Volume 19, Issue 12, December 2007

The threedimensional RayleighTaylor instability is studied in a low Atwood number miscible fluid system. The two fluids are contained within a Plexiglas tank that is mounted on vertical rails and accelerated downward by a weight and pulley system. A net acceleration between 13 and can be maintained, resulting in an effective body force equivalent to 0.33–1.35 times Earth’s gravity. A singlemode, threedimensional perturbation is produced by oscillating the tank, which has a square cross section, along its diagonal. Early time measured growth rates are shown to have good agreement with linear stability theory. At late time, the instability exhibits a nonconstant vertical interfacial velocity in agreement with the recent numerical computations of Ramaprabhu et al. [Phys. Rev. E74, 066308 (2006)]. Both the latetime bubble and spike velocities have values greater than those predicted by both the simple buoyancydrag model developed by Oron et al. [Phys. Plasmas8, 2883 (2001)] and the potential flow model of Goncharov [Phys. Rev. Lett.88, 134502 (2002)]. The disagreement with the models can be attributed to the formation of vortices, in this case vortex rings, observed in the experiments but not accounted for by the models.
 LETTERS


Scalar flux spectrum in isotropic steady turbulence with a uniform mean gradient
View Description Hide DescriptionThe scaling law of a scalar flux spectrum (velocityscalar cospectrum) in the inertial convective range of passive scalar turbulence under a uniform mean scalar gradient is examined using direct numerical simulation with a resolution of up to grid points. When the Reynolds number is increased up to , the scalar flux spectrum tends to obey the power law , as predicted by Lumley [J. Atmos. Sci.21, 99 (1964);Phys. Fluids10, 855 (1967)], with a nondimensional constant of at . The effect on the scaling of the scalar flux spectrum is well compensated using the mean molecular destruction of the scalar flux . The dependence of is also compared with the results of previous studies, and its asymptotic state at an infinite Reynolds number is discussed.

Electrically tunable viscosity of dilute suspensions of carbon nanotubes
View Description Hide DescriptionThe apparent viscosity of a dilute (volume fraction ), singlewallcarbonnanotube(SWNT)/terpineol suspension was experimentally found to more than double at moderate shear rates under an external electric field of strength . The electrorheological response is interpreted in terms of an electrostaticpolarization model, where the governing parameter is a modified Mason number giving the ratio of viscous to dipoledipole forces. Analysis of the hydrodynamic and electrostatic forces suggests that the magnitude of the electrorheological response in the dilute SWNTsuspension, which is much higher than conventional electrorheological fluids at comparable volume fractions, is due to the high aspect ratio of the nanotubes. Comparison is made to a suspension of glassy carbon spheres, in which a threeorderofmagnitudehigher volume fraction is required to achieve similar increases in the apparent viscosity under the same conditions.
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 ARTICLES

 Interfacial Flows

Linear instability of pressuredriven channel flow of a Newtonian and a HerschelBulkley fluid
View Description Hide DescriptionThe linear stability characteristics of pressuredriven twolayer channel flow are considered, wherein a Newtonian fluid layer overlies a layer of a HerschelBulkley fluid. A pair of coupled OrrSommerfeld eigenvalueequations are derived and solved using an efficient spectral collocation method for cases in which unyielded regions are absent. An asymptotic analysis is also carried out in the longwave limit, the results of which are in excellent agreement with the numerical predictions. Our analytical and numerical results indicate that increasing the dimensionless yield stress, prior to the formation of unyielded plugs below the interface, is destabilizing. Increasing the shearthinning tendency of the lower fluid is stabilizing.

Bifurcation and hysteresis phenomena in twodimensional sailsailflow interactions
View Description Hide DescriptionThis is the first report of complex bifurcation and manyfold hysteresis phenomena in the physics behind a pair of sails set in a flow of two dimensions. The formalism is based on inviscid vorticitytheory, and the basic equations consist of a pair of integrodifferential equations subject to a pair of nonlinear integral constraints. The method of solution is built up with the boundary element method for the integrodifferential equations as well as the Newton–Raphson method for parameter search by use of the nonlinear integral constraints. Three types of configurations are considered as case studies on effects due to overlapping of the jib sail upon the main sail. Numerical analyses predict three sets of solutions: The convexconvex sail shapes, the concaveconvex sail shapes, and the concaveconcave sail shapes as well as two or threefold hysteresis in aerodynamic and structural characteristics. Experimental observations confirm the existence of all the three solution sets, and moreover another set, the convexconcave sail shapes, is found by the experiment. The three case studies show that too much overlap of sails fails to obtain high lift.

