Volume 13, Issue 5, May 2001
 ARTICLES


Transient growth: A factor in bypass transition
View Description Hide DescriptionTransient growth arises through the coupling between slightly damped, highly oblique (nearly streamwise) T–S and Squire modes leading to algebraic growth followed by exponential decay in a region that is subcritical with respect to the T–S neutral curve. A weak transient growth can also occur for two dimensional or axisymmetric modes since the Orr–Sommerfeld operator and its compressible counterpart are not selfadjoint, therefore their eigenfunctions are not strictly orthogonal. So transient growth is a candidate mechanism for many examples of bypass transition. The original transient growth theories were all temporal. However spatial growth formulations are now emerging including their extension to compressible flow with pressure gradient and heat transfer. The relevance to bypass transition is examined through several examples including Poiseuille pipe flow, the hypersonic blunt body paradox and distributed roughness effects.

Network model for deep bed filtration
View Description Hide DescriptionWe study deep bed filtration, where particles suspended in a fluid are trapped while passing through a porous medium, using numerical simulations in various network models for flow in the bed. We first consider cellular automatamodels, where filtrate particles move in a fixed background flow field, with either nomixing or completemixing rules for motion at a flow junction. The steadystate and timedependent properties of the trapped particle density and filter efficiency are studied. The complete mixing version displays a phase transition from open to clogged states as a function of the mean particle size, while such a transition is absent in the (more relevant) nomixing version. The concept of a trapping zone is found to be useful in understanding the timedependent properties. We next consider a more realistic hydrodynamic network model, where the motion of the fluid and suspended particles is determined from approximate solutions of the timedependent Stokes equation, so that the pressure field constantly changes with particle movement. We find that the steadystate and timedependent behavior of the network model is similar to that of the corresponding cellular automatamodel, but the long computation times necessary for the simulations make a quantitative comparison difficult. Furthermore, the detailed behavior is extremely sensitive to the shape of the pore size distribution, making experimental comparisons subtle.

The coalescence of two equalsized drops in a twodimensional linear flow
View Description Hide DescriptionA fourroll mill was used to experimentally investigate the coalescence of two equalsized drops in general linear flows. The experimental system consisted of polybutadiene drops suspended in polydimethylsiloxane. Under the experimental conditions studied, the bulkphase rheological properties of both fluids are Newtonian. We studied both headon collisions for a purely extensional “hyperbolic” flow that always lead to coalescence, and collisions with a finite offset from the inflow axis for several different flow types produced in the fourroll mill. The experimental results have been compared with approximate theoretical predictions of coalescence, based on an asymptotic theory for small capillary number, where the drops are spherical apart from a small planar deformation at the frontal surfaces between the two drops. In headon collisions, it was found experimentally that the product of the film drainage time and strain rate is independent of capillary number (Ca) and drop radius at very low Ca. This scaling behavior agrees qualitatively with theoretical predictions from the approximate small Ca theory. At higher values of Ca, this dimensionless drainage time varies as This scaling is also consistent with theoretical predictions. For collisions with a nonzero offset from the headon configuration, coalescence occurs only for capillary numbers below a critical value, Measurements were made of as a function of the drop size and the flow type, for various values of the offset. The critical capillary number for coalescence was found to decrease with increasing offset, in qualitative agreement with predictions from the theoreticalmodel. However, these versus offset results do not agree quantitatively with the theoretical predictions. In the model the minimum film thickness occurs when the twodrop pair has rotated to the angle at which the external flow just begins to pull them apart. However, for configurations with small but nonzero offsets, it is found experimentally that coalescence occurs earlier in the collision process. Thus, the actual time available for film drainage is shorter, and the critical capillary number is smaller than what is predicted by the model. At the same time, for larger offsets, it is shown experimentally that the collision and initiation of film drainage is delayed relative to what is predicted, and thus there are offset values where the model predicts that coalescence is possible, whereas, in fact, no coalescence is observed.

