Volume 23, Issue 12, December 2011

We present a study on pattern formation in a Newtonian liquid during lifting of a circular Hele–Shaw cell. When a confined layer of oil is subject to such a stretch flow, air penetrates into the liquid from the sides and a fingering instability, a variant of the classical Saffman–Taylor instability, evolves. This setting has the particularity that the finger growth takes place in a conserved volume of liquid and that the dimensionless surface tension, the control parameter which governs the Saffman–Taylor instability, is changing with time. This leads to a constantly evolving pattern, which we investigate with regard to number of fingers and finger amplitude. We distinguish in the pattern at each instant growing fingers and stagnant fingers. Systematically varying the properties of the viscous oil and the geometry of the Hele–Shaw cell, we show that the number of growing fingers is at each moment well described by a simple approach based on linear stability analysis and depends only on the dimensionless surface tension. In contrast, the finger amplitude and consequently the total number of fingers (growing and stagnant fingers) depend also on the cell confinement. We demonstrate that the finger amplitude has a distinct influence on the debonding force. Higher finger amplitude and number of fingers lead to lower forces.
 ANNOUNCEMENTS
 LETTERS


A new unstable mode in the wake of a circular cylinder
View Description Hide DescriptionThe flow past a circular cylinder looses stability at Re ∼ 47, via the primary wake (PW) mode. Linear stability analysis of the steady base flow, in two dimensions, is conducted using a stabilized finite element formulation. A new mode, referred to as the secondary wake (SW) mode, is discovered which is found to be unstable for Re ≥ 110.8. The relative roles of the PW and SW mode in the development of Karman vortex shedding are also investigated.

The relationship between the velocity skewness and the amplitude modulation of the small scale by the large scale in turbulent boundary layers
View Description Hide DescriptionA defining feature of the innerouter interactions in wallbounded turbulent flows is the imprint of the outer largescale motions on the inner small scale. Recently, Mathis et al. [“Largescale amplitude modulation of the smallscale structures in turbulent boundary layers,” J. Fluid Mech. 628, 311 (2009)] quantified this imprint by applying the Hilbert transform to smallscale components of the fluctuating streamwise velocity, u. They found that the wallnormal profile of the amplitude modulation between the large scale and the envelope of the small scale exhibits strong resemblance to the skewness profile of u. In this study, we assess this apparent relationship and show that the Reynolds number trend in the skewness profile of u is strongly related to the amplitude modulation effect of the small scales by the large. This observation also leads to an alternative diagnostic for the amplitude modulation effect, which is one component of the skewness factor based on a scale decomposition.
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 ARTICLES

 Biofluid Mechanics

The dynamics of a vesicle in a wallbound shear flow
View Description Hide DescriptionThe threedimensional dynamics of a lipid vesicle in a wallbound shear flow is simulated by a Stokesflow boundary integralequation method. When the vesicle is far away from the wall, the wall induces a lift velocity that is proportional to the wall normal component of the particle stresslet and is inversely proportional to the square of its centroid height. When the vesicle is in close contact with the wall under the action of gravity, its bottom surface height scales linearly with the shear rate, with a scaling constant that depends strongly on its nonsphericity. The numerical results are in quantitative agreement with the experimental measurements. The wall boundary causes the particle shear stress and normal stress differences to increase, but the effect diminishes when the centroid height is more than twice the vesicle radius. The simulation shows that the presence of the wall delays the transition (i.e., creates higher critical viscosity ratios) from the tanktreading motion to trembling and tumbling.

Correlations and fluctuations of stress and velocity in suspensions of swimming microorganisms
View Description Hide DescriptionActive systems, which are driven out of equilibrium, can produce long range correlations and large fluctuations that are not restricted by the fluctuationdissipation theorem. We consider here the fluctuations and correlations in suspensions of swimming microorganisms that interact hydrodynamically. Modeling the organisms as force dipoles in Stokes flow and considering runandtumble and rotational diffusion models of their orientational dynamics allow derivation of closed form results for the stress fluctuations in the longwave limit. Both of these models lead to Lorentzian distributions, in agreement with some experimental data. These fluctuations are not restricted by the fluctuationdissipation theorem, as is explicitly verified by comparing the fluctuations with the viscosity of the suspension. In addition to the stress fluctuations in the suspension, we examine correlations between the organisms. Because of the hydrodynamic interactions, the velocities of two organisms are correlated even if the positions and orientations are uncorrelated. We develop a theory of the velocity correlations in this limit and compare with the results of computer simulations. We also formally include orientational correlations in the theory; and comparing with simulations, we are able to show that these are important even in the dilute limit and are responsible in large part for the velocity correlations. While the orientation correlations cannot as yet be predicted from this theory, by inserting the results from simulations into the theory it is possible to properly determine the form of the swimmer velocity correlations. These correlations of orientations are also the key to understanding the spatial correlations of the fluid velocity. Through simulations we show that the orientational correlations decay as r ^{−2} with distance—inserting this dependence into the theory leads to a logarithmic dependence of the velocity fluctuations on the size of the system.
 Micro and Nanofluid Mechanics

