Volume 11, Issue 10, October 1999
 LETTERS


Domain of convergence of perturbative solutions for HeleShaw flow near interface collapse
View Description Hide DescriptionRecent work [Phys. Fluids 10, 2701 (1998)] has shown that for HeleShaw flows sufficiently near a finitetime pinching singularity, there is a breakdown of the leadingorder solutions perturbative in a small parameter ε controlling the largescale dynamics. To elucidate the nature of this breakdown we study the structure of these solutions at higher order. We find a finite radius of convergence that yields a new length scale exponentially small in ε. That length scale defines a ball in space and time, centered around the incipient singularity, inside of which perturbation theory fails. Implications of these results for a possible matching of outer solutions to inner scaling solutions are discussed.

Rarefaction–undercompressive fronts in driven films
View Description Hide DescriptionWe consider experiments by Ludviksson and Lightfoot [AIChE J. 17, 1166 (1971)] on thin liquid films driven up a vertical plate by a thermally induced surface tension gradient with a counteracting gravitational force, and revisit their theoretical analysis, which neglects the effects of curvature, for predicting the climbing rate of the front. We present a new theory for the lubrication model with curvature effects, and get rising rates that depend on the microscopic length scale at the contact line. The predictions are, in general, in better agreement with the experiment.

Flow state multiplicity in convection
View Description Hide DescriptionPattern formation in a layer of fluid heated from below is an example of macroscopic ordering in continuous media. Here we show that in a relatively compact experimental version of the problem, a rich and diverse set of stable flows can be found. These flows, many of which are novel, can be categorized and understood in terms of their symmetry properties. This approach shows promise for providing insight into the more complicated fluid motion that occurs as the lateral dimension of the layer is increased.

New instability modes of a diffusion flame near extinction
View Description Hide DescriptionThe linear temporal stability of a diffusionflame, modeled with a simplified onedimensional approach, in a twodimensional mixing layer is investigated. In addition to the wellknown “shear” or Kelvin–Helmholtz mode, which is also present in nonreacting flows, the current analysis reveals new “heat release” or combustion modes for flames near the extinction limit, i.e., when the Damköhler number is low. For conditions near extinction, the shear mode is found to be damped and only the combustion modes are unstable. The fundamental destabilizing mechanism of these combustion modes is elucidated.
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 ARTICLES


Pressuredriven channel flows of a model liquidcrystalline polymer
View Description Hide DescriptionShear flows disrupt molecular orientation in liquidcrystallinepolymers(LCPs) through director tumbling, and this causes difficulty in controlling the polymer structure and properties in injection molding and extrusion. In this paper we simulate LCP channel flows using the Doi theory. A Bingham closure is used to preserve director tumbling and wagging. The objective is to examine how contractions and expansions in a channel affect LCP orientation and to explore the possibility of using the channel geometry as a means of manipulating LCP order. A finiteelement method is used to solve the coupled equations for fluid flow and polymer configuration. Results show that a contraction aligns the director with the streamline and improves molecular order, while an expansion drives the director away from the flow direction and reduces molecular order. If the expansion is strong enough, an instability develops downstream as disturbances in the flow and polymer configuration reinforce each other through the polymer stress. This instability generates a wave that spans roughly the central half of the channel and propagates downstream at the centerline velocity. For abrupt contractions or expansions, disclinations of ±1/2 strength arise in the corner vortex. The numerical results agree qualitatively with experiments when comparisons can be made. In particular, the wavy pattern following a sudden expansion is remarkably similar to previous experimental observations. The simulations suggest that using contractions and expansions may be a feasible strategy for controlling LCP order and morphology in processing.

Circulationinduced shape deformations of drops and bubbles: Exact twodimensional models
View Description Hide DescriptionIn this paper simple twodimensional mathematical models for understanding the fluid dynamical problem of how circulation affects the free surface shapes of inviscid drops and bubbles with surface tension are presented. This theoretical paradigm is of interest in many areas of science including largescale transport processes in chemical engineering. Exact solutions for the finiteamplitude steadystate equilibria of the mathematical models are found. Equilibrium states are shown to exist right up to steady capillary pinchoff in the case of a bubble, the bubbles just before pinchoff having large perimetertoarea ratios.