Marangoni instability in ultrathin twolayer films
View Description Hide DescriptionThe development of instabilities under the joint action of the van der Waals forces and Marangoni stresses in a twolayer film on a heated or cooled substrate, is considered. The problem is solved by means of a linear stability theory and nonlinear simulations. Nontrivial change of the droplet shape in the presence of the Marangoni effect, which manifests itself as the deformation of a “plateau” into an “inkpot,” is observed. The appearance of the threshold oscillations predicted by the linear stability theory is confirmed by nonlinear simulations.

PhotoMarangoni convection in a thin liquid film
View Description Hide DescriptionMarangoni convection caused by a photochemical reaction of the type in a thin liquid film with deformable interface is studied. A system of two coupled nonlinear evolution equations for the film thickness and the reactant concentration is derived in the longwave approximation. Linear stability analysis is performed and the conditions for Marangoni convection to occur are obtained. It is shown that the type of instability depends on the ratio of diffusivities of the reactant and the product of the photochemical reaction: If the diffusivities are equal, the instability is always monotonic, while when they are significantly different the instability can be oscillatory. Numerical simulations of the derived system of equations are performed. It is shown that in the case of the monotonic instability, the system develops a spotty pattern that ultimately leads to the film rupture. In the case of oscillatory instability, it is shown that photoMarangoni convection can result in sustained wavy patterns with a square symmetry.

Gravitational effects on the deformation of a droplet adhering to a horizontal solid surface in shear flow
View Description Hide DescriptionIn this paper we investigate the gravitational effects on the deformation of a threedimensional droplet adhering to a horizontal rough solid surface in steady shear Stokes flows. Our study considers both positive and negative Bond numbers for viscous and inviscid droplets. When the interfacial system is initially at hydrostatic equilibrium, our study shows that the Bond number affects the deformation of viscousdroplets with moderate and large initial contact angles in a different way than those for small angles owing to the interplay between the viscous and surface tension forces. Inviscid droplets with different initial contact angles show similar behavior as the Bond number increases, i.e., their deformation is monotonically decreased owing to the monotonic decrease of the droplets’ height and thus the exerted pressure force. Our study identifies the gravitational effects of the onset of interfacial sliding, i.e., on the portions of the contact line which slide first due to violation of the hysteresis condition. When the interfacial system is not at hydrostatic equilibrium at the flow initiation, its dynamic evolution is more complicated owing to the combined action of the shear flow with the gravitational forcing due to the difference between the initial shape with the hydrostatic one. Our computational results are accompanied with an analysis of the forces on the droplet which provides physical insight and identifies the threedimensional nature of the interfacialdeformation.

Decomposition driven interface evolution for layers of binary mixtures. I. Model derivation and stratified base states
View Description Hide DescriptionA dynamical model is proposed to describe the coupled decomposition and profile evolution of a free surfacefilm of a binary mixture. An example is a thin film of a polymer blend on a solid substrate undergoing simultaneous phase separation and dewetting. The model is based on modelH describing the coupled transport of the mass of one component (convective CahnHilliard equation) and momentum (NavierStokesKorteweg equations) supplemented by appropriate boundary conditions at the solid substrate and the free surface. General transport equations are derived using phenomenological nonequilibrium thermodynamics for a general nonisothermal setting taking into account Soret and Dufour effects and interfacial viscosity for the internal diffuse interface between the two components. Focusing on an isothermal setting the resulting model is compared to literature results and its base states corresponding to homogeneous or vertically stratified flat layers are analyzed.
 Viscous and NonNewtonian Flows

Measurement of tack of Newtonian liquids on porous substrates
View Description Hide DescriptionProbetack experiments of Tirumkudulu et al. [Phys. Fluids15, 1588 (2003)] have shown that squeeze flow of Newtonian liquids on flat, impermeable substrates can be successfully modeled using the lubrication approximation. Here, we present a model for squeeze flow of Newtonian liquids on porous substrates where the flow in the gap is coupled to the fluid flow in the porous media. The competition of spreading and imbibition of liquid on a partially saturated porous substrate determines the force versus gap profile in both the squeeze (compression) and pulloff (tension) modes. The finite difference method was used to discretize the lubrication equation in the gap while boundary element method was employed to solve for flow in the porous substrate. The model predicts a lower magnitude of force for porous substrates in both compression and tension modes compared to that for impermeable substrates. Experiments on porous alumina substrates with Newtonian liquids show close agreement with the model predictions in both compression and tension modes when the gap is corrected for the obliqueness of the confining surfaces.Cavitation is predicted for some cases in the tension mode when the pressure in the gap reduced below the vapor pressure of liquid.
 Particulate, Multiphase, and Granular Flows

Phase diagram of vertically shaken granular matter
View Description Hide DescriptionA shallow, vertically shaken granular bed in a quasitwodimensional container is explored experimentally yielding a wider variety of phenomena than in any previous study: (1) bouncing bed, (2) undulations, (3) granular Leidenfrost effect, (4) convection rolls, and (5) granular gas. These phenomena and the transitions among them are characterized by dimensionless control parameters and combined in a full experimental phase diagram.

