Volume 24, Issue 1, January 2012

We simulate the crossflow migration of rigid particles such as platelets in a red blood cell (RBC) suspension using the Stokes flowboundary integralequation method. Two types of flow environments are investigated: a suspension undergoing a bulk shear motion and a suspension flowing in a microchannel or duct. In a cellularsuspension undergoing bulk shear deformation, the crossflow migration of particles is diffusional. The velocity fluctuations in the suspension, which are the root cause of particle migration, are analyzed in detail, including their magnitude, the autocorrelation of Lagrangian tracer points and particles, and the associated integral time scales. The orientation and morphology of red blood cells vary with the shear rate, and these in turn cause the dimensionless particle diffusivity to vary nonmonotonically with the flow capillary number. By simulating RBCs and platelets flowing in a microchannel of 34 μm height, we demonstrate that the velocity fluctuations in the core cellularflow region cause the platelets to migrate diffusively in the wall normal direction. A mean lateral velocity of particles, which is most significant near the edge of the cellfree layer, further expels them toward the wall, leading to their excess concentration in the cellfree layer. The calculated shearinduced particle diffusivity in the cellladen region is in qualitative agreement with the experimental measurements of micronsized beads in a cylindrical tube of a comparable diameter. In a smaller duct of 10 × 15 μm cross section, the volume exclusion becomes the dominant mechanism for particle margination, which occurs at a much shorter time scale than the migration in the bigger channel.
 ARTICLES

 Biofluid Mechanics

A twosphere model for bacteria swimming near solid surfaces
View Description Hide DescriptionWe present a simple model for bacteria like Escherichia coli swimming near solid surfaces. It consists of two spheres of different radii connected by a dragless rod. The effect of the flagella is taken into account by imposing a force on the tail sphere and opposite torques exerted by the rod over the spheres. The hydrodynamic forces and torques on the spheres are computed by considering separately the interaction of a single sphere with the surface and with the flow produced by the other sphere. Numerically, we solve the linear system which contains the geometrical constraints and the forcefree and torquefree conditions. The dynamics of this swimmer near a solid boundary is very rich, showing three different behaviors depending on the initial conditions: (1) swimming in circles in contact with the wall, (2) swimming in circles at a finite distance from the wall, and (3) swimming away from it. Furthermore, the order of magnitude of the radius of curvature for the circular motion is in the range m, close to values observed experimentally.

Shearinduced particle migration and margination in a cellular suspension
View Description Hide DescriptionWe simulate the crossflow migration of rigid particles such as platelets in a red blood cell (RBC) suspension using the Stokes flowboundary integralequation method. Two types of flow environments are investigated: a suspension undergoing a bulk shear motion and a suspension flowing in a microchannel or duct. In a cellularsuspension undergoing bulk shear deformation, the crossflow migration of particles is diffusional. The velocity fluctuations in the suspension, which are the root cause of particle migration, are analyzed in detail, including their magnitude, the autocorrelation of Lagrangian tracer points and particles, and the associated integral time scales. The orientation and morphology of red blood cells vary with the shear rate, and these in turn cause the dimensionless particle diffusivity to vary nonmonotonically with the flow capillary number. By simulating RBCs and platelets flowing in a microchannel of 34 μm height, we demonstrate that the velocity fluctuations in the core cellularflow region cause the platelets to migrate diffusively in the wall normal direction. A mean lateral velocity of particles, which is most significant near the edge of the cellfree layer, further expels them toward the wall, leading to their excess concentration in the cellfree layer. The calculated shearinduced particle diffusivity in the cellladen region is in qualitative agreement with the experimental measurements of micronsized beads in a cylindrical tube of a comparable diameter. In a smaller duct of 10 × 15 μm cross section, the volume exclusion becomes the dominant mechanism for particle margination, which occurs at a much shorter time scale than the migration in the bigger channel.
 Micro and Nanofluid Mechanics

Scaling laws for slippage on superhydrophobic fractal surfaces
View Description Hide DescriptionWe study the slippage on hierarchical fractalsuperhydrophobicsurfaces and find an unexpected rich behavior for hydrodynamicfriction on these surfaces. We develop a scaling law approach for the effective slip length, which is validated by numerical resolution of the hydrodynamic equations. Our results demonstrate that slippage does strongly depend on the fractal dimension and is found to be always smaller on fractalsurfaces as compared with surfaces with regular patterns. This shows that in contrast to naive expectations, the value of effective contact angle is not sufficient to infer the amount of slippage on a fractalsurface: depending on the underlying geometry of the roughness, strongly superhydrophobicsurfaces may, in some cases, be fully inefficient in terms of drag reduction. Finally, our scaling analysis can be directly extended to the study of heat transfer at fractalsurfaces, in order to estimate the Kapitsa surface resistance on patternedsurfaces, as well as to the question of trapping of diffusing particles by patchy hierarchical surfaces, in the context of chemoreception.

