Volume 22, Issue 7, July 2010

We investigate experimentally the dynamics of a liquid drop that impacts on a solid surface whose wettability is patterned at the microscopic scale. The target surface is patterned such that hydrophilic(hydrophobic) microscale spokes radiate from the center on a hydrophobic(hydrophilic) background. Following the initial spreading stages, the drop recoils on the hydrophobic region while being arrested on the hydrophilic area, thereby resulting in a micropatterned liquid footprint. We also find that the fingering instability of the drop edge is affected by the wettability patterns in the initial spreading stages. The number of fingers depends on a combination of the impact Weber number and the number of spokes. We suggest that the present scheme of patterning liquid deposits at microscales could be exploited in various microfluidic applications.
 LETTERS


The effect of rotational shear on granular discharge rates
View Description Hide DescriptionThe mass flow rate of a packed granular column subject to rotational shear discharging though an aperture is investigated computationally. We show that the flow rate increases in response to the applied shear. This increase obeys a powerlaw relation based on the rotational Froude number, , where is the angular rotation speed and is a system length scale. The exponent of this relation is found to be independent of the particle diameter, intergrain friction, and the geometry of the setup.

Global mode analysis of a pipe flow through a 1:2 axisymmetric sudden expansion
View Description Hide DescriptionWe report the results of the global mode analysis to characterize the onset of unsteadiness in a circular pipe flow through an axisymmetric sudden expansion of inlettooutlet diameter ratio of . We find that the axisymmetric state becomes linearly unstable at a significantly higher critical Reynolds number than the one reported in previous experimental works. This unstable global mode corresponds to an oscillatory bifurcation with wavenumber located at the end of the recirculation region.

Sectional lift coefficient of a flapping wing in hovering motion
View Description Hide DescriptionWe investigate the behavior of sectional lift coefficient of a flapping wing of a fruitfly in hovering motion. Through threedimensional numerical simulations, we show that during the stroke, the sectional lift coefficient significantly varies in time as well as in the spanwise direction owing to complex interactions between the wing and vortices in the wake. However, the timeaveraged sectional lift force coefficient is inversely proportional to the spanwise distance from the rotation center except very near the wingtip region. This is because the wingtip vortex significantly decreases the lift force on the wingtip region during and after midstroke.

Hierarchy of minimal flow units in the logarithmic layer
View Description Hide DescriptionThe minimal simulation boxes of the buffer layer of turbulent channels can be extended to the logarithmic and outer regions, where they contain a segment of streamwise velocity streak, and a vortex cluster. Smaller boxes restrict “healthy” turbulence closer to the wall, to a layer whose thickness scales with the spanwise size of the box. These minimal boxes burst quasiperiodically, and the bursting period for a band of wall distances grows linearly away from the wall, independently of the box size within the limits within which turbulence is well represented.
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 ARTICLES

 Biofluid Mechanics

Bioconvection in a suspension of isotropically scattering phototactic algae
View Description Hide DescriptionPhototaxis is a directed swimming response toward a light source sensed by microorganisms. Positive phototaxis represents swimming toward the source of light intensity and negative phototaxis is the swimming away from it. In this paper we develop a new model for phototaxis that incorporates the effects of absorption and scattering by the microorganisms. This model is then used to analyze the linear stability of a suspension of phototactic algae illuminated by a collimated radiation at the top. A comprehensive numerical study of the linear stability is presented with particular emphasis on the scattering effect. As a result of scattering, for some parameter values, the microorganisms accumulate in two horizontal layers at different depths in the basic equilibrium state. Examples of oscillatory instabilities are also found.
 Micro and Nanofluid Mechanics

Slow gas microflow past a sphere: Analytical solution based on moment equations
View Description Hide DescriptionThe regularized 13moment equations are solved analytically for the microflow of a gas past a sphere in the case of low Mach numbers. The result is given in fully explicit expressions and shows nontrivial behavior for all fluid fields including stress, heat flux, and temperature. Various aspects of the flow such as temperature polarization and total force are reproduced correctly for moderate Knudsen number. The analytical solution allows studying the rise of Knudsen layers and their interaction and coupling to the fluid variables in the bulk. Additionally, based on the regularized 13moment equations system, hybrid boundary conditions are given for the standard Stokes equations in order to enable them to predict nonequilibrium effects in the flow past a sphere.

