Volume 11, Issue 5, May 1999
 LETTERS


Nonlinear interactions of chemical reactions and viscous fingering in porous media
View Description Hide DescriptionNonlinear interactions of chemical reactions and viscous fingering are studied in porous media by direct numerical simulations of Darcy’s law coupled to the evolution equation for the concentration of a chemically reacting solute controlling the viscosity of misciblesolutions.Chemical kinetics introduce important topological changes in the fingering pattern: new robust pattern formation mechanisms such as droplet formation and enhanced tip splitting are evidenced and analyzed.

Stability of a growing end rim in a liquid sheet of uniform thickness
View Description Hide DescriptionWe study the stability of a viscousliquid layer of uniform thickness subject only to viscous stresses and surface tension. We show that the growing cylindrical end rim does not typically break into droplets for moderate wavelengths. Thus, other mechanisms are needed to cause the instabilities, which, for instance, lead to the famous milk crown. The problem remains open for very large wavelengths.

Stability of periodically compressed vortices at low Mach number
View Description Hide DescriptionStabilityanalysis of circular and elliptical vortices periodically compressed axially in their plane reveals, at low Mach number, two distinct mechanisms of threedimensional instability. The first one is a manifestation of the elliptical instability, modified by compression. The second one, which exists also in the circular case, is a resonance between the frequency of compression and the intrinsic rotation rate of the uncompressed vortex.
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 ARTICLES


Blade coating of a powerlaw fluid
View Description Hide DescriptionIn this paper we reexamine the problem of applying a thin layer of a powerlaw fluid to a solid substrate by means of a simple blade coater. Specifically we use lubrication theory to examine steady plane flow of a powerlaw fluid in the narrow nonuniform channel formed between a fixed blade of prescribed shape and a plane substrate moving parallel to itself. The firstorder asymptotic solution for the case of a weakly nonNewtonian fluid is presented. An explicit expression is obtained for the firstorder pressure gradient from which the firstorder contributions to several important physical quantities including the thickness of the applied fluid layer and the forces on the blade are calculated for both plane and exponentially shaped blades. In particular, we find that, depending on the shape and height ratio of the coater, the effect of weakly nonNewtonian behavior can be either to increase or to decrease both the pressure and the load from their Newtonian values. We also reexamine the approximate solutions of Hwang [Trans. ASME J. Fluids Eng. 104, 469 (1982)] and Dien and Elrod [Trans. ASME J. Lubrication Technol. 105, 385 (1983)] and make a detailed comparison between their predictions and those of the exact solution in the weakly nonNewtonian limit. We find that in this limit the Dien and Elrod approximation is usually in significantly better agreement with the exact solution than Hwang’s approximation. In the Appendix we reexamine the Dien and Elrod approximate solution for the flow of a generalized Newtonian fluid.

Experimental trajectories of two drops in planar extensional flow
View Description Hide DescriptionIn this paper we map the experimental trajectories of two deformable drops in planar extensional flow and compare the experimental results with theoretical calculations for spherical drops. We examine the effects that deformation, initial position, and viscosity ratio have on the interaction of two drops and the necessity of incorporating deformation into trajectory calculations, which can be used to estimate the collision rates, the collision efficiencies, and the collision interaction times. For drops which do not come into close contact, the existing theoretical calculations for spherical drops accurately predict the symmetric trajectories and capture the increased hydrodynamic interaction for higher viscosity ratios. For drops which come into close contact, the spherical drop theory accurately predicts the approach and exit trajectories and with a slight empirical modification adequately predicts the interaction times for deformable drops with a Taylor deformation parameter up to 0.22. The experimental results show that for drops with close contact, the collision trajectories are asymmetric and irreversible with a minimum separation between the centers of mass that is less than the minimum separation of two spheres. This minimum separation corresponds to the minor axis of the deformed drop and is not captured by the spherical theory. However, overall, the modified trajectory theory based upon the hydrodynamic mobility for spherical drops does provide a reasonable estimate for the trajectories and the interaction times for two deformable drops in planar extensional flow.

The spreading of a nonisothermal liquid droplet
View Description Hide DescriptionThe effect of the slip coefficient and the mobility capillary number on the spreading of a thin axisymmetric liquid droplet with uniform heating/cooling of the solid surface is examined. The results show that increasing the slip coefficient reduces the spreading/shrinking behavior of the droplet and that the final equilibrium states are slip dependent. These results are explained by the development of a return flow inside the droplet. We show how a speeddependent slip coefficient can be used to remove the dependence of the final state on the slip coefficient. It is also shown that increasing the mobility capillary number decreases the spreading/shrinking rate of the droplet. For thermocapillarydriven droplets, there is a capillarynumberdependent time delay for the onset of motion. The entire effect of the mobility capillary number on the spreading process is explained in terms of the deformability of the free surface.