Multiscale modeling of particle in suspension with smoothed dissipative particle dynamics
View Description Hide DescriptionWe apply smoothed dissipative particle dynamics (SDPD) [Español and Revenga, Phys. Rev. E 67, 026705 (2003)] to model solid particles in suspension. SDPD is a thermodynamically consistent version of smoothed particle hydrodynamics (SPH) and can be interpreted as a multiscale particle framework linking the macroscopic SPH to the mesoscopic dissipative particle dynamics (DPD) method. Rigid structures of arbitrary shape embedded in the fluid are modeled by frozen particles on which artificial velocities are assigned in order to satisfy exactly the noslip boundary condition on the solidliquid interface. The dynamics of the rigid structures is decoupled from the solvent by solving extra equations for the rigid body translational/angular velocities derived from the total drag/torque exerted by the surrounding liquid. The correct scaling of the SDPD thermal fluctuations with the fluidparticle size allows us to describe the behavior of the particle suspension on spatial scales ranging continuously from the diffusiondominated regime typical of submicronsized objects towards the nonBrownian regime characterizing macrocontinuum flow conditions. Extensive tests of the method are performed for the case of two/three dimensional bulk particlesystem both in Brownian/nonBrownian environment showing numerical convergence and excellent agreement with analytical theories. Finally, to illustrate the ability of the model to couple with external boundary geometries, the effect of confinement on the diffusional properties of a single sphere within a microchannel is considered, and the dependence of the diffusion coefficient on the wallseparation distance is evaluated and compared with available analytical results.

Numerical demonstration of the reciprocity among elemental relaxation and drivenflow problems for a rarefied gas in a channel
View Description Hide DescriptionRelaxations from a uniform mass/heat flow and flows driven by an external force/temperaturegradient for a rarefied gas between two parallel plates are studied on the basis of the kinetic theory of gases. By numerical computations of the linearized Bhatnagar–Gross–Krook model of the Boltzmann equation, it is demonstrated that the reciprocity among these elemental flows derived from a general reciprocity theory for timedependent problems [S. Takata, J. Stat. Phys. 140, 985 (2010)] holds at any time and any Knudsen numbers. Moreover, a propagation of the discontinuity of the velocity distribution function (VDF) in the relaxation problems and that of the derivative discontinuity of the VDF in the drivenflow problems are demonstrated. Their relation is also clarified.

Dipolophoresis of dielectric spheroids under asymmetric fields
View Description Hide DescriptionNonspherical particles are common in colloidal science. Spheroidal shapes are particularly convenient for the analysis of the pertinent electrostatic and hydrodynamic problems and are thus widely used to model the manipulation of biological cells as well as deformed drops and bubbles. We study the rotary motion of a dielectric spheroidal microparticle which is freely suspended in an unbounded electrolyte solution in the presence of a uniform applied electric field, assuming a thin Debye layer. For the common case of a uniform distribution of the native surfacecharge density, the rotary motion of the particle is generated by the contributions of the inducedcharge electroosmotic (ICEO) slip and the dielectrophoresis associated with the distribution of the Maxwell stress, respectively. Series solutions are obtained by using spheroidal (prolate or oblate) coordinates. Explicit results are presented for the angular velocity of particles spanning the entire spectrum from rodlike to disklike shapes. These results demonstrate the nonmonotonic variation of the angular speed with the eccentricity of particle shape and the singularity of the multiple limits corresponding to conducting (ideally polarizable) particles of extreme eccentricity (e ≈ 1). The nonmonotonic variation of the angular speed with the particle dielectric permittivity is related to the inducedcharge contribution. We apply these results to describe the motion of particles subject to a uniform field rotating in the plane. For a sufficiently slow rotation rate, prolate particles eventually become “locked” to the external field with their stationary relative orientation in the plane of rotation being determined by the particle eccentricity and dielectric constant. This effect may be of potential use in the manipulation of polydisperse suspensions of dielectric nonspherical particles. Oblate spheroids invariably approach a uniform orientation with their symmetry axes directed normal to the externalfield plane of rotation.