Electrospraying insulating liquids via charged nanodrop injection from the Taylor cone of an ionic liquid
View Description Hide DescriptionCharge is injected into the bulk of an insulating liquid in the form of nanodrops produced by an immersed Taylor cone of an ionic liquid. The charge then drifts onto the insulator surface, destabilizing it and leading to the formation of an electrified jet that atomizes into approximately monodisperse micron size insulator drops. The approach is similar to those previously based on field injection of ions from sharp tungsten tips, but the continuous renewal and selfsharpening of the liquidchargeinjector permits longterm stable operation. Using heptane as the insulator and 1ethyl3methylimidazolium as the ionic liquid we produce approximately monodisperse drops with average diameters ranging from less than 4 up to , injecting in some cases as little as 0.0002% by volume of ionic liquid. No fundamental limitation restricting the possibility of forming even smaller drops is apparent. The scaling law of Kim and Turnbull [“Generation of chargeddrops of insulating liquids by electrostatic spraying,” J. Appl. Phys.47, 1964 (1976)] where the drop diameter varies as the 2/3 power of the liquid flow rate and the −2/3 power of the spray current is confirmed, implying that the drops are on the average charged to 50%–60% of the Rayleigh limit.

Stokes flow past a compound drop in a circular tube
View Description Hide DescriptionMicrofluidics could generate drops or bubbles with controllable size and frequency at this stage. However, analytical work on such problem is less reported in the literature. In this study, we study the motion of a compound drop, consisting of a fluid drop engulfed in a larger drop, confined in a circular tube. The analysis is based on the low Reynolds numberStokes flow theory. Interfaces are assumed to be spherical due to large surface tension. Stream functions in one bipolar and two cylindrical coordinate systems are developed in series form. Our new contribution is the transformation between cylindrical and bipolar coordinate systems. Flow patterns are mainly dependent on the relative motion and the size of the inner drop. Four types of flow patterns are identified. Drag force on the inner or outer drops is in proportion to the product of the drop radius and viscosity of the phase encapsulating the drop. Drag force on the inner or outer spheres is finally expressed as linear combinations of velocities of the three phases (i.e., the inner drop, the outer drop, and the continuous flow), respectively. Our results show that those coefficients of the linear combinations for the drag forces depend on several parameters: eccentricity of the compound drop,viscosity ratio of two neighboring phases, radius ratio of the inner drop to the outer drop, and the radius ratio of the outer drop to the tube. The two radius ratios have largest effects on the coefficients of the inner or outer drop, respectively. Stability of the compound drop in a circular tube is analyzed. It is found that though the compound drop cannot reach an absolutely steady state, it will enter a quasisteady state where the inner sphere is adjacent to the shell of the outer sphere in practice.

On the effect of hydrodynamic slip on the polarization of a nonconducting spherical particle in an alternating electric field
View Description Hide DescriptionThe polarization of a charged, dielectric, spherical particle with a hydrodynamically slipping surface under the influence of a uniform alternating electric field is studied by solving the standard model (the Poisson–Nernst–Planck equations). The dipole moment characterizing the strength of the polarization is computed as a function of the double layer thickness, the electric field frequency, the particle’s surface charge, and the slip length. Our studies reveal that two processes contribute to the dipole moment: ion transport inside the double layer driven by the electric field and the particle’s electrophoretic motion. The hydrodynamic slip will simultaneously impact both processes. In the case of a thick double layer, an approximate analytical expression for the dipole moment of a weakly charged particle with an arbitrary slip length and a small zeta potential [normalized with the thermal voltage ], accurate within , shows that the polarization is dominated by the particle’s electrophoretic motion and the enhancement of the polarization due to the hydrodynamic slip is primarily attributed to the enhancement of the electrophoretic mobility from the slip. In contrast, for a thin double layer, the dipole moment is governed by ion transport inside the double layer. Asymptotical analytical models conclude that the hydrodynamic slip has more complicated influence on the polarization. At the highfrequency range where the surface conduction is important, the dipole moment is predicted to increase for any zeta potential. On the contrary, at the lowfrequency range where the bulk diffusion is significant, the enhancement of the dipole moment due to the slip is lost at large zeta potentials.
 Interfacial Flows