Critical behavior of drop breakup in axisymmetric viscous flow
View Description Hide DescriptionThe critical behavior of a liquid droplet immersed in a host fluid under external axisymmetric viscousflow is studied. It is well known that when the external extensional flow is weak the system approaches a steadystate flow, but when the shear rate is increased beyond some critical value a steady state is never attained and the droplet is stretched to infinity. This behavior is explained qualitatively by a simple semianalytic argument. The critical power law behavior of the droplet shape and its time dependence when the shear rate approaches the critical value is studied and is verified by numerical simulations for linear axisymmetric flows. For biaxial extensional flow (negative elongational flow) it is known that another critical point appears, and the droplet goes over into a toroidal shape. Similar critical behavior is predicted at that point also.

Effects of insoluble surfactants on the nonlinear deformation and breakup of stretching liquid bridges
View Description Hide DescriptionDuring the emission of single drops and the atomization of a liquid from a nozzle, threads of liquid are stretched and broken. A convenient setup for studying in a controlled manner the dynamics of liquid threads is the socalled liquid bridge, which is created by holding captive a volume of liquid between two solid disks and pulling apart the two disks at a constant velocity. Although the stability of static bridges and the dynamics of stretching bridges of pure liquids have been extensively studied, even a rudimentary understanding of the dynamics of the stretching and breakup of bridges of surfactantladen liquids is lacking. In this work, the dynamics of a bridge of a Newtonian liquid containing an insoluble surfactant are analyzed by solving numerically a onedimensional set of equations that results from a slenderjet approximation of the Navier–Stokes system that governs fluid flow and the convectiondiffusion equation that governs surfactant transport. The computational technique is based on the methodoflines, and uses finite elements for discretization in space and finite differences for discretization in time. The computational results reveal that the presence of an insoluble surfactant can drastically alter the physics of bridge deformation and breakup compared to the situation in which the bridge is surfactant free. They also make clear how the distribution of surfactant along the free surface varies with stretching velocity, bridge geometry, and bulk and surface properties of the liquid bridge. Gradients in surfactant concentration along the interface give rise to Marangoni stresses which can either retard or accelerate the breakup of the liquid bridge. For example, a highviscosity bridge being stretched at a low velocity is stabilized by the presence of a surfactant of low surface diffusivity (high Peclet number) because of the favorable influence of Marangoni stresses on delaying the rupture of the bridge. This effect, however, can be lessened or even negated by increasing the stretching velocity. Large increases in the stretching velocity result in interesting changes in their own right regardless of whether surfactants are present or not. Namely, it is shown that whereas bridges being stretched at low velocities rupture near the bottom disk, those being stretched at high velocities rupture near the top disk.

Buoyancydriven viscous interaction of a rising drop with a smaller trailing drop
View Description Hide DescriptionAn axisymmetric boundaryintegral method was developed and used to study the interaction of two deformable drops (or bubbles) rising (or settling) due to gravity in a viscous medium under conditions of small Reynolds number. The focus is on cases where the smaller drop trails behind the larger drop. When the Bond number is small, interfacial tension keeps the drops nearly spherical, and they separate with time. At higher Bond numbers, however, deformation is significant and the trailing drop is stretched due to the flow created by the leading drop; it may form one or more necks and break when one of these pinches off. The leading drop is flattened due to the flow created by the trailing drop; it may form a depression on its underside which evolves into a plume that rises through its center. Moreover, at sufficiently high Bond numbers, the larger leading drop does not leave the trailing drop behind, but instead may entrain and engulf it within the depression or plume. Systematic results for the parameter ranges which demarcate impending breakup and coalescence are presented.

Fluid dynamics of a double emulsion droplet in an electric field
View Description Hide DescriptionOne of general free boundary problems concerning the electrohydrodynamiceffects on a concentric double emulsiondrop is studied theoretically for the three constituent phases of leaky dielectric fluids. In order to proceed the problem analytically, the domain perturbation procedure is utilized in the small deformation limit. The patterns of electricfielddriven flow are successfully characterized by examining the distribution of induced surface charges at the inner and outer drop interfaces. The second recirculating flow is generated in the annular phase when the inner and outer interfaces are charged with the same sense. The deformation type of inner and outer interfaces can be roughly interpreted by the flow patterns, although the exact description on the deformation requires consideration of the combined contributions from both electric and flow fields. In addition, the presence of double emulsiondroplets alters the stress field of the continuous phase. The electricfieldinduced “particle stress” not only changes the effective viscosity of dispersion of the double emulsiondroplets but yields the normal stress difference, which is typical of a viscoelastic fluid. Finally, the heat transfer rate enhanced by the electricfielddriven flow is also considered.