Interaction of a liquid flow around a micropillar with a gas jet
View Description Hide DescriptionAn experimental study was conducted to investigate twophase flow characteristics resulting from gas jet injection into a 225 μm high by 1500 μm wide microchannel. The jet was injected from a 25 μm wide slit on the downstream side of a 150 μm diameter pillar. The liquidReynolds number (Re = ρUD/μ) based on pillar diameter ranged from 100 to 700, and the average gas momentum coefficient (ρ_{jet} U_{jet}A_{jet} /ρ_{main} U_{main}A_{ref} ), defined as the ratio of gas momentum to liquid momentum, ranged from 1.6 × 10^{−5} to 3.368 × 10^{−1}. Flow visualization, micro particle image velocimetry (μPIV), and micro particle tracing velocimetry (μPTV) were used to elucidate the twophase flow patterns, liquid velocity field, and bubble dynamics. Two modes of gas jets were observed in which bubbles either formed and detached at the pillar or formed an attached ligament that sheared bubbles from its trailing edge. The modes were determined to be primarily Reynolds number dependent. Both modes were observed to positively affect turbulent kinetic energy in the microchannel. The momentum coefficient of the gas jet had the most significant effect at low Reynolds numbers, when bubble formation took place at the pillar.

Dipolophoresis of interacting conducting nanoparticles of finite electric double layer thickness
View Description Hide DescriptionA general integral method is presented for calculating the dipolophoretic velocities of two interacting, ideally polarizable colloids of arbitrary electric double layer thickness under weak AC electric forcing. The 12 nonlinear mobilities are comprised of inducedchargeelectrophoresis (ICEP), dielectrophoresis(DEP), and FaxénStokes contributions. The explicit integral scheme, based on the Teubner [J. Chem. Phys. 76, 5564 (1982)] formulation, is demonstrated for the case of twosphere interaction. Further simplifications using the remotesphere approximation are employed and the asymptotic results thus obtained are compared against those recently obtained by Saintillan [Phys. Fluids 20, 067104 (2008)] and extend the latter for finite Debye scales and forcing frequencies. It is also shown that the same methodology can be used to determine the mobility of a polarized particle in the proximity of an insulating or conducting plane boundary. The case of a spherical colloid near an uncharged insulating planar wall is of special interest and by using the Lorentz image solution, we readily recover the largespacing approximation of Yariv [Proc. R. Soc. A. London Ser. A 465, 709 (2009)] as a limiting case.
 Interfacial Flows

Asymptotic behavior of a retracting twodimensional fluid sheet
View Description Hide DescriptionTwodimensional (2D) capillary retraction of a viscousliquid film is studied using numerical and analytical approaches for both diphasic and free surface flows. Full 2D NavierStokes equations are integrated numerically for the diphasic case, while onedimensional (1D) free surface modelequations are used for free surface flows. No pinchoff is observed in the film in any of these cases. By means of an asymptotic matching method on the 1D model, we derive an analytical expansion of the film profile for large times. Our analysis shows that three regions with different timescales can be identified during retraction: the rim, the film, and an intermediate domain connecting these two regions. The numerical simulations performed on both models show good agreement with the analytical results. Finally, we report the appearance of an instability in the diphasic retracting film for small Ohnesorge number. We understand this as a KelvinHelmholtz instability arising due to the formation of a shear layer in the neck region during the retraction.