A reducedorder model of diffusive effects on the dynamics of bubbles
View Description Hide DescriptionWe propose a new reducedorder model for spherical bubble dynamics that accurately captures the effects of heat and mass diffusion. The objective is to reduce the full system of partial differential equations to a set of coupled ordinary differential equations that are efficient enough to implement into complex bubbly flow computations. Comparisons to computations of the full partial differential equations and of other reducedorder models are used to validate the model and establish its range of validity.
 Laminar Flows

Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries
View Description Hide DescriptionWe investigate the hydrodynamicfrictionproperties of superhydrophobicsurfaces and quantify their superlubricating potential. On such surfaces, the contact of the liquid with the solid roughness is minimal, while most of the interface is a liquidgas one, resulting in strongly reduced friction. We obtain scaling laws for the effective slip length at the surface in terms of the generic surfacecharacteristics (roughness length scale, depth, solid fraction of the interface, etc.). These predictions are successfully compared to numerical results in various geometries (grooves, posts or holes). This approach provides a versatile framework for the description of slip on these composite surfaces. Slip lengths up to are predicted for an optimized patternedsurface.

Maximizing mixing and alignment of orientable particles for reaction enhancement
View Description Hide DescriptionWe present a model for the evolution of concentrations of orientable species undergoing a collisional binary reaction and examine the dependence of the concentration of the reaction product on flow parameters in Poiseuille flow. Interesting patterns of concentration are obtained depending on parameters. We use the model to investigate the reaction in a microfluidic device known as the shear superposition micromixer. Simulation results over a range of Péclet, Damköhler, and rotational Péclet numbers indicate that this micromixer is well suited to enhance the rate of reaction via the mechanism of simultaneous mixing and alignment of the orientable species. Connections to biological systems are discussed.

Transient buoyancydriven front dynamics in nearly horizontal tubes
View Description Hide DescriptionThe interpenetration of light and heavy liquids has been studied in a long tube inclined at small angles to the horizontal. For angles greater than a critical angle (whose value decreases when the density contrast measured by the Atwood number increases), the velocity of the interpenetration front is controlled by inertia and takes the steady value , with . At lower angles, the front is initially controlled by inertia, but later limited by viscous effects. The transition occurs at a distance , which increases indefinitely as increases to . Once the viscous effects act, the velocity of the front decreases in time to a steady value which is proportional to . For a horizontal tube in the viscous regime, the velocity of the front decreases to zero as . At the same time, the profile of the interface only depends on the reduced variable . A quasiunidirectional model reproduces well the variation of the velocity of the front and the profiles of the interface, both in inclined and horizontal tubes. In the inclined tube, the velocity of the front is determined by matching rarefaction waves to a shock wave.

Rollover instability due to double diffusion in a stably stratified cylindrical tank
View Description Hide DescriptionDouble diffusion of a viscous fluid is simulated for heat leakage driven by buoyant convection under cryogenic storage conditions in a cylindrical tank with laminar flow. If the tank is stably stratified, there is a potential instability due to the inability of the fluid in the lower layer to release heat to the top vapor space, whereas the upper liquid layer can exchange heat and mass through sensible heat transfer and evaporation with the vapor space. Eventually, the lower layer becomes less dense due to thermal expansion and is no longer constrained in the stratification. The rapid rise and overturning of the fluid is termed rollover, and can be accompanied by a potentially explosive release of vapor. In this paper, hydrodynamics and heat and mass transport are used to study the stability characteristics of rollover. The transient state is used as a base state for a linear stability analysis which shows the transition from a “corner eddy” mode spinning down to spinning up is the driver for the rollover instability. Four different vaporliquid interfacial boundary conditions are tested, with similar results for the time to rollover. Surprisingly, the long time prerollover state is dominated in the laminar flow regime by heat conduction and diffusion, as the expected double roll structure is suppressed and advection plays a small roll in the majority of the prerollover period. Scalings are suggested for controlling dimensionless groups on this prerollover basis that can be used as a guideline to determine the regime of double diffusion—a single roll or a double roll stratification, as well as the severity of the eventual rollover event. An energy analysis demonstrates the switch from practically advection free to free convection regimes.
 Instability and Transition

Rayleigh–Bénard flow of a rarefied gas and its attractors. III. Threedimensional computer simulations
View Description Hide DescriptionWe investigate the threedimensional Rayleigh–Bénard flow for a set of different Knudsen and Froude numbers at a fixed temperature ratio , as well as for different aspect ratios. We observe a variety of stable vortex structures in the form of rolls, squares, and more complicated polygonal patterns. For sufficiently low Knudsen numbers, the existence of irregular regimes was confirmed. Two numerical approaches (the direct simulation Monte Carlo method and a finitedifference method for the Navier–Stokes equation) are used and the results show a satisfactory agreement.