Drop impact on microwetting patterned surfaces
View Description Hide DescriptionWe investigate experimentally the dynamics of a liquid drop that impacts on a solid surface whose wettability is patterned at the microscopic scale. The target surface is patterned such that hydrophilic(hydrophobic) microscale spokes radiate from the center on a hydrophobic(hydrophilic) background. Following the initial spreading stages, the drop recoils on the hydrophobic region while being arrested on the hydrophilic area, thereby resulting in a micropatterned liquid footprint. We also find that the fingering instability of the drop edge is affected by the wettability patterns in the initial spreading stages. The number of fingers depends on a combination of the impact Weber number and the number of spokes. We suggest that the present scheme of patterning liquid deposits at microscales could be exploited in various microfluidic applications.

The sensitivity of drop motion due to the density and viscosity ratio
View Description Hide DescriptionThe effect of the density and viscosity ratio on the motion of single drops rising in immiscible liquids is computationally investigated. The density and viscosity ratio play an important role in droplet morphology, unstable droplet behavior, and terminal droplet characteristics. The numerical method used in this investigation is a coupled levelset and volumeoffluid method together with a sharp interface treatment for the interfacial jump conditions. The computations assume an axisymmetric geometry. Drop rise motion is highly dependent on the viscosity ratio. The results reported in this paper augment the information provided by the correlation table for bubble rise motion by Bhaga and Weber [“Bubbles in viscousliquids: Shapes, wakes and velocities,” J. Fluid Mech.105, 61 (1981)]. A dropsystem with a large viscosity ratio is susceptible to exhibiting unstable motion in the large Eötvös number regions; an unstable drop can show complicated behavior with various breakup modes that are dependent on the Morton number. With regard to the effect of the density ratio, it is observed that the difference between a bubble and a drop with “equivalent” properties is not prominent except in the low Morton number regions. The results of investigating the effect of the density and viscosity ratio on dropmotion indicate that the Morton number, Eötvös numbers, and viscosity ratio are the primary governing parameters and the density ratio is a secondary governing parameter.
 Laminar Flows

Spatially localized binary fluid convection in a porous medium
View Description Hide DescriptionThe origin and properties of timeindependent spatially localized binary fluid convection in a layer of porous material heated from below are studied. Different types of single and multipulse states are computed using numerical continuation, and the results related to the presence of homoclinic snaking of single and multipulse states.

Inertia dominated thinfilm flows over microdecorated surfaces
View Description Hide DescriptionWe analyze the inertia dominated flow of thin liquid films on microtextured substrates, which here are assemblies of micronsize posts arranged on regular lattices. We focus on situations for which the thinfilm thickness and the roughness characteristic length scale are of the same order of magnitude, i.e., a few hundred microns. We assume that the liquidflows isotropically through the roughness at a flow rate that depends on the geometrical features of the porous layer; above the texture, the flow is characterized by a larger Reynolds number and modeled using a boundary layer approach. The influence of the microtexture on the thinfilmflow above the microposts is captured by a reduction of the flow rate due to the leakage flow through the texture and a slip boundary condition, which depends on the flow direction as well as on the lattice properties. In this way, the velocity field in the free surface flow adopts the symmetry of the microtexture underneath. The results of this model are in good agreement with experimental observations obtained for thinfilmflows formed upon jet impact on microtextures. The characteristics of the polygonal hydraulic jumps that we obtain depend on both the jet parameters and the topographical features of the surface roughness. We use the measurements and the numerical predictions to estimate the flow rate through the shallow porous layer and the effective slip length for this inertia dominated flow regime. We also discuss the limitations of the model.
 Instability and Transition

Instabilities and transient behaviors of a liquid film flowing down a porous inclined plane
View Description Hide DescriptionThe nonmodal linear stability of a falling film over a porous inclined plane has been investigated. The base flow is driven by gravity. We use Darcy’s law to describe the flow in the porous medium. A simplified onesided model is used to describe the fluid flow. In this model, the influence of the porous layer on the flow in the film can be identified by a parameter . The instabilities of a falling film have traditionally been investigated by linearizing the governing equations and testing for unstable eigenvalues of the linearized problem. However, the results of eigenvalue analysis agree poorly in many cases with experiments, especially for shear flows. In the present paper, we have studied the linear stability of threedimensional disturbances using the nonmodal stability theory. Particular attentions are paid to the transient behavior rather than the long time behavior of eigenmodes predicted by traditional normal mode analysis. The transient behaviors of the response to external excitations and the response to initial conditions are studied by examining the pseudospectral structures and the energy growth function . Before we study the nonmodal stability of the system, we extend the results of longwave analysis in previous works by examining the linear stabilities for streamwise and spanwise disturbances. Results show that the critical conditions of both the surface mode and the shear mode instabilities are dependent on for streamwise disturbances. However, the spanwise disturbances have no unstable eigenvalue.