Gravitydriven flow over heated, porous, wavy surfaces
View Description Hide DescriptionThe method of weighted residuals for thin film flow down an inclined plane is extended to include the effects of bottom waviness, heating, and permeability in this study. A bottom slip condition is used to account for permeability and a constant temperature bottom boundary condition is applied. A weighted residual model (WRM) is derived and used to predict the combined effects of bottom waviness, heating, and permeability on the stability of the flow. In the absence of bottom topography, the results are compared to theoretical predictions from the corresponding Benney equation and also to existing OrrSommerfeld predictions. The excellent agreement found indicates that the model does faithfully predict the theoretical critical Reynolds number, which accounts for heating and permeability, and these effects are found to destabilize the flow. Floquet theory is used to investigate how bottom waviness influences the stability of the flow. Finally, numerical simulations of the modelequations are also conducted and compared with numerical solutions of the full NavierStokes equations for the case with bottom permeability. These results are also found to agree well, which suggests that the WRM remains valid even when permeability is included.

On the validity of a universal solution for viscous capillary jets
View Description Hide DescriptionIn this paper, we assess the validity of a universal solution based on the slenderness approximation to describe the velocity and shape of viscous capillary jets produced by two very different mechanisms: the action of the constant gravity force and the focusing effect of a coflowing gas stream. In the gravitational case, the jet’s velocity distribution given by the universal solution is compared with that calculated numerically from the NavierStokes equations. The universal solution provides remarkably good predictions for the wide range of parameters considered in this work. Its accuracy generally improves as the Reynolds number increases and/or the Froude number decreases, probably because the jet viscous region decreases in this case. The flow focusing method was examined experimentally by acquiring and processing images of the tapering liquidmeniscus formed between the feeding capillary and the discharge orifice. In this case, the universal solution provides satisfactory results for sufficiently slender liquidmeniscus (i.e., for sufficiently large liquidviscosities and flow rates and small applied pressure drops), provided that the ratio capillarytoorifice distance H to orifice diameter D takes sufficiently small values. If these conditions are not satisfied, the universal solution underestimates the jet radius close to the feeding capillary, but it still provides accurate predictions beyond the discharge orifice. For small H/D values, the accuracy of the universal solution is mainly limited by radial momentum effects associated with the sharp contraction of the meniscus shape, which becomes less slender as the liquidviscosity and flow rate decrease, or the pressure drop increases. For large H/D values, the driving force significantly deviates from its assumed constant value in the universal solution, giving rise to larger discrepancies between that solution and the experimental results even for slender shapes.

Sustained inertialcapillary oscillations and jet formation in displacement flow in a tube
View Description Hide DescriptionWe study inertial effects in the displacement of a fluid in a capillary by a more viscous fluid, using a numerical method. A levelset approach is employed to track the meniscus, and a Navier slip boundary condition is imposed in order to alleviate a stress singularity at the moving contact line. Various flow regimes are identified with a Reynolds number and a capillary number as the main parameters. At relatively low Reynolds number, the meniscus forms a steady shape, and the interfacial curvature at the tube centre can change from being concave to convex upon increasing the Reynolds number if the displacing fluid is wetting to the tube surface. For wetting displacing fluids, beyond a critical Reynolds number, a quasisteady solution is no longer found: instead, the interface undergoes nondampened periodic oscillations and, at even larger values of the Reynolds number, quasiperiodically, and the interface evolves from simple wavy shapes to complex shapes with multiple wavy units. This oscillating state is observed for sufficiently small contact angle values defined from the displacing fluid (<80°). Beyond a second critical Reynolds number, the displacing fluid forms a jet and the meniscus advances with a nearly constant speed which decreases with Re. This is also observed at large contact angle values. In a developing jet, however, the interface shape remains partially quasisteady, near the contact line region and the tube centre. The flow behaviour is shown to be robust over a range of other governing parameters, including the capillary number and the slip length. The potential implications of the work on network models of twophase flow through porous media are discussed.

Formation of organized nanostructures from unstable bilayers of thin metallic liquids
View Description Hide DescriptionDewetting of pulsedlaser irradiated, thin (<20 nm), optically reflective metallic bilayers on an optically transparent substrate with a reflective support layer is studied within the lubrication equations model. A steadystate bilayer film thickness (h) dependent temperature profile is derived based on the mean substrate temperature estimated from the elaborate thermal model of transient heating and melting/freezing. Large thermocapillary forces are observed along the plane of the liquidliquid and liquidgas interfaces due to this hdependent temperature, which, in turn, is strongly influenced by the hdependent laser light reflection and absorption. Consequently the dewetting is a result of the competition between thermocapillary and intermolecular forces. A linear analysis of the dewetting length scales established that the nonisothermal calculations better predict the experimental results as compared to the isothermal case within the bounding Hamaker coefficients. Subsequently, a computational nonlinear dynamics study of the dewetting pathway was performed for Ag/Co and Co/Ag bilayer systems to predict the morphology evolution. We found that the systems evolve towards formation of different morphologies, including coreshell, embedded, or stacked nanostructure morphologies.

