field, with the corresponding outer scale turbulentReynolds number following a similar trend. The Taylor scale Reynolds number is found to be roughly constant throughout the flow field at about 200, with a value virtually independent of the data analysis technique. The novel conditional statistics allowed for the identification of highly convoluted stagnation lines and, in some cases, of strong threedimensional effects, that can be screened, as they typically yield more than one stagnation line in the flow field. The ability to lock on the instantaneous stagnation line, at the intersection of the stagnation surface with the PIV measurement plane, is particularly useful in the combustion context, since the flame is aerodynamically stabilized in the vicinity of the stagnation surface. Estimates of the ratio of the mean residence time (inverse strain rate) to the vortex turnover time yield values greater than unity. The conditional mean velocity gradient suggests that, in contrast to the existing literature, the highest gradients are around the system centerline, which would result in a higher probability of flameextinction in that region under chemically reacting conditions. The compactness of the domain and the short mean residence time render the system well suited to direct numerical simulation, more so than conventional jet flames.

Analysis of turbulent transport and mixing in transitional Rayleigh–Taylor unstable flow using direct numerical simulation data
View Description Hide DescriptionData from a direct numerical simulation (DNS) [N. J. Mueschke and O. Schilling, “Investigation of Rayleigh–Taylor turbulence and mixing using direct numerical simulation with experimentally measured initial conditions. I. Comparison to experimental data,” Phys. Fluids21, 014106 (2009)] of a transitional Rayleigh–Taylor mixing layer modeled after a small Atwood number water channel experiment is used to comprehensively investigate the structure of mean and turbulenttransport and mixing. The simulation had physical parameters and initial conditions approximating those in the experiment. The budgets of the mean vertical momentum, heavyfluid mass fraction, turbulent kinetic energy, turbulent kinetic energy dissipation rate, heavyfluid mass fraction variance, and heavyfluid mass fraction variance dissipation rate equations are constructed using Reynolds averaging applied to the DNS data. The relative importance of mean and turbulent production, turbulent dissipation and destruction, and turbulenttransport are investigated as a function of Reynolds number and across the mixing layer to provide insight into the flowdynamics not presently available from experiments. The analysis of the budgets supports the assumption for small Atwood number, Rayleigh–Taylor driven flows that the principal transport mechanisms are buoyancy production, turbulent production, turbulent dissipation, and turbulent diffusion (shear and mean field production are negligible). As the Reynolds number increases, the turbulent production in the turbulent kinetic energy dissipation rate equation becomes the dominant production term, while the buoyancy production plateaus. Distinctions between momentum and scalar transport are also noted, where the turbulent kinetic energy and its dissipation rate both grow in time and are peaked near the center plane of the mixing layer, while the heavyfluid mass fraction variance and its dissipation rate initially grow and then begin to decrease as mixing progresses and reduces density fluctuations. All terms in the transport equations generally grow or decay, with no qualitative change in their profile, except for the pressure flux contribution to the total turbulent kinetic energy flux, which changes sign early in time (a countergradient effect). The productiontodissipation ratios corresponding to the turbulent kinetic energy and heavyfluid mass fraction variance are large and vary strongly at small evolution times, decrease with time, and nearly asymptote as the flow enters a selfsimilar regime. The latetime turbulent kinetic energy productiontodissipation ratio is larger than observed in sheardriven turbulent flows. The order of magnitude estimates of the terms in the transport equations are shown to be consistent with the DNS at latetime, and also confirms both the dominant terms and their evolutionary behavior. These results are useful for identifying the dynamically important terms requiring closure, and assessing the accuracy of the predictions of Reynoldsaveraged Navier–Stokes and largeeddy simulation models of turbulenttransport and mixing in transitional Rayleigh–Taylor instabilitygeneratedflow.