Breakup of drops and bubbles translating through cylindrical capillaries
View Description Hide DescriptionWe examine the shape deformation and breakup of air bubbles and viscousdrops moving through vertical cylindrical capillaries under the action of pressure and/or buoyancy forces. Experimental observations of fluid particle shape are reported over a wide range of particle sizes and capillary numbers in a variety of twophase systems. Four different modes of breakup are identified, and the critical conditions for the onset of various modes are examined. It is found that buoyancy forces can have a stabilizing effect on the breakup mechanism observed by Olbricht and Kung [Phys. Fluids 4, 134, (1992)] for low viscosityratio drops, wherein a growing indentation at the trailing end of the drop develops into a penetrating jet of outer phase fluid.

Axisymmetric thermal wake interaction of two bubbles in a uniform temperature gradient at large Reynolds and Marangoni numbers
View Description Hide DescriptionAn analysis is performed on the thermocapillary motion of two bubbles in a continuous phase in which a linear temperature field is imposed in the undisturbed state. The bubbles are moving in the direction of the temperature gradient and are assumed to interact axisymmetrically via the influence of the thermal wake of the leading bubble on the trailing bubble. The flowfield interaction of the bubbles is neglected. The Reynolds number of the motion is assumed to be sufficiently large that a potential flow field prevails around each bubble. The energy equation for the temperature field around the trailing bubble is solved by matched asymptotic expansions for large values of the Marangoni number. It is shown that the thermal wake of the leading bubble induces a nonmonotonic temperature field on the surface of the trailing bubble. The effective temperature gradient on the trailing bubble is weakened. Hence its migration speed is reduced compared to the case when it is isolated. This result is in qualitative agreement with experimental results for a pair of interacting drops obtained from a space flight experiment that has been reported in the literature.

Observations of shearinduced particle migration for oscillatory flow of a suspension within a tube
View Description Hide DescriptionSuspensions of noncolloidal, neutrally buoyant, spherical particles were subjected to oscillating displacements at low Reynolds number along the axis of a circular tube. Using nuclear magnetic resonance imaging(NMRI), the phase distribution of a suspension with a particle volume fraction 0.4 was assessed for a variety of conditions. The variables studied included ratio of particle to tube diameter, amplitude of oscillation, and number of oscillations. Consistent with macroscopic theories of shearinduced particle migration, the particles preferentially moved away from the walls and to the center of the pipe for amplitudes of oscillation much greater than the particle diameter when the ratios of particle radius to tube radius were and However, for a ratio of particle radius to tube radius of the images showed that the suspension was not uniform along the tube length for an amplitude of oscillation equivalent to one pipe diameter. For a larger ratio of particle radius to tube radius of the suspension remained uniform along the pipe for similar conditions. For the smaller ratio of particle to tube radius of and an amplitude of oscillation of five particle radii, the particles migrated to the wall of the pipe as predicted by the Stokesian dynamics simulations of Morris [“Anomalous particle migration in oscillatory pressuredriven suspension flow,” presented at the 1997 Annual Meeting of the AICHE (unpublished)]. These phenomena, which have not previously been observed experimentally, are not described by any existing theories of shearinduced particle migration.

Orientation distribution in a dilute suspension of fibers subject to simple shear flow
View Description Hide DescriptionThe orientation distribution in a dilute suspension of Brownian fibers subject to simple shear flow is investigated theoretically. The fiber has a large aspect ratio r and may carry charge. The flow strength is characterized by the rotary Peclet number where γ is the shear rate and D is the rotary diffusivity of the fiber. Emphasis is placed on the microstructure of the suspension in strong flows, in which the advection and diffusion are of equal importance in a small region of angular space near the flow direction. A new computational method based on a finite difference scheme is developed to calculate the orientation distribution function and then evaluate the orientation moments. For the case of uncharged rods and the obtained orientation moments compare favorably with those deduced from the spherical harmonic method. The effects of the fiber aspect ratio and charge are also investigated when they become crucial. It is found that at a given large Pe, the orientation distribution of uncharged fibers with becomes broad and its peak shifts down towards the flow axis, compared with the case of thin rods with The microstructure of a suspension of charged fibers depends on the charge distribution and the double layer thickness relative to the fiber length.