Surfactantdriven dynamics of liquid lenses
View Description Hide DescriptionSessile liquid lenses spreading over a fluid layer, in the presence of Marangoni stresses due to surfactants, show a surprisingly wide range of interesting behaviour ranging from complete spreading of the lens, to spreading followed by retraction, to sustained pulsating oscillations. Models for the spreading process, the effects of surfactant at the moving contact line, sorption kinetics above and below the critical micelle concentration, are all incorporated into the modelling. Numerical results cast light upon the physical processes that drive these phenomena, and the regular oscillatory beating of lenses is shown to occur in specific limits.

Scaling percolation in thin porous layers
View Description Hide DescriptionPercolation in porous media is a complex process that depends on the flow rate, material, and fluids properties as well as the boundary conditions. Traditional methods of characterizing percolation rely upon visual observation of a flow pattern or a pressuresaturation relation valid only in the limit of no flow. In this paper, the dynamics of fluid percolation in thin porous media is approached through a new scaling. This new scaling in conjunction with the capillary number and the viscosity ratio has resulted in a linear nondimensional correlation of the percolation pressure and wetted area in time unique to each porous media. The effect of different percolationflow patterns on the dynamic pressuresaturation relation can be condensed into a linear correlation using this scaling. The general trend and implications of the scaling have been analyzed using an analytical model of a fluid percolating between two parallel plates and by experimental testing on thin porous media. Cathode porous transport layers (PTLs), also known as gas diffusion layers, of a proton exchange membrane (PEM) fuel cell having different morphological and wetting properties were tested under drainage conditions. Images of the fluid percolation evolution and the percolation pressure in the PTLs were simultaneously recorded. A unique linear correlation is obtained for each type of PTL samples using the new scaling. The correlation derived from this new scaling can be used to quantitatively characterize porous media with respect to percolation. While the characterization method discussed herein was developed for the study of porous materials used in PEM fuel cells, the method and scaling are applicable to any porous media.

Wicking flow through microchannels
View Description Hide DescriptionWe report numerical simulations of wicking through micropores of two types of geometries, axisymmetric tubes with contractions and expansions of the cross section, and twodimensional planar channels with a Yshaped bifurcation. The aim is to gain a detailed understanding of the interfacial dynamics in these geometries, with an emphasis on the motion of the threephase contact line. We adopt a diffuseinterface formalism and use CahnHilliard diffusion to model the moving contact line. The Stokes and CahnHilliard equations are solved by finite elements with adaptive meshing. The results show that the liquid meniscus undergoes complex deformation during its passage through contraction and expansion. Pinning of the interface at protruding corners limits the angle of expansion into which wicking is allowed. For sufficiently strong contractions, the interface negotiates the concave corners, thanks to its diffusive nature. Capillary competition between branches downstream of a Yshaped bifurcation may result in arrest of wicking in the wider branch. Spatial variation of wettability in one branch may lead to flow reversal in the other.
 Viscous and NonNewtonian Flows

Dynamic evolution of fingering patterns in a lifted Hele–Shaw cell
View Description Hide DescriptionWe present a study on pattern formation in a Newtonian liquid during lifting of a circular Hele–Shaw cell. When a confined layer of oil is subject to such a stretch flow, air penetrates into the liquid from the sides and a fingering instability, a variant of the classical Saffman–Taylor instability, evolves. This setting has the particularity that the finger growth takes place in a conserved volume of liquid and that the dimensionless surface tension, the control parameter which governs the Saffman–Taylor instability, is changing with time. This leads to a constantly evolving pattern, which we investigate with regard to number of fingers and finger amplitude. We distinguish in the pattern at each instant growing fingers and stagnant fingers. Systematically varying the properties of the viscous oil and the geometry of the Hele–Shaw cell, we show that the number of growing fingers is at each moment well described by a simple approach based on linear stability analysis and depends only on the dimensionless surface tension. In contrast, the finger amplitude and consequently the total number of fingers (growing and stagnant fingers) depend also on the cell confinement. We demonstrate that the finger amplitude has a distinct influence on the debonding force. Higher finger amplitude and number of fingers lead to lower forces.