Direct simulation Monte Carlobased expressions for the gas mass flow rate and pressure profile in a microscale tube
View Description Hide DescriptionThe direct simulation Monte Carlo (DSMC) method of Bird is used to develop simple closedform expressions for the mass flow rate and the pressure profile for the steady isothermal flow of an ideal gas through a microscale tube connecting two infinite reservoirs at different pressures but at the temperature of the tube wall. Gas molecules reflect from the tube wall according to the Maxwell model (a linear combination of specular and diffuse reflections at the wall temperature) with a unity or subunity value of the accommodation coefficient (the probability that molecules reflect diffusely from the wall). The DSMCbased expressions have four parameters. Two parameters are specified so that the mass flow rate reduces to the known expression in the freemolecular regime. One parameter was previously determined by comparison to DSMC simulations in the slip regime. The remaining parameter is determined by comparison to DSMC simulations for pressures spanning the transition regime with several values of the accommodation coefficient. The expressions for the mass flow rate and the pressure profile agree well with the DSMC simulations (rms and maximum differences of 2% and 5% for all cases examined), with other more complicated expressions and with recent experiments involving microscale tubes and channels for all flow regimes.
 Interfacial Flows

Transient reduction of the drag coefficient of charged droplets via the convective reversal of stagnant caps
View Description Hide DescriptionDroplets are frequently observed to move as if they were solid rather than liquid, i.e., with no slip at the liquidliquid interface. This behavior is usually explained in terms of the socalled “stagnant cap” model, in which surfactants accumulate at the trailing edge of the droplet, immobilizing the surface and increasing the observed drag coefficient. Here, we show that the drag coefficient for chargeddroplets is temporarily reduced by reversing the direction of an electric driving force. Using high speed video, we simultaneously track the velocity and relative interfacial velocity of individual aqueous droplets moving electrophoretically through oil. The observed velocity behavior is highly sensitive to the concentration of surfactant. For sufficiently low or sufficiently high concentration, upon reversal of the electric field the droplet rapidly accelerates in the opposite direction but then decelerates, concurrent with a transient rearrangement of tracer particles on the dropletsurface. In contrast, droplets with intermediate surfactant concentrations exhibit neither deceleration nor significant tracer particle rearrangement. We interpret the observations in terms of convectively dominated rearrangement of the stagnant cap, and we discuss the implications for precise electrophoretic control of dropletmotion in labonachip devices and industrial electrocoalescers.

Stability and breakup of confined threads
View Description Hide DescriptionA boundaryintegral method for periodic arrays of drops, threads or sheets between parallel walls is presented. The Green’s functions take the form of a farfield HeleShaw description, which is used to generate periodic Green’s functions for the parallelwall configuration. The method is applied to study the effect of confinement on the breakup of threads. A comparison is made with classical Tomotika’s theory and growth rates parallel and perpendicular to the walls are determined as a function of confinement ratio. Contrary to existing belief, we find that confined threads are not stable, but that the time for breakup increases with confinement and viscosity ratio, at least for threads whose diameter is smaller than the wallspacing. We also show the inphase and outofphase breakup for an array of threads, as well as the stabilizing effect of shear flow.
 Particulate, Multiphase, and Granular Flows

Particulate mixing in a turbulent serpentine duct
View Description Hide DescriptionDirect numerical simulations of particles in a serpentine duct were conducted at bulk flow Stokes numbers between 0.125 and 6. The geometrical curvature causes particles to depart direction from the mean flow. Above a Stokes number of about unity, a reflection layer forms along the outer curve of the bend. Reflectional mixing creates regions of nearly uniform particle mean velocity and kinetic energy. Particles leave the inner bend in a plume that separates from the inner wall at low Stokes number. At higher Stokes number, the plume splits in two, adding an upper part consisting of ballistic particles, that do not follow the geometrical curvature. When the Stokes number is low, the instantaneous 3D distribution of particles visualizes wall streaks. But at higher Stokes number, particles disperse out of the reflection layer and form large scale puffs in the central portion of the duct.