Reversing buoyancy of particledriven gravity currents
View Description Hide DescriptionParticleladen flows exhibit reversing buoyancy behavior if the density of the ambient through which they propagate is greater than that of the interstitial fluid, though less than the initial bulk density of the suspension. In this case a gravity current is initiated above the underlying boundary until sufficient particles have sedimented from the flow, at which time the particleladen fluid becomes less dense than the surrounding ambient. The buoyancy of the residual suspension reverses and it lifts off the boundary to ascend through the ambient. Such phenomena are encountered in industrial and natural situations. This study presents a laboratory investigation of finite volume releases of particleladen fluid which undergo reversing buoyancy. A simple box model theory is proposed to describe the flow and to predict the distance from the source at which liftoff occurs. The predictions of the model agree well with both our experiments and those of previous studies. Additionally, we investigate these flows using the shallowwater equations which are analyzed using asymptotic series. These reveal the structure of the internal dynamics within the currents and predict liftoff distances which verify the validity of those obtained from the less rigorous box model.

Investigation of the Weissenberg effect in suspensions of magnetic nanoparticles
View Description Hide DescriptionWe have investigated the appearance of normal stress differences in suspensions of magnetic nanoparticles by observation of the rise of the fluid at a rotating axis. The combination of the low shear rate limits of the normal stress coefficients and has been varied by means of the action of a magnetic field applied to the fluid. Its value has been determined for different shear rate and magnetic field strength. To enhance observability of the rise of the fluids, the experiments have been carried out under conditions of reduced gravity during parabolic flights.

On mass transport of progressive edge waves
View Description Hide DescriptionThe boundarylayer solutions for progressive edge waves on a uniformly sloping beach are derived. The Larangian mass transport in the crossshore direction is found to be caused by the viscous effect and it changes direction within the boundary layer. For validation of the theoretical model, laboratory experiments are performed. The measuredmass transports of mode 0 and 1 edge waves are in good agreement with the theory qualitatively but not quantitatively. The discrepancies appear to be caused by the failure of the theory near the shoreline, where the boundary layer is no longer thin. The inviscid Stokes drift caused by the nonlinear interactions of different edgewave modes within the potential flow field is also investigated theoretically and verified by experiments. It is found that nearshore circulation cells can be formed by the nonlinear interactions between different progressive edgewave modes.

A new upper bound on the growth rate of linear instability of baroclinic zonal flows in a twolayer model on a betaplane
View Description Hide DescriptionTemporally growing modes of the linearized equations of motion for the baroclinic zonal flows in the twolayer model on a betaplane are considered. The classical result for the bound on growth rate derived by Joseph Pedlosky for an arbitrary unstable mode has been improved considerably. Further, a rigorous mathematical proof of Howard’s conjecture which states that the growth rate of an arbitrary unstable wave must approach zero as the wavelength approaches zero, is presented in this case for the first time under the restriction of the boundedness of the basic potential vorticity gradient of the mean zonal current in each layer with respect to the horizontal coordinate in the concerned flow domain.

Flow control of vortex shedding by a short splitter plate asymmetrically arranged downstream of a cylinder
View Description Hide DescriptionThe flow around a circular cylinder (diameter d) was investigated, behind which a short thin splitter plate with a chord of was inserted horizontally as an interference element. The plate was traversed upstream along the wake; hence its relative position is defined by gap G from the cylinder base to the plate tip and by level Z from the wake centerline. The variation in both base suction coefficient and Strouhal number with significantly depends on In the circular cylinder case with as the plate approaches the cylinder, the base suction coefficient exhibits a critical fall in a similar fashion to Roshko’s experiment where Interestingly, however, unlike his experiment, the Strouhal number exceeds the natural one for some range of beyond the critical gap. To further examine the flow mechanism, a similar situation was investigated by using a rectangular cylinder (height h, depth ) in place of the circular cylinder. A rise in the Strouhal number is observed for These rises in Strouhal number are explained by the flow mode in which the shear layer separated from the gap side is forced to flow into the gap and to interact in the nearer wake by the approach of the splitter plate.

Coexisting acousticrotational flow in a cylinder with axisymmetric sidewall mass addition
View Description Hide DescriptionTimedependent flow dynamics within a cylinder with sidewall mass injection are investigated. A timedependent injection velocity, prescribed along the sidewall boundary of a long, narrow, halfopen cylinder, induces a low Mach number, high Reynolds number flow. The injection is the source of planar acoustic disturbances which interact with the injected fluid to produce vorticity on the sidewall in an inviscid manner. The analysis of these flow processes is based on the Navier–Stokes equations, which are reduced to simpler forms using a multiplescale asymptotic analysis. The equations that arise from the analysis describe the leadingorder vorticitydynamics. These nonlinear equations possess both wave and diffusion properties and are solved in an initial value sense. The results show that the vorticity produced at the sidewall convects toward the center of the cylinder, diffuses radially, and convects downstream.