On a variational principle for Beltrami flows
View Description Hide DescriptionIn a previous paper [R. González, L. G. Sarasua, and A. Costa, “Kelvin waves with helical Beltrami flow structure,” Phys. Fluids20, 024106 (2008)] we analyzed the formation of Kelvin waves with a Beltrami flow structure in an ideal fluid. Here, taking into account the results of this paper, the topological analogy between the role of the magnetic field in Woltjer’s theorem [L. Woltjer, “A theorem on forcefree magnetic fields,” Proc. Natl. Acad. Sci. U.S.A.44, 489 (1958)] and the role of the vorticity in the equivalent theorem is revisited. Via this analogy we identify the forcefree equilibrium of the magnetohydrodynamics with the Beltrami flow equilibrium of the hydrodynamic. The stability of the last one is studied applying Arnold’s theorem. We analyze the role of the enstrophy in the determination of the equilibrium and its stability. We show examples where the Beltrami flow equilibrium is stable under perturbations of the Beltrami flow type with the same eigenvalue as the basic flow one. The enstrophy variation results invariant with respect to a uniform rotating and translational frame and the stability is conserved when the flow experiences a transition from a Beltrami axisymmetric flow to a helical one of the same eigenvalue. These results are discussed in comparison with that given by Moffatt in 1986 [H. K. Moffatt, “Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 2. Stability considerations,” J. Fluid Mech.166, 359 (1986)].

Linear spatial instability of viscous flow of a liquid sheet through gas
View Description Hide DescriptionThe present paper focuses on the linear spatialinstability of a viscous twodimensional liquidsheet bounded by two identical viscous gas streams. The Orr–Sommerfeld differential equations and the boundary conditions of the flow configuration are numerically solved using Chebyshev series expansions and the collocation method. The strong dependence of the instability parameters on the velocity profiles is proven by using both quadratic and error functions to define the base flow in the liquidsheet and the gas shear layer. The sensitivity of the spatialinstability growth rate to changes in the dimensionless parameters of the problem is assessed. Regarding the liquidsheetReynolds number, it has been observed that, when this parameter increases, both the most unstable growth rate and the corresponding wavenumber decrease, whereas the cutoff wavenumber increases. The results of this analysis are compared with temporal theory through Gaster transformation. The effects liquid and gas viscosity have on instability are studied by comparing the instability curves given by the presented model with those from an inviscid liquidsheet and a viscousliquidsheet in an inviscid gaseous medium. The model presented in this paper features a variation in the cutoff wavenumber with all the governing parameters of the problem, whereas that provided by cases that account for an inviscid surrounding gas depends only on the liquidsheet Weber number and the ratio of gas to liquid densities. Results provided by the presented model have been experimentally validated and show that quadratic profiles have a greater capacity to predict the disturbance wavelength.

Shortwavelength stability analysis of Hill’s vortex with/without swirl
View Description Hide DescriptionThe stability of Hill’s vortex with/without swirl is studied by the shortwavelength stabilityanalysis or WKB analysis. It is shown that the classical Hill’s spherical vortex is subjected not only to the Widnall instability but also to the curvature instability found for thin vortex rings and helical vortex tubes. A new “combined” mode of instability caused by the two instabilities is discovered. The magnitude of the exponential growth rate of the combined mode is similar with the curvature instability around the stagnation point; it exceeds the Widnall instability near the boundary. The effects of swirl on the instabilities are investigated using a family of solutions obtained by Moffatt [“The degree of knottedness of tangled vortex lines,” J. Fluid Mech.35, 117 (1969)]. As the swirl parameter increases, a stable region appears around the stagnation point; the maxima of the growth rates decrease; the combined mode region disappears for . As increases further, however, the region of the generalized centrifugal instability emerges from the stagnation point.
 Turbulent Flows