Rheological measurements of large particles in high shear rate flows
View Description Hide DescriptionThis paper presents experimental measurements of the rheological behavior of liquidsolid mixtures at moderate Stokes and Reynolds numbers. The experiments were performed in a coaxial rheometer that was designed to minimize the effects of secondary flows. By changing the shear rate, particle size, and liquid viscosity, the Reynolds numbers based on shear rate and particle diameter ranged from 20 to 800 (Stokes numbers from 3 to 90), which is higher than examined in earlier rheometric studies. Prior studies have suggested that as the shear rate is increased, particleparticle collisions also increase resulting in a shear stress that depends nonlinearly on the shear rate. However, over the range of conditions that were examined in this study, the shear stress showed a linear dependence on the shear rate. Hence, the effective relative viscosity is independent of the Reynolds and Stokes numbers and a nonlinear function of the solid fraction. The present work also includes a series of roughwall experiments that show the relative effective viscosity is also independent of the shear rate and larger than in the smooth wall experiments. In addition, measurements were made of the nearwall particle velocities, which demonstrate the presence of slip at the wall for the smoothwalled experiments. The depletion layer thickness, a region next to the walls where the solid fraction decreases, was calculated based on these measurements. The relative effective viscosities in the current work are larger than found in lowReynolds number suspension studies but are comparable with a few granular suspension studies from which the relative effective viscosities can be inferred.

A fully resolved numerical simulation of turbulent flow past one or several spherical particles
View Description Hide DescriptionThe flow past one or nine spheres arranged in a plane lattice held fixed in a stream of decaying homogeneous isotropic turbulence is studied by means of fully resolved NavierStokes simulations. The particle radius is 3–5 times the Kolmogorov length and about 1/3 of the integral length scale. The mean particle Reynolds number is 80 and the turbulence intensity 17% and 33%. Several features of the flow are described: the mean and fluctuating dissipation and its spatial distribution, the mean and fluctuatinghydrodynamic forces on the spheres, stimulated vortex shedding, and others. A special attention is paid to the relation between the work done on the fluid by the particles (in the reference frame of the former) and the total dissipation. It is shown that these quantities, which are assumed to balance in many pointparticle models, can actually be very different when inertial effects are important.
 Laminar Flows

Uniformly valid asymptotic flow analysis in curved channels
View Description Hide DescriptionThe laminar incompressible flow in a twodimensional curved channel having at its upstream and downstream extremities two tangent straight channels is considered. A global interactive boundary layer (GIBL) model is developed using the approach of the successive complementary expansions method (SCEM) which is based on generalized asymptotic expansions leading to a uniformly valid approximation. The GIBL model is valid when the non dimensional number is O(1) and gives predictions in agreement with numerical NavierStokes solutions for Reynolds numbersR _{ e } ranging from 1 to 10^{4} and for constant curvatures ranging from 0.1 to 1, where H is the channel width and R _{ c } the curvature radius. The asymptotic analysis shows that μ, which is the ratio between the curvature and the thickness of the boundary layer of any perturbation to the Poiseuille flow, is a key parameter upon which depends the accuracy of the GIBL model. The upstream influence length is found asymptotically and numerically to be .

Mechanism of drag generation by surface corrugation
View Description Hide DescriptionDrag generated by periodic corrugation has been determined analytically in the limit of long corrugation wavelength. Three physical mechanisms have been identified, i.e., the additional shear drag due to an increase of the wetted area and the rearrangement of the shear stress distribution, the pressure form drag associated with the mean pressure gradient, and the pressure interaction drag associated with the phase difference between the surface geometry and the periodic part of the pressure field. The total drag increases rapidly with increase of the corrugation amplitude, with the form and interaction drags contributing up to 45% and 30% of this increase, respectively.
 Instability and Transition

Threedimensional swirling flows in a tall cylinder driven by a rotating endwall
View Description Hide DescriptionThe onset and nonlinear dynamics of swirling flows in relatively tall cylinders driven by the rotation of an endwall are studied numerically. These flows are distinguished from the more widely studied swirling flows in shorter cylinders; the instability in the taller cylinders is direct to threedimensional flows rather than to unsteady axisymmetric flows. The simulations are in very good agreement with recent experiments in terms of the critical Reynolds number, frequency, and azimuthal wavenumber of the flows, but there is disagreement in the interpretation of these flows. We show that these flows are indeed rotating waves and that they have the same vorticity distributions as the flowsmeasured using particle image velocimetry in the experiments. Identifying these as rotating waves gives a direct connection with prior linear stabilityanalysis and the threedimensional flows found in shorter cylinders as secondary instabilities leading to modulated rotating waves.

