Volume 15, Issue 10, October 2003
 ARTICLES


Sensitivity of binary liquid thermal convection to confinement
View Description Hide DescriptionThe stable axisymmetric convective states of a binary liquid enclosed in a vertical cylinder heated from below are exhaustively and accurately identified by pseudospectral numerical integration. In order to gain some insight on the influence that nearby boundaries can exert on flow dynamics, three aspect ratios (1/2, 1, and 2), as well as two types of lateral kinematicboundary conditions (either noslip or freeslip) are investigated. The ranges over which stable quiescent, oscillatory and steady convective states extend and coexist are given. The bifurcations leading to transitions from one branch of solutions to another, as well as those that occur along the oscillatory branch, are analyzed. The most significant effect of varying boundary conditions and aspect ratio involves the route from oscillatory to steady convection. For a given configuration, that route consists of a period doubling cascade followed by chaos, or a subcritical generalized Hopf (or Neimark–Sacker) bifurcation, or a homoclinic bifurcation. The dynamics of thermal convection of enclosed binary mixtures is clearly very sensitive to both boundary conditions and aspect ratio.

Experimental study of nonnormal nonlinear transition to turbulence in a rotating magnetic field driven flow
View Description Hide DescriptionTransition to turbulence of a rotating magnetic field driven liquid metalflow in a cylinder is studied experimentally by a probeless method. A sudden, wide frequency band transition as well as intermittency are observed just at the theoretically predicted linear instability limit. A small geometrical imperfection triggers the transition in the linearly stable regime in a control parameter range where additional unstable steady solutions have been detected numerically. We discuss possibilities to predict theoretically the nonnormal nonlinear transition and its characteristics.

The averaging of gravity currents in porous media
View Description Hide DescriptionWe explore the problem of a moving free surface in a watersaturated porous medium that has either a homogeneous or a periodically heterogeneous permeability field. We identify scaling relations and derive similarity solutions for the homogeneous, constant coefficient case in both a Cartesian and an axisymmetric, radial coordinate system. We utilize these similarity scalings to identify halfheight slumping time scales as a rough guide for field groundwater cleanup strategies involving injected brines. We derive averaged solutions using homogenization for a vertically periodic, a horizontally periodic, and a twodimensional periodic case—the solution of which requires solving a cell problem. Using effective coefficients, we connect the first two of these homogenized solutions to the similarity scaling solution derived for the homogeneous case. By simplifying to a thin limit, retaining variations of the porous media in the horizontal direction, we derive a homogenization solution in agreement with the general horizontally layered solution and an expression for the leadingorder correction. Finally, we implement two numerical solution approaches and show that selfsimilar scaling and agreement with leadingorder averaging emerge in finite time, and demonstrate the accuracy and convergence rate of the leading order correction for both the interior and the boundary of the domain.

Stability of liquid bridges between an elliptical and a circular supporting disk
View Description Hide DescriptionA numerical method has been developed to determine the stability limits for liquid bridges held between noncircular supporting disks, and the application to a configuration with a circular and an elliptical disk subjected to axial acceleration has been made. The numerical method led to results very different from the available analytical solution, which has been revisited and a better approximation has been obtained. It has been found that just retaining one more term in the asymptotic analysis the solution reproduces the real behavior of the configuration and the numerical results.

Evidence of an inverted temperature gradient during evaporation/condensation of a LennardJones fluid
View Description Hide DescriptionMolecular dynamics simulations of the LennardJones fluid have been performed to study the vapor flow between two liquid slabs kept at slightly different temperatures. For the first time, direct evidence is found of the onset of inverted gradient temperature profiles in the vapor. The simulations results also show good agreement with a kinetic theory analysis of the vapor phase flow field.