Turbulence without Richardson–Kolmogorov cascade
View Description Hide DescriptionWe investigate experimentally wind tunnelturbulence generated by multiscale/fractal grids pertaining to the same class of lowblockage spacefilling fractal square grids. These grids are not active but nevertheless produce very much higher turbulence intensities and Reynolds numbers than higher blockage regular grids. Our hot wire anemometry confirms the existence of a protracted production region where turbulence intensity grows followed by a decay region where it decreases, as first reported by Hurst and Vassilicos [“Scalings and decay of fractalgenerated turbulence,”Phys. Fluids19, 035103 (2007)]. We introduce the wakeinteraction length scale and show that the peak of turbulence intensity demarcating these two regions along the centerline is positioned at about . The streamwise evolutions on the centerline of the streamwise mean flow and of various statistics of the streamwise fluctuatingvelocity all scale with . Mean flow and turbulence intensity profiles are inhomogeneous at streamwise distances from the fractal grid smaller than , but appear quite homogeneous beyond . The velocityfluctuations are highly nonGaussian in the production region but approximately Gaussian in the decay region. Our results confirm the finding of Seoud and Vassilicos [“Dissipation and decay of fractalgenerated turbulence,”Phys. Fluids19, 105108 (2007)] that the ratio of the integral lengthscale to the Taylor microscale remains constant even though the Reynolds number decreases during turbulence decay in the region beyond . As a result, the scaling , which follows from the scaling of the dissipation rate in boundaryfree shear flows and in usual gridgenerated turbulence, does not hold here. This extraordinary decoupling is consistent with a noncascading and instead selfpreserving singlelength scale type of decaying homogeneous turbulence proposed by George and Wang [“The exponential decay of homogeneous turbulence,”Phys. Fluids21, 025108 (2009)], but we also show that is nevertheless an increasing function of the inlet Reynolds number. Finally, we offer a detailed comparison of the main assumption and consequences of the George and Wang theory against our fractalgenerated turbulence data.

Direct numerical simulations of magnetic field effects on turbulent flow in a square duct
View Description Hide DescriptionMagnetic fields are crucial in controlling flows in various physical processes of industrial significance. One such process is the continuous casting of steel, where different magnetic field configurations are used to control the turbulent flow of steel in the mold in order to minimize defects in the cast steel. The present study has been undertaken to understand the effects of a magnetic field on mean velocities and turbulence parameters in turbulent molten metal flow through a square duct. The coupled Navier–Stokes magnetohydrodynamic equations have been solved using a threedimensional fractionalstep numerical procedure. The Reynolds number was kept low in order to resolve all the scales in the flow without using a subgrid scale turbulence model. Computations were performed with three different grid resolutions, the finest grid having cells. Because liquid metals have low magnetic Reynolds number, the induced magnetic field has been considered negligible and the electric potential method for magnetic fieldflow coupling has been implemented. After validation of the computer code, computations of turbulent flow in a square duct with different Hartmann numbers were performed until complete laminarization of the flow. The timedependent and timeaveraged nature of the flow has been examined through distribution of mean velocities, turbulent fluctuations,vorticity, and turbulent kinetic energy budgets.

Finite time Lagrangian analysis of an unsteady separation induced by a near wall wake
View Description Hide DescriptionFollowing the Lagrangian theory of unsteady flow separation on slip boundaries proposed by Lekien and Haller [“Unsteady flow separation on slip boundaries,” Phys. Fluids20, 097101 (2008)], we use finite time Lagrangian analysis in order to educe large scale, unsteady flow separation downstream a near wall obstacle at a significant Reynolds number. By large scale flow separation, we mean here the ejection of fluid and vorticity outside a neighborhood of the wall at the scale of the obstacle. Indeed, while the separation point at the wall is not spatially resolved by the high speed particle image velocimetrymeasurements, freeslip boundary conditions are applied before educing unstable manifolds in the near wall region using finite time Lyapunov exponents. For this turbulent flow, conditional statistics are presented in order to discuss the relative contributions of the unsteady aerodynamics and of the turbulence in the separation region. The dynamics of the corresponding separation point has a very clear link with the fluctuating wall pressure induced by this unsteady turbulent flow.