Experimental study of the singlemode threedimensional RayleighTaylor instability
View Description Hide DescriptionThe threedimensional RayleighTaylor instability is studied in a low Atwood number miscible fluid system. The two fluids are contained within a Plexiglas tank that is mounted on vertical rails and accelerated downward by a weight and pulley system. A net acceleration between 13 and can be maintained, resulting in an effective body force equivalent to 0.33–1.35 times Earth’s gravity. A singlemode, threedimensional perturbation is produced by oscillating the tank, which has a square cross section, along its diagonal. Early time measured growth rates are shown to have good agreement with linear stability theory. At late time, the instability exhibits a nonconstant vertical interfacial velocity in agreement with the recent numerical computations of Ramaprabhu et al. [Phys. Rev. E74, 066308 (2006)]. Both the latetime bubble and spike velocities have values greater than those predicted by both the simple buoyancydrag model developed by Oron et al. [Phys. Plasmas8, 2883 (2001)] and the potential flow model of Goncharov [Phys. Rev. Lett.88, 134502 (2002)]. The disagreement with the models can be attributed to the formation of vortices, in this case vortex rings, observed in the experiments but not accounted for by the models.

Turbulent spots and waves in a model for plane Poiseuille flow
View Description Hide DescriptionThe structure of a turbulent spot in plane Poiseuille flow is investigated using a model derived from the Navier–Stokes equations through a Galerkin method. The mean profile of the streamwise velocity inside the turbulent spot has the characteristic flat profile of a turbulent Poiseuille flow. The waves developing at the wing tips of the spot have an asymmetric streamwise velocity with respect to the channel centerline, whereas their associated wallnormal velocity component is symmetric. On the outskirts of the spot, a largescale flow occupying the full gap between the plates is observed. It is characterized by a streamwise inflow toward the spot and a spanwise outflow from the spot. A detailed comparison with the numerical simulations and the experiments in the literature shows that these results are in fair agreement with the main features of the transitional plane Poiseuille flow.

Analytical and numerical stability analysis of Soretdriven convection in a horizontal porous layer
View Description Hide DescriptionWe present an analytical and numerical stability analysis of Soretdriven convection in a porous cavity saturated by a binary fluid. Both the mechanical equilibrium solution and the monocellular flow obtained for particular ranges of the physical parameters of the problem are considered. The porous cavity, bounded by horizontal infinite or finite boundaries, is heated from below or from above. The two horizontal plates are maintained at different constant temperatures while no mass flux is imposed. The influence of the governing parameters and more particularly the role of the separation ratio, , characterizing the Soret effect and the normalized porosity, , are investigated theoretically and numerically. From the linear stability analysis, we find that the equilibrium solution loses its stability via a stationary bifurcation or a Hopf bifurcation depending on the separation ratio and the normalized porosity of the medium. The role of the porosity is important, when it decreases, the stability of the equilibrium solution is reinforced. For a cell heated from below, the equilibrium solution loses its stability via a stationary bifurcation when the separation ratio , while for , it loses stability via a Hopf subcritical bifurcation. The oscillatory solution is unstable and becomes stationary. For a cell heated from above, the equilibrium solution is linearly stable if , while a stationary or an oscillatory bifurcation occurs if . The results obtained from the linear stability analysis are widely corroborated by direct 2D numerical simulations. In the case of longwave disturbances, for and for higher than a particular value called , we observe that the monocellular flow leads to a separation of the species between the two ends of the cell. First, we determined the velocity, temperature, and concentration fields analytically for monocellular flow. Then we studied the stability of this flow. For a cell heated from below and for the monocellular flow loses stability via a Hopf bifurcation. As the Rayleigh number increases, the resulting oscillatory solution evolves to a stationary multicellular flow. For a cell heated from above and , the monocellular flow remains linearly stable. We verified numerically that this problem admits other stable multicellular stationary solutions for this range of parameters.
 Turbulent Flows

Properties of d and ktype roughness in a turbulent channel flow
View Description Hide DescriptionRoughness is classified by the socalled roughness function, which represents the downward shift of the velocity profile relative to a smooth wall. The dependence of the roughness function on the Reynolds number is discussed with the aim of clarifying the difference between dtype and ktype behaviors. This difference has been traditionally associated with the stability of the flow within the roughness elements. The present direct numerical simulation results indicate that the difference more correctly reflects the different contributions from the frictional drag and pressure drag to the total stress.