On threedimensional linear stability of Poiseuille flow of Bingham fluids
View Description Hide DescriptionPlane channel Poiseuille flow of a Bingham fluid is characterized by the Bingham number, B, which describes the ratio of yield and viscous stresses. Unlike purely viscousnonNewtonian fluids, which modify hydrodynamic stability studies only through the dissipation and the basic flow, inclusion of a yield stress additionally results in a modified domain and boundary conditions for the stability problem. We investigate the effects of increasing B on the stability of the flow, using eigenvalue bounds that incorporate these features. As we show that threedimensional linear stability can be achieved for a Reynolds number bound of form for all wavelengths. For long wavelengths this can be improved to which compares well with computed linear stability results for twodimensional disturbances [J. Fluid Mech. 263, 133 (1994)]. It is also possible to find bounds of form which derive from purely viscous dissipation acting over the reduced domain and are comparable with the nonlinear stability bounds in J. NonNewt. Fluid Mech. 100, 127 (2001). We also show that a Squirelike result can be derived for the plane channel flow. Namely, if the equivalent eigenvalue bounds for a Newtonian fluid yield a stability criterion, then the same stability criterion is valid for the Bingham fluid flow, but with reduced wavenumbers and Reynolds numbers. An application of these results is to bound the regions of parameter space in which computational methods need to be used.

The effect of surfactant monolayers on vortex rings formed from an impacting water drop
View Description Hide DescriptionThe impact of a falling drop with a flat liquid surface can result in the formation of a subsurface vortex ring that penetrates a significant distance into the liquid bulk. Herein an experimental study of these vortex rings is presented for water drops striking a water surface. The effect of surfactantmonolayers was investigated by measuring the vortex velocity for water surfaces that were free of surfactants, as well as surfaces covered with surfactantmonolayers. A soluble and an insoluble monolayer were investigated, Triton X100 and oleyl alcohol, respectively. Oleyl alcohol was investigated for concentrations ranging from 0 to and Triton X100 was investigated for concentrations ranging from 0 to 4.0 mg/L. For both surfactants the vortex velocity displayed a maximum at intermediate surfactant concentrations. In all cases the drop fluid was free of surfactants. A possible mechanism based on capillary wave damping is presented to explain these results. The relevance of this work to rain over lakes and oceans is discussed.

Instability of creeping Couette flow past a neoHookean solid
View Description Hide DescriptionFluid flow over a deformable solid can become unstable due to the fact that waves may propagate along the solid–fluid interface. In order to understand the role that nonlinear rheological properties of the solid play in these elastohydrodynamic instabilities, we apply linear stability analysis to investigate creeping Couette flow of a Newtonian fluid past an incompressible and impermeable neoHookean solid of finite thickness. As inertial effects are assumed to be negligible, the problem is governed by three dimensionless parameters: an imposed strain, a thickness ratio, and an interfacial tension. In the base state, there is a first normal stress difference in the neoHookean solid, and this leads to instability behavior that is significantly different from what is observed with a linear constitutive equation. In the absence of interfacial tension, the first normal stress difference gives rise to a shortwave instability. For sufficiently thin solids, a large range of highwavenumber modes becomes unstable first as the strain that is imposed on the system increases, while for sufficiently thick solids, a small band of wavenumbers first becomes unstable. The presence of interfacial tension removes the shortwave instability and leads to larger critical imposed strains and smaller critical wavenumbers. In comparison to the linear elasticconstitutive equation, the neoHookean model leads to smaller values of the critical imposed strain and larger values of the critical wavenumber, but the difference rapidly diminishes as the solid thickness increases. Analysis of the continuity of velocity boundary condition at the interface reveals that at the critical conditions, the mean flow tends to amplify horizontal interface perturbations, while horizontal velocity perturbations tend to suppress them. The results of this work highlight the importance of properly accounting for large displacement gradients when modeling elastohydrodynamic instabilities.