Dynamic generation of capillary waves
View Description Hide DescriptionWe investigate the dynamic generation of capillary waves in twodimensional, inviscid, and irrotational water waves with surface tension. It is well known that short capillary waves appear in the forward front of steep water waves. Although various experimental and analytical studies have contributed to the understanding of this physical phenomenon, the precise mechanism that generates the dynamic formation of capillary waves is still not well understood. Using a numerically stable and spectrally accurate boundary integral method, we perform a systematic study of the time evolution of breaking waves in the presence of surface tension. We find that the capillary waves originate near the crest in a neighborhood, where both the curvature and its derivative are maximum. For fixed but small surface tension, the maximum of curvature increases in time and the interface develops an oscillatory train of capillary waves in the forward front of the crest. Our numerical experiments also show that, as time increases, the interface tends to a possible formation of trapped bubbles through selfintersection. On the other hand, for a fixed time, as the surface tension coefficient τ is reduced, both the capillary wavelength and its amplitude decrease nonlinearly. The interface solutions approach the profile. At the onset of the capillaries, the derivative of the convection is comparable to that of the gravity term in the dynamic boundary condition and the surface tension becomes appreciable with respect to these two terms. We find that, based on the wave, it is possible to estimate a threshold value such that if then no capillary waves arise. On the other hand, for τ sufficiently large, breaking is inhibited and pure capillary motion is observed. The limiting behavior is very similar to that in the classical KdV equation. We also investigate the effect of viscosity on the generation of capillary waves. We find that the capillary waves still persist as long as the viscosity is not significantly greater than surface tension.

Branching behavior of standing waves—The signatures of resonance
View Description Hide DescriptionArclength continuation methods are used to conduct a detailed branching study of standing wave solutions for fluids in a rectangular container, using depth and crest acceleration as control parameters. At each depth the applicable acceleration range extends between zero and one, and a number of multiple solution structures are uncovered. An intimate connection is established between these structures and the phenomenon of harmonic resonance.

Determination of particle size distributions from acoustic wave propagation measurements
View Description Hide DescriptionThe wave equations for the interior and exterior of the particles are ensemble averaged and combined with an analysis by Allegra and Hawley [J. Acoust. Soc. Am. 51, 1545 (1972)] for the interaction of a single particle with the incident wave to determine the phase speed and attenuation of sound waves propagating through dilute slurries. The theory is shown to compare very well with the measuredattenuation. The inverse problem, i.e., the problem of determining the particle size distribution given the attenuation as a function of frequency, is examined using regularization techniques that have been successful for bubbly liquids. It is shown that, unlike the bubbly liquids, the success of solving the inverse problem is limited since it depends strongly on the nature of particles and the frequency range used in inverse calculations.

Propagation and reflection of internal waves
View Description Hide DescriptionFully nonlinear numerical simulations are performed to examine the behavior of largeamplitude internal gravity waves incident upon a level where the Dopplershifted frequency of the waves is comparable to the background buoyancy frequency. Although linear theory predicts that the waves should reflect if the Dopplershifted frequency is greater than the buoyancy frequency, it is found that nonlinear effects may greatly enhance the transmission of a wave packet across a reflecting level. If the Dopplershifted frequency is moderately less than the buoyancy frequency, then nonlinear effects may greatly enhance the reflection of waves. A range of simulations is performed to characterize the reflection coefficient as a function of the amplitude and spatial extent of the wave packet. In comparison with horizontally periodic wave packets, it is found that the nonlinearly enhanced transmission of wave packets is more significant if they are horizontally compact. This occurs because the waveinduced mean flow effectively increases and decreases the horizontal phase speed of the waves on the incident and trailing flank of the wave packets, respectively, and this significantly broadens the frequency spectrum of the waves.

Timedependent simulations of point explosions with heat conduction
View Description Hide DescriptionA hydrodynamicdiffusion code is used to simulate a point explosion. The gas motion is governed by both hydrodynamics and nonlinear heat conduction and is a combination of the wellknown, selfsimilar Taylor–Sedov spherically expanding shock wave and the spherically expanding thermal wave. Two problems are discussed. In the first problem, a similarity solution exists if the diffusion coefficient is given in terms of powers of density and temperature which also define the ambient spatial density profile. If the initial explosion energy is small, the diffusive effect is limited to a region behind the shock. However, if the explosion energy is large, the thermal front precedes the hydrodynamic front, which is then an isothermal shock. In the second problem, the initial density is constant and the diffusion coefficient depends on only a power of the temperature. In this case, the solution is not selfsimilar; in early times, heat conduction dominates; in late times—hydrodynamics. The problems were previously analyzed by Reinicke and MeyerterVehn in terms of similarity variables.