Dynamics of a condensing liquid film under conjoining/disjoining pressures
View Description Hide DescriptionThe dynamics of a condensing apolar ultrathin liquid film is studied in the framework of longwave theory in the cases of both horizontal and slightly tilted solid coated surfaces. When condensation is slow, the film on a horizontal substrate passes through the stages of hole opening driven by the “reverse reservoir effect,” hole closing, eventual thickness equilibration and further spatially uniform growth of the condensate. When condensation is faster and the resistance to phase change is lower, secondary droplet(s) may emerge within the hole. During the film evolution the thickness of the microlayer covering the hole remains practically constant due to the “reverse reservoir effect.” The total heat flux across the condensate film is found to decrease with the absolute value of the condensation constant. When the solid substrate is tilted, the filmdynamics exhibits the formation of multidrop structures and their coarsening along with the stages typical for the horizontal case. The increase of the tilt angle leads to faster transition from dropwise to filmwise condensation and to the increase of the total heat flux through the condensate.

Nuclear magnetic resonance measurements of fluctuations in air–water twophase flow: Pipe flow with and without “disturbing” section
View Description Hide DescriptionIn this paper we describe a new NMR sequence designed to measure the liquid mass flux and the liquid momentum flux fluctuations in a twophase flow, in order to study the action of the flow on transverse mechanical structures. An upward air–water bubbly flow is investigated in a circular pipe under atmospheric pressure and room temperature for two geometrical configurations: with and without a “disturbing” section. Results show that the addition of the “disturbing” section on the experimental setup led to the appearance of an instability phenomenon that greatly influences the relative intensity of both mass and momentum liquid flux fluctuations. Moreover, the spectral analysis of the former quantities demonstrates that the fluctuations of the liquid space fraction play a major role in the behavior of the liquid momentum flux fluctuations.

Rupture of thin viscous films by van der Waals forces: Evolution and selfsimilarity
View Description Hide DescriptionThe van der Waals driven rupture of a freely suspended thin viscous sheet is examined using a longwavelength model. Dimensional analysis shows the possibility of firsttype similarity solutions in which the dominant physical balance is between inertia, extensional viscous stresses and the van der Waals disjoining pressure, while surface tension is negligible. For both line rupture and point rupture the film thickness decreases like and the lateral length scale like where is the time remaining until rupture. In each geometry these scalings are confirmed by numerical simulations of the timedependent behavior, and a discrete family of similarity solutions is found. The “lowestorder” mode in the family is the one selected by the timedependent dynamics.

Measurement of the recoalescence flux into the rear of a Taylor bubble
View Description Hide DescriptionMost of the theoretical models on vertical slug flow assume the mass balance of a Taylor bubble to depend only on the incoming gas flux at the top of the Taylor bubble and on the outgoing entrainment flux at the bottom. This means that the recoalescence flux, which is defined as the fraction of the entrainment flux that coalesces back into the bubble, is neglected. Only in Fernandes et al. [AIChE J. 29, 981 (1983)] is a model proposed for this recoalescence flux but their model has never been verified by measurements. Therefore, we set out in the present research to measure and quantify the recoalescence flux. Our experiments have been carried out in a recirculating flow facility with a vertical cylindrical test section with inner diameter mm. In this test section a Taylor bubble is kept at a fixed vertical position by a constant downward liquid flow A continuous stream of small helium bubbles is injected into the wake of the Taylor bubble. The recoalescence flux is then determined by measuring the concentration of helium in the Taylor bubble. Our experiments show that there is a recoalescence flux and that in general it cannot be neglected in the mass balance of the Taylor bubble. The total gas loss from the Taylor bubble, and the recoalescence flux increase both strongly with the Taylor bubble length, The fraction of entrained bubbles that recoalesces back into the Taylor bubble increases from 10% of the entrainment flux at cm to 45% at cm

Leakage through filtercake into a fluid sampling probe
View Description Hide DescriptionPore fluid can be withdrawn from reservoir rock by means of a probe lowered down a well and clamped against the rock surface. The rest of the rock surface is covered by a drilling fluid filtercake which impedes, but does not totally prevent, flow of filtrate from the wellbore into the rock and thence into the probe. The magnitude of this filtrate flow is investigated in an idealized geometry in which the porous rock, with permeability k, occupies the halfspace The probe covers the circular region of the plane and the rest of the plane is covered by a thin filtercake of permeability and thickness h. The fluid is assumed incompressible and obeys Darcy’s law, so that the fluid pressure p in the porous rock satisfies the Laplace equation. The pressure in the probe is and in the wellbore and in the pore fluid at infinity. This mixed boundary value problem depends only on If the problem is equivalent to that of an electrified disc at constant potential in unbounded space, and pore fluid is drawn from the rock at infinity. If fluid leaks from the wellbore into the reservoir, and the volume of fluid withdrawn by the probe is equal to the volume of fluid which passes from the wellbore into the rock. When fluid streamlines within the rock are similar to those for close to the probe, but emanate from the filtercake on on a length scale Estimates of the hydraulic resistance of filtercakes usually encountered when drilling for petroleum indicate that this leakage flux is sufficiently small to be neglected over typical time scales for fluid sampling.