Computational study of a weakly compressible mixing barrier in low Prandtl number, strongly stratified fluids
View Description Hide DescriptionThe effects of weak compressibility in the evolution of a fluid governed by the anelastic fluid equations are explored computationally. The basis for this study is a careful determination of the role which the anelastic divergence constraint plays in the evolution of a periodic array of interacting vortices. Our numerical studies address a blocking phenomenon occurring in strongly stratified flows with small Prandtl numbers. Computationally, we first document this blocking event which strongly limits vertical mixing. This is achieved using idealized equations of fluid motion which do not excite a density perturbation and exhibits that the presence of a strong density transition layer, consistently modeled in the anelastic mass balance, may lead to a dramatic modification of vortex interactions when compared with the incompressible analog. These modifications are evidenced by the formation of a weakly compressible mixing barrier. We subsequently isolate this particular blocking phenomenon as emerging in the limit of small Prandtl number through a sequence of computational simulations of the complete anelastic fluid equations which retain a density perturbation. It is shown that a sequential reduction of the Prandtl number yields much weaker vertical mixing as evidenced by passive tracer statistics.

Coherent vortex extraction in threedimensional homogeneous turbulence: Comparison between CVSwavelet and PODFourier decompositions
View Description Hide DescriptionThe coherent vortex simulation (CVS) decomposes each realization of a turbulent flow into two orthogonal components: An organized coherent flow and a random incoherent flow. They both contribute to all scales in the inertial range, but exhibit different statistical behaviors. The CVS decomposition is based on the nonlinear filtering of the vorticity field, projected onto an orthonormal wavelet basis made of compactly supported functions, and the computation of the induced velocity field using Biot–Savart’s relation. We apply it to a threedimensional homogeneous isotropic turbulent flow with a Taylor microscale Reynolds number computed by direct numerical simulation at resolution Only wavelet modes correspond to the coherent flow made of vortex tubes, which contribute 99% of energy and 79% of enstrophy, and exhibit the same energy spectrum as the total flow. The remaining wavelet modes correspond to a incoherent random flow which is structureless, has an equipartition energy spectrum, and a Gaussian velocity probability distribution function (PDF). For the same flow and the same compression rate, the proper orthogonal decomposition (POD), which in this statistically homogeneous case degenerates into the Fourier basis, decomposes each flow realization into large scale and small scale flows, in a way similar to large eddy simulation(LES) filtering. It is shown that the large scale flow thus obtained does not extract the vortex tubes equally well as the coherent flow resulting from the CVS decomposition. Moreover, the small scale flow still contains coherent structures, and its velocity PDF is stretched exponential, while the incoherent flow is structureless, decorrelated, and its velocity PDF is Gaussian. Thus, modeling the effect of the incoherent flow discarded by CVSwavelet shall be easier than modeling the effect of the small scale flow discarded by PODFourier or LES.

Apparent slip flows in hydrophilic and hydrophobic microchannels
View Description Hide DescriptionThe slip effects of water flow in hydrophilic and hydrophobic microchannels of 1 and 2 μm depth are examined experimentally. Highprecision microchannels were treated chemically to enhance their hydrophilic and hydrophobicproperties. The flow rates of pure water at various applied pressure differences for each surface condition were measured using a highprecision flow metering system and compared to a theoretical model that allows for a slip velocity at the solid surface. The slip length was found to vary approximately linearly with the shear rate with values of approximately 30 nm for the flow of water over hydrophobicsurfaces at a shear rate of The existence of slip over the hydrophilicsurface remains uncertain, due to the sensitivity of the current analysis to nanometer uncertainties in the channel height.

Ghost effect and bifurcation in a gas between coaxial circular cylinders with different temperatures
View Description Hide DescriptionA gas in a timeindependent state between rotating coaxial circular cylinders with different temperatures is considered. The bifurcation of the field in the continuum limit is studied on the basis of the system of fluiddynamictype equations and their boundary conditions derived from the Boltzmann system [Phys. Fluids 8, 628 (1996)]. When the ratio of the temperatures of the two cylinders is not close to unity, the bifurcation occurs at infinitesimal speeds of rotation of the cylinders of the first order of the Knudsen number. The temperature field in the continuum limit is determined together with the infinitesimal velocity field, and a bifurcated temperature field, as well as an axially uniform and symmetric field, exists in the absence of the flow velocity (ghost effect). Further, thermal stress, as well as viscous stress, produces a finite effect on the bifurcated temperature field (a nonNavier–Stokes effect). For example, when the ratio of the temperatures of the two cylinders is ten or one tenth, the nonNavier–Stokes effect on the bifurcated temperature field amounts to 10%.