Nonlinear viscous fluid patterns in a thin rotating spherical domain and applications
View Description Hide DescriptionWe study the nonlinear incompressible fluid flows within a thin rotating spherical shell. The model uses the twodimensional NavierStokes equations on a rotating threedimensional spherical surface and serves as a simple mathematical descriptor of a general atmospheric circulation caused by the difference in temperature between the equator and the poles. Coriolis effects are generated by pseudoforces, which support the stable westtoeast flows providing the achievable meteorological flows rotating around the poles. This work addresses exact stationary and nonstationary solutions associated with the nonlinear NavierStokes. The exact solutions in terms of elementary functions for the associated Euler equations (zero viscosity) found in our earlier work are extended to the exact solutions of the NavierStokes equations (nonzero viscosity). The obtained solutions are expressed in terms of elementary functions, analyzed, and visualized.
 Particulate, Multiphase, and Granular Flows

Turbulence modification and heat transfer enhancement by inertial particles in turbulent channel flow
View Description Hide DescriptionWe present results of direct numerical simulation of turbulence modification and heat transfer in turbulent particleladen channel flow and show an enhancement of the heat transfer and a small increase in the frictionvelocity when heavy inertial particles with high specific heat capacity are added to the flow. The simulations employ a coupled EulerianLagrangian computational model in which the momentum and energy transfer between the discrete particles and the continuous fluid phase are fully taken into account. The effect of turbophoresis, resulting in an increased particle concentration near a solid wall due to the inhomogeneity of the wallnormal velocityfluctuations, is shown to be responsible for an increase in heat transfer. As a result of turbophoresis, the effective macroscopic transport properties in the region near the walls differ from those in the bulk of the flow. To support the turbophoresis interpretation of the enhanced heat transfer, results of simulations employing no particlefluid coupling and simulations with twoway coupling at considerably lower specific heat, or considerably lower particle concentration are also included. The combination of these simulations allows distinguishing contributions to the Nusselt number due to mean flow,turbulent fluctuations and explicit particle effects. We observe an increase in Nusselt number by more than a factor of two for heavy inertial particles, which is the net result of a decrease in heat transfer by turbulentvelocityfluctuations and a much larger increase in heat transfer stemming from the mean temperature difference between the fluid and the particles close to the walls.

Inertial migration of deformable capsules in channel flow
View Description Hide DescriptionUsing threedimensional computer simulations, we study the crossstream inertial migration of neutrally buoyant deformable particles in a pressuredriven channel flow. The particles are modeled as elastic shells filled with a viscous fluid. We show that the particles equilibrate in a channel flow at offcenter positions that depend on particle size, shell compliance, and the viscosity of encapsulated fluid. These equilibrium positions, however, are practically independent of the magnitude of channel Reynolds number in the range between 1 and 100. The results of our studies can be useful for sorting, focusing, and separation of micrometersized synthetic particles and biological cells.

Threedimensional numerical simulation of drops suspended in Poiseuille flow at nonzero Reynolds numbers
View Description Hide DescriptionA finite difference/front tracking method is used to study the motion of threedimensional deformable drops suspended in plane Poiseuille flow at nonzero Reynolds numbers. A parallel version of the code was used to study the behavior of suspension on a reasonable grid resolution ( grids). The viscosity and density of drops are assumed to be equal to that of the suspending medium. The effect of Capillary number, the Reynolds number, and volume fraction are studied in detail. It is found that drops with small deformation behave like rigid particles and migrate to an equilibrium position about half way between the wall and the centerline (the SegreSilberberg effect). However, for highly deformable drops there is a tendency for drops to migrate to the middle of the channel, and the maximum concentration occurs at the centerline. The concentration profile obtained across the channel is in agreement with that measured by Kowalewski (T. A. Kowalewski, “Concentration and velocity measurement in the flow of dropletsuspensions through a tube,” Exp. Fluids 2, 213 (1984)) experimentally for viscosity ratios less than or equal to one. The effective viscosity of suspension decreases with Capillary number in agreement with the creeping flow limit. Also, the effective viscosity increases with the Reynolds number of the flow.