Negative Magnus lift on a rotating sphere at around the critical Reynolds number
View Description Hide DescriptionNegative Magnus lift acting on a sphere rotating about the axis perpendicular to an incoming flow was investigated using largeeddy simulation at three Reynolds numbers of 1.0 × 10^{4}, 2.0 × 10^{5}, and 1.14 × 10^{6}. The numerical methods used were first validated on a nonrotating sphere, and the spatial resolution around the sphere was determined so as to reproduce the laminar separation, reattachment, and turbulenttransition of the boundary layer observed in the vicinity of the critical Reynolds number. The rotating sphere exhibited a positive or negative Magnus effect depending on the Reynolds number and the imposed rotating speed. At Reynolds numbers in the subcritical or supercritical regimes, the direction of the Magnus lift force was independent of the rotational speed. In contrast, the lift force was negative in the critical regime when particular rotating speeds were imposed. This negative Magnus effect was investigated in the context of suppression or promotion of boundary layertransition around the separation point.

An experimental study of transitional pulsatile pipe flow
View Description Hide DescriptionThe transitional regime of a sinusoidal pulsatile flow in a straight, rigid pipe is investigated using particle imagevelocimetry. The main aim is to investigate how the critical Reynolds number is affected by different pulsatile conditions, expressed as the Womersley number and the oscillatory Reynolds number. The transition occurs in the region of Re = 22503000 and is characterized by an increasing number of isolated turbulence structures. Based on velocity fields and flow visualizations, these structures can be identified as puffs, similar to those observed in steady flowtransition.Measurements at different Womersley numbers yield similar transition behavior, indicating that pulsatile effects do not play a role in the regime that is investigated. Variations of the oscillatory Reynolds number also appear to have little effect, so that the transition here seems to be determined only by the mean Reynolds number. For larger mean Reynolds numbers, a second regime is observed: here, the flow remains turbulent throughout the cycle. The turbulence intensity varies during the cycle, but has a phase shift with respect to the mean flow component. This is caused by a growth of kinetic energy during the decelerating part and a decay during the accelerating part of the cycle. Flow visualization experiments reveal that the flow develops localized turbulence at several random axial positions. The structures quickly grow to fill the entire pipe in the decelerating phase and (partially) decay during the accelerating phase.

Falling liquid films on longitudinal grooved geometries: Integral boundary layer approach
View Description Hide DescriptionFalling thin liquid film on a substrate with complex topography is modeled using a three equation integral boundary layer system. Linear stability and nonlinear dynamics of the film in the framework of this model are studied on a topography with sinusoidal longitudinal grooves aligned parallel in the direction of the main flow. The linear stability theory reveals the stabilizing nature of the surface tension force and the groove measure on the film, and the pronounced destabilizing effects of inertia. The evolution of the film thickness is tracked numerically for a vertically falling film on a grooved geometry by choosing wavenumbers corresponding to the unstable mode where the growth rate of instability is maximum. The effect of surface geometry on the temporal evolution of the film dynamics is analyzed on a periodic domain. Numerical investigations agree with the linear stability predictions and show that the longitudinal grooves exert a stabilizing effect on the film and the waviness is suppressed when the steepness of the longitudinal groove measure increases.

Experimental evidence of a triadic resonance of plane inertial waves in a rotating fluid
View Description Hide DescriptionPlane inertial waves are generated using a wavemaker, made of oscillating stacked plates, in a rotating water tank. Using particle image velocimetry, we observe that, after a transient, the primary plane wave is subject to a subharmonic instability and excites two secondary plane waves. The measured frequencies and wavevectors of these secondary waves are in quantitative agreement with the predictions of the triadic resonance mechanism. The secondary wavevectors are found systematically more normal to the rotation axis than the primary wavevector: this feature illustrates the basic mechanism at the origin of the energy transfers towards slow, quasi twodimensional, motions in rotating turbulence.

The linear stability of oscillating pipe flow
View Description Hide DescriptionAn investigation is made of the threedimensional linear stability of the Stokes layer generated within a fluid contained inside a long oscillating cylinder. Both longitudinal and torsional vibrations are examined and the system of disturbance equations derived using Floquet theory are solved using pseudospectral methods. Critical parameters for instability are obtained for an extensive range of pipe radii and longitudinal and azimuthal wavenumbers. For sufficiently small pipe diameters, threedimensional perturbations are sometimes found to be more unstable than their twodimensional counterparts. In contrast, at larger radii, the threedimensional disturbance modes are less important and the twodimensional versions are expected to be observed in practice. These results imply constraints on experiments that are designed to exhibit shear modes in oscillatory flow.