Fingering instabilities in driven thin nematic films
View Description Hide DescriptionMotivated by experimental work (Cazabet et al., unpublished), we consider the possibility of fingering instabilities in thin films of nematic liquid crystals. We use lubrication theory on the flow equations for nematic liquid crystals to derive a simple model describing the evolution of the film height. As far as we are aware, this is the first time such a systematically derived, timedependent thin film model for nematics has been presented. Simple “leadingorder” solutions (depending on only one spatial coordinate) are found for two different flow driving mechanisms: (i) gravity perpendicular to the film and (ii) gravity parallel to the film (capillarity is also included in both cases). The effect of imposing twodimensional perturbations to these solutions is studied. We find that for case (i) instability is possible, depending on whether or not there is complete wetting (i.e., whether or not the equilibrium contact angle of the droplet with the substrate is zero). For case (ii) we always have instability, as we would expect from the analogous result for Newtonian fluids [Phys. Fluids 8, 460 (1996); Europhys. Lett. 10, 25 (1989)].

The action of pressureradiation forces on pulsating vapor bubbles
View Description Hide DescriptionThe action of pressureradiation (or Bjerknes) forces on gas bubbles is well understood. This paper studies the analogous phenomenon for vapor bubbles, about which much less is known. A possible practical application is the removal of boiling bubbles from the neighborhood of a heated surface in the case of a downward facing surface or in the absence of gravity. For this reason, the case of a bubble near a plane rigid surface is considered in detail. It is shown that, when the acoustic wave fronts are parallel to the surface, the bubble remains trapped due to secondary Bjerknes force caused by an “image bubble.” When the wave fronts are perpendicular to the surface, on the other hand, the bubble can be made to slide laterally.

The effect of slight deformation on droplet coalescence in linear flows
View Description Hide DescriptionA trajectory analysis is used to determine the effect of small deformations and van der Waals attractions on the collision efficiency of two nonBrownian drops freely suspended in a linear flow at small Reynolds number. Simple shear flow and uniaxial compressional and extensional flow are considered. Treating the capillary number (Ca) as a small parameter permits an approach similar to matched asymptotic expansions. For the analysis shows that the deformation is mainly axisymmetric and that the tangential motion of the drops in apparent contact is unaffected to leading order by the small deformation. A comparison with full threedimensional boundaryintegral calculations confirms the accuracy of the asymptotic approach. In the dimensionless parameter space, results for the collision efficiency are mapped for four parameters: Ca, size ratio, droptomedium viscosity ratio, and a dimensionless Hamaker parameter. For spherical drops in uniaxial compression and extension, the collision efficiencies are identical due to the reversibility of Stokes flow. When small deformation is introduced, however, the collision efficiencies are lower for compression than for extension. For slightly deformable drops in simple shear flow, the critical capture cross section upstream is no longer a circle, in contrast to the behavior of spherical drops in the absence of van der Waals forces. For all flow types, a key result is that the collision efficiency decreases rapidly from the corresponding value for spherical drops, as the capillary number increases beyond a critical value, due to small deformations. Consequently, droplet growth by coalescence will be arrested when the drops reach a prescribed size, as shown by population dynamics simulations for a model physical system.