A geometric/kinematic interpretation of scalar mixing
View Description Hide DescriptionScalar mixing is interpreted in terms of the growth or decay with time of an infinitesimal spatial volume located between two neighboring isoscalar surfaces. The divergence of the velocity of propagation of isoscalar surfaces relative to the fluid and perpendicular to them yields a variable which may be interpreted as the inverse of a scalar characteristic mixing time. Two contributions, the planar isoscalar surface part and the one due to its curvature, are distinguished. The former is directly linked to the diffusive and chemical mechanisms contributing to the evolution of the scalar dissipation rate. The second rank symmetric tensor, seems to play an important role in these definitions. Direct numerical simulations (DNS) help to identify possible dependences of on its invariants.

Steady streaming of fluid in the entrance region of a tube during oscillatory flow
View Description Hide DescriptionTo gain understanding and insight of entranceflow phenomena, we solve a fluid dynamics problem involving an oscillating flat velocity profile (with zero mean flow) at the end (entrance) of a semiinfinite, rigid tube. A successiveapproximation method is used. Analytic solutions are given for the firstorder (linear) approximation. Time averages of products of these firstorder solutions for velocity components and their spatial derivatives are used to approximate the nonlinear terms in the secondorder equations, which are solved numerically. The timeaveraged secondorder results demonstrate steady bidirectional streaming in the entrance region. Solutions are given to an illustrative problem with a Reynolds number of 20 and a Womersley unsteadiness parameter of At this low Reynolds number, the streaming effects are small, with the magnitudes of the calculated streaming velocities less than three percent of the amplitude of the velocity specified at the entrance. The calculated magnitudes of streaming velocities would be expected to increase at higher Reynolds numbers; however, the convergence of the secondorder solution would be less certain at higher Reynolds numbers. The steady streaming is related to firstorder velocity variations that exist near the entrance of the tube.

The geometry and statistics of mixing in aperiodic flows
View Description Hide DescriptionThe relationship between statistical and geometric properties of particle motion in aperiodic, twodimensional flows is examined. Finitetimeinvariant manifolds associated with transient hyperbolic trajectories are shown to divide the flow into distinct regions with similar statistical behavior. In particular, numerical simulations of simple, eddyresolving barotropic flows indicate that there exists a close correlation between such geometric structures and patchiness plots that describe the distribution of Lagrangian average velocity over initial conditions. For barotropic turbulence, we find that Eulerian velocity correlation time scales are significantly longer than their Lagrangian counterparts indicating the existence of welldefined Lagrangian structures. Identification of such structures shows a similar, close relationship between the invariant manifold geometry and patchiness calculations at intermediate time scales, where anomalous dispersion rates are found.

Experimental study of nonBoussinesq Rayleigh–Bénard convection at high Rayleigh and Prandtl numbers
View Description Hide DescriptionA set of experiments is performed, in which a layer of fluid is heated from below and cooled from above, in order to study convection at high Rayleigh numbers (Ra) and Prandtl numbers (Pr). The working fluid, corn syrup, has a viscosity that depends strongly on temperature. Viscosity within the fluid layer varies by a factor of 6 to in the various experiments. A total of 28 experiments are performed for and Pr sufficiently large, that the Reynolds number (Re) is less than 1; here, values of Ra and Pr are based on material properties at the average of the temperatures at the top and bottom of the fluid layer. As Ra increases above flow changes from steady to timedependent. As Ra increases further, large scale flow is gradually replaced by isolated rising and sinking plumes. At there is no evidence for any large scale circulation, and flow consists only of plumes. Plumes have mushroomshaped “heads” and continuous “tails” attached to their respective thermal boundary layers. The characteristic frequency for the formation of these plumes is consistent with a scaling. In the experiments at the largest Ra, the Nusselt number (Nu) is lower than expected, based on an extrapolation of the Nu–Ra relationship determined at lower Ra; at the highest Ra, and the lowerthanexpected Nu is attributed to inertial effects that reduce plume head speeds.