The stability of steady, helical vortex filaments in a tube
View Description Hide DescriptionThe nonlinear conditions for the development of helical vortex filaments in a circular tube are considered. The helical flow is assumed to be irrotational, except in a vortex filament of infinitesimal core area. By introducing an appropriate image for this helical vortex filament, the boundary condition on the material frontier is satisfied. By assuming an axisymmetric flow upstream and imposing the conservation laws, a dependence between the helix pitch and the nonlinear amplitude of the helical vortex developed downstream is obtained. Our results show that only helical flows with the pitch in a certain range of values are allowed. The dependence of this range on the swirl ratio and on the tube cross section is considered. We discuss the usefulness of the nonlinear analysis of the allowed flows to explain experimental results and to complement the usual linear study of stability.

Pulsed gradient spin echo nuclear magnetic resonance measurements of hydrodynamic instabilities with coherent structure: Taylor vortices
View Description Hide DescriptionPulsed gradient spin echo (PGSE) nuclear magnetic resonance(NMR) is applied to the characterization of hydrodynamicinstabilities. It is demonstrated theoretically and experimentally that for Taylor vortexflow in a Couette cell the PGSE NMR data is coherently modulated in an interference pattern dependent upon the vortex size, or wavelength, and velocity intensity. Spatially resolved NMR velocity images of all three velocity components for water in supercritical Taylor number flow and NMR velocity image data of the axial disturbance velocity for pentane at three supercritical values of the Taylor number are presented. For the short column used the NMR velocity data clearly show axial asymmetry with maximum velocities and the center (eye) of the vortex drawn toward the radial outflow boundary.

Nonparallel linear stability analysis of Long’s vortex
View Description Hide DescriptionA nonparallel linear stabilityanalysis of a family of selfsimilar vortex cores which includes Long’s vortex as a particular member is performed using parabolized stability equations (PSE). The resulting streamwise variation of both the spatial growth rate and the axial wave number of the different unstable modes is compared with the results from a local spatialstabilityanalysis which also takes into account the effects of viscosity and of the streamwise variation of the basic flow, so that the effect of the history of the disturbances on their stability is quantified. It is shown that this last effect is negligible for high Reynolds numbers, but becomes increasingly important as the Reynolds number decreases, especially for very small growth rates. The marching method used to solve the PSE is computationally much faster than the standard methods for solving the nonlinear eigenvalue problem resulting from the local stability equations. As a new result, the local spatial calculations reveal the existence of unstable counterrotating spiral modes with negative group velocities for Type II Long’s vortices (that is, vortices with negative streamwise velocity at the axis), thus showing that these flows are subcritical in Benjamin’s sense. This kind of instability does not appear for Type I vortices, which can only sustain nonaxisymmetric convective instabilities, and are therefore supercritical. Thus, the spatialstabilityanalysis establishes a fundamental distinction between Type I and Type II Long’s vortices.

Experiments on the Richtmyer–Meshkov instability: Wall effects and wave phenomena
View Description Hide DescriptionExperiments examining the interaction of shock waves with an interface separating two gases of different densities are reported. Flow visualization by the schlieren method and xray densitometry reveals that important secondary effects are introduced by the experimental apparatus, especially at the walls of the shock tube from shock wave/boundary layer interaction below, above, and at the interface itself. These effects can impair the observation of the primary phenomenon under study and can lead to the overall deformation of the interface. In particular, the thickness of the viscous boundary layer at the interface is computed using a familiar shock tube turbulent boundary layer model and the occurrence of bifurcation of reflected waves below and above the interface is successfully predicted based on classical bifurcation arguments. The formation of wall vortical structures at the interface is explained in terms of baroclinic vorticity deposition resulting from the interaction of reflected waves with the interface distorted by the boundary layer. This mechanism of wall vortex formation can also explain observed test gas contamination in reflected shock tunnels when shock wavebifurcation is absent. In general, it is found that most of the side effects of the experimental investigation of the Richtmyer–Meshkov instability can be alleviated by performing experiments in large test sections near atmospheric initial pressure.

On the threedimensional Rayleigh–Taylor instability
View Description Hide DescriptionThe threedimensional Rayleigh–Taylor instability is studied using a lattice Boltzmann scheme for multiphase flow in the nearly incompressible limit. This study focuses on the evolution of the threedimensional structure of the interface. In addition to the bubble and spike fronts, a saddle point is found to be another important landmark on the interface. Two layers of heavyfluid rollups, one at the spike tip and the other at the saddle point, were observed. The secondary instability in the horizontal planes entangles the already complicated structure of the interface. Parallel computations are utilized to accommodate the massive computational requirements of the simulations.