Pattern formation in nonNewtonian Hele–Shaw flow
View Description Hide DescriptionWe study theoretically the Saffman–Taylor instability of an air bubble expanding into a nonNewtonian fluid in a Hele–Shaw cell, with the motivation of understanding suppression of tipsplitting and the formation of dendritic structures observed in the flow of complex fluids, such as polymeric liquids or liquid crystals. A standard viscoelastic flow model is simplified in the case of flow in a thin gap, and it is found that there is a distinguished limit where shear thinning and normal stress differences are apparent, but elastic response is negligible. This observation allows formulation of a generalized Darcy’s law, where the pressure satisfies a nonlinear elliptic boundary value problem. Numerical simulation shows that shearthinning alone modifies considerably the pattern formation and can produce fingers whose tipsplitting is suppressed, in agreement with experimental results. These fingers grow in an oscillating fashion, shedding “sidebranches” from their tips, closely resembling solidification patterns. A careful analysis of the parametric dependencies of the system provides an understanding of the conditions required to suppress tipsplitting, and an interpretation of experimental observations, such as emerging lengthscales.

On dense granular flows down flat frictional inclines
View Description Hide DescriptionWe consider dense, relatively shallow flows of 3 mm glass spheres moving down a chute with a flat, frictional base of 3.6 m length. Sustained flows are observed at inclinations corresponding to an effective friction between the static and dynamic friction of individual grains. A capacitance instrument records the formation of waves with a dominant component traveling upstream. Simultaneous measurements of granular temperature at the base using a load cell reveal that the waves are accompanied by substantial reduction in granular agitation. A theory incorporating contributions from impulsive and enduring interactions with the base produces quantitative predictions for the range of sustained flows observed in the experiments. Closure of the theory is achieved using a balance between the production and dissipation of angular momentum in a narrow basal shear layer. A linear stability analysis of the corresponding hydraulic equations further suggests the origin of the waves.

Effect of membrane bending stiffness on the axisymmetric deformation of capsules in uniaxial extensional flow
View Description Hide DescriptionThe effect of interface bending stiffness on the axisymmetric deformation of liquid capsules enclosed by elastic membranes subject to uniaxial extensional flow is considered. Flowinduced deformation causes the development of membrane inplane elastic tensions and bending moments due to the noninfinitesimal thickness of the membrane or to a preferred equilibrium configuration of an interfacial molecular network, accompanied by transverse shear tensions. The elastic tensions are related to the surfacedeformation by means of Mooney’s linear constitutive law for thin elastic sheets, and the bending moments are related to the membrane resting shape and to the instantaneous principal curvatures by means of constitutive equations. Interfacial force and torque balances are used to relate the jump in hydrodynamic traction across the interface to the elastic tensions and bending moments. A numerical procedure is implemented for simulating the capsule deformation in uniaxial extensional Stokes flow based on a boundaryintegral method. Results on the transient and asymptotic deformation for capsules with spherical unstressed shapes illustrate the effect of bending stiffness expressed by an interface modulus of bending for a broad range of elasticity capillary numbers. It is shown that bending stiffness, however large, is not able to restrain continued deformation beyond a critical capillary number, and its main effect is to cause highly deformed capsules with pointed shapes to develop instead nearly cylindrical shapes with rounded caps.

Beadbreak instability
View Description Hide DescriptionFlow is considered in a “fluid bead” located in the nip between two contrarotating rolls, and bounded by two curved menisci. Such a flow arises in meniscus roll coating where fluid is transferred from the lower applicator roll to a substrate, in contact and moving with the upper metering roll, by means of a transfer jet (snake). Equilibrium of the bead is maintained through a balance of hydrodynamic and capillary stresses, the stability of which is considered experimentally by increasing the speed of the metering roll while that of the applicator roll remains constant. At a critical speed ratio, the upstream meniscus becomes unstable; the bead contracts as the meniscus accelerates forward and merges with its downstream counterpart—giving rise to “beadbreak.” A mathematical model, based on lubrication theory, exhibits multiple solutions and a limit point for the existence of steady solutions. A linear stability analysis identifies the stable solution and shows that the flow becomes unstable at the limit point—which is taken to be the onset of beadbreak. The criterion for beadbreak is at the upstream meniscus, where P, σ, and represent fluid pressure, surface tension, and radius of curvature of the upstream meniscus, respectively. The effect of the pressure gradient is shown to stabilize the fluid bead whereas that of the surface tension term is to destabilize once the upstream meniscus—and hence the whole bead—is located downstream of the nip. For a given geometry and flow rate, the critical speed ratio decreases with increasing characteristic capillary number. Theoretical predictions compare well with experimental data.