Shock wave induced interaction of microbubbles and boundaries
View Description Hide DescriptionIn the present study we experimentally investigate bubble dynamics after laser induced shock wave exposure in the vicinity of salt crystals suspended in water. Highspeed microscopic images show aspherical collapse and rebound of single and multiple bubbles with initial radii between 5 and 150 μm. Radius time curves of bubbles close to one boundary are compared to the bubble dynamics of a spherical model. The bubble dynamics strongly depends on the position of neighboring bubbles and on the number of boundaries given by the surrounding salt grains. After excitation bubbles are drawn to the closest particles in their vicinity. Subsequent application of shock waves leads to jet formation against the rigid boundaries. The bubbles often tend to form in or migrate into cracks on the crystal surfaces and sometimes lead to the breakage of particles due to rapid bubble dynamics. Similar behavior may occur in other cases where material damage is induced by shock waves in liquids such as lithotripsy or shock wavecleaning applications.

Lubrication analysis of thermocapillaryinduced nonwetting
View Description Hide DescriptionRecent interest in the phenomenon of thermocapillaryinduced noncoalescence and nonwetting has produced experimental evidence of the existence of a film of lubricating gas that prevents the two surfaces in question (liquid–liquid for noncoalescence; liquid–solid for nonwetting) from coming into contact with one another. Measurements further indicate that the pressure distribution in this film creates a dimpled liquid freesurface. Lubrication theory is employed to investigate the coupled effects of liquid and gas flows for a twodimensional nonwetting case of a hot droplet pressed toward a cold wall. The analysis focuses on the respective roles of viscous and inertial forces on droplet deformation. Resultant droplet shapes show an influence of gas viscosity maintaining nonwetting and of inertia contributing to a dimple. Previous analyses of thermocapillarydriven flow in liquid layers and droplets model the gas as purely passive which cannot be the case in the present application.

Evolution equations for strongly nonlinear internal waves
View Description Hide DescriptionThis paper is concerned with shallowwater equations for strongly nonlinear internal waves in a twolayer fluid, and comparison of their solitary solutions with the results of fully nonlinear computations and with experimental data. This comparison is necessary due to a contradictory nature of these equations which combine strong nonlinearity and weak dispersion. First, the Lagrangian (Whitham’s) method for dispersive shallowwater waves is applied to derivation of equations equivalent to the Choi–Camassa (CC) equations. Then, using the Riemann invariants for strongly nonlinear, nondispersive waves, we obtain unidirectional, evolution equations with nonlinear dispersive terms. The latter are first derived from the CC equations and then introduced semiphenomenologically as quasistationary generalizations of weakly nonlinear Korteweg–de Vries and Benjamin–Ono models. Solitary solutions for these equations are obtained and verified against fully nonlinear computations. Comparisons are also made with available observational data for extremely strong solitons in coastal zones with well expressed pycnoclines.

Transition of a moving contact line from smooth to angular
View Description Hide DescriptionWe consider the motion of a small droplet sliding under gravity down an inclined plane. Experimentally [see T. Podgorski, Thèse, Université Paris 6 (October 2000); T. Podgorski et al., Phys. Rev. Lett. 87, 036102 (2001)] it has been observed that such a droplet will develop an angular point in the contact line at its rear if its velocity is sufficiently high (i.e., if the plane is inclined sufficiently steeply to the horizontal). The angular point first appears at some critical, or threshold speed, and the angular discontinuity observed increases monotonically from zero at We seek to describe this phenomenon using a simple mathematical model based on lubrication theory. In the subcritical regime we find an exact solution to our model: a drop of circular perimeter sliding steadily down the inclined plane. This allows us to predict the formation of a nonanalyticity in the contact line at a welldefined critical droplet speed. For speeds just beyond this, we construct a local solution valid near the singular point, for which the angle in the contact line predicted by our theory agrees with the experimental results. We also construct a local solution in the fully developed critical regime, for sliding speeds well above critical, which is suggestive of a further bifurcation, also observed in the experiments [T. Podgorski, Thèse, Université Paris 6 (October 2000)].

A gas phase displacing a liquid with soluble surfactants out of a small conduit: The plane case
View Description Hide DescriptionIn this work we report the actions of solublesurfactants when a liquid contained between parallel plates is being displaced by a steadily moving gas phase. For that purpose the full Navier–Stokes equations are solved together with surfactant mass balances at the interface, as well as in the liquid phase. The influence of the different relevant dimensionless parameters is described and analyzed for capillary numbers within the range these results proved to be in good agreement with those obtained by Ratulowski and Chang for very low values of the capillary number. Besides, the bulk and interfacial distribution of surfactants along with the corresponding flow fields are shown for several values of the elasticity and Péclet numbers.

Developing flow of a powerlaw liquid film on an inclined plane
View Description Hide DescriptionDeveloping flow of a liquid film along a stationary inclined wall is analyzed for a powerlaw constitutive equation. For films with appreciable inertia and therefore small interfacial slopes, the boundarylayer approximation may be used. The boundarylayer equations are solved numerically through the von Mises transformation that gives a partial differential equation over a semiinfinite strip and approximately by the method of von Kármán and Polhausen that gives an ordinary differential equation for the film thickness, called a film equation. Film equations derived from selfsimilar velocity profiles fail when the film thickens and the flow undergoes a supercritical to subcritical transition; a nonremovable singularity arises at the critical point, the location of the flow transition. A film equation is developed that accommodates this transition. Predictions exhibit a standing wave where hydrostatic pressure becomes important and opposes inertia. This thickening effect is accentuated for small angles of inclination at moderate Reynolds numbers. In the limit of small film thickness in which gravitational effects are negligible, the thickness profile is nonlinear in agreement with an independent and new similarity solution. This result contrasts with the established linear thickness profile for a Newtonian liquid. The circumstances in which the film equation gives results close to the full boundary layer equation are identified.

Electrohydrodynamically driven chaotic mixing in a translating drop. II. Experiments
View Description Hide DescriptionWe investigate experimentally the mixing inside of settling drops in spatially uniform timedependent electric fields. Though the Reynolds and Stokes numbers are small, the combined flow field can produce complex patterns via chaotic advection. These complex structures are visualized and examined by illuminating passive particles injected in translating drops, and the results are compared with numerical particle simulations. A silicone oil/castor oil system is used for the dispersed and continuous phases, respectively. Drop sizes are typically on the order of 5–6 mm in diameter under nearly isopycnic conditions, the small difference yielding a finite settling velocity. The electric field values are typically modulated between 3–10 kV at frequencies ranging from 0.01–0.05 Hz. The results presented show excellent agreement with the numerical results despite several conditions that need to be satisfied for the quasistatic approximation used in the analysis.

Two statistical models for predicting collision rates of inertial particles in homogeneous isotropic turbulence
View Description Hide DescriptionThe objective of the paper is to present and compare two models for the collision rate of inertial particles immersed in homogeneous isotropic turbulence. The merits and demerits of several known collision models are discussed. One of the models proposed in the paper is based on the assumption that the velocities of the fluid and a particle obey a correlated Gaussian distribution. The other model stems from a kinetic equation for the probability density function of the relative velocity distribution of two particles. The predictions obtained by means of these two models are compared with numerical simulations published in the literature.