Geometric and statistical properties in the evolution of material surfaces in threedimensional chaotic flows
View Description Hide DescriptionIn this article we analyze the invariant geometric properties of threedimensional (3D) chaotic flows. Attention is focused on the statistical (measuretheoretical) characterization of the asymptotic evolution of material surfaces forming the boundary between fluid elements, which can be characterized quantitatively in terms of intermaterial contact area density. The approach developed by Giona and Adrover [Phys. Rev. Lett. 81, 3864 (1998)] for diffeomorphisms (Poincaré map of twodimensional periodically forced flows) is extended to threedimensional autonomous systems, for which a relation is obtained between intermaterial contact area density and stretching field. The Arnold–Beltrami–Childress flow is considered as a model system. The statistical and singular properties of the intermaterial contact area measure are addressed and some as yet unsolved fundamental issues related to nonautonomous threedimensional flows are discussed.

Experimental study of Rayleigh–Taylor instability: Low Atwood number liquid systems with singlemode initial perturbations
View Description Hide DescriptionSinglemode Rayleigh–Taylor instability is experimentally studied in low Atwood number fluid systems. The fluids are contained in a tank that travels vertically on a linear rail system. A singlemode initial perturbation is given to the initially stably stratified interface by gently oscillating the tank in the horizontal direction to form standing internal waves. A weight and pulley system is used to accelerate the fluids downward in excess of the earth’s gravitational acceleration. Weight ranging from 90 to 450 kg produces body forces acting upward on the fluid system equivalent to those produced by a gravitational force of 0.33–1.35 times the earth’s gravity. Two fluid combinations are investigated: A miscible system consisting of a salt water solution and a water–alcohol solution; and an immiscible system consisting of a salt solution and heptane to which surfactant has been added to reduce the interfacial tension. The instability is visualized using planar laserinduced fluorescence and is recorded using a video camera that travels with the fluid system. The growth in amplitude of the instability is determined from the digital images and the body forces on the fluid system are measured using accelerometers mounted on the tank. Measurements of the initial growth rate are found to agree well with linear stability theory. The average of the latetime bubble and spike velocities is observed to be constant and described by where A is the Atwood number, k is the wave number, and G is the apparent gravity of the fluid system (i.e., the fluid system acceleration minus the earth’s gravity).

The dispersion relation for internal acousticgravity waves in a baroclinic fluid
View Description Hide DescriptionThe dispersion relation for internal waves in a fluid is generalized from the barotropic approximation to the baroclinic case to allow for the inclination of surfaces of constant density to surfaces of constant pressure. This generalization allows the barotropic approximation to be tested in a variety of situations. The dispersion relation applies to both acoustic waves and internal gravity waves propagating in either the ocean or the atmosphere. Imaginary terms in the dispersion relation proportional to the baroclinic vector indicate energy exchange between the wave and the mean flow, a result of buoyancy being a nonconservative force in a baroclinic fluid. A buoyancy calculation shows that the baroclinic generalization of the Brunt–Väisälä frequency N is given by where is density, is potential density, and p is pressure. The baroclinicity in a weather front or cyclonic ring can sometimes have as large an effect as the Earth’s rotation on the propagation of internal gravity waves.

Mixing and available potential energy in stratified flows
View Description Hide DescriptionMixing plays an important role in atmospheric and oceanic flows. It occurs on the small scales, is due to molecular diffusion, and is irreversible. On the other hand, stirring is a kinematic process that enhances mixing but is reversible. Budgets of the available potential energy, which require that the reference potential energy be computed, are used to study these processes. We develop an approach for calculating the available potential energy from the probability density function that is more efficient than existing methods, especially in two and three dimensions. It is suitable for application to both numerical simulations and experiments. A new length scale is defined which quantifies stirring and provides a measure of the strength of overturns resulting from stirring as well as their size. Simulations of liddriven cavityflow and stratified homogeneous turbulentshear flow provide illustrations of the method. The new length scale is similar to Thorpe scale in liddriven cavityflow and closely related to the Ellison scale in homogeneous sheared turbulence.

The continuous spectrum for a boundary layer in a streamwise pressure gradient
View Description Hide DescriptionSolutions of the Orr–Sommerfeld equation belonging to the continuous spectrum are presented for boundary layers developing in the presence of a streamwise pressure gradient. Although the continuous spectrum has received considerable attention in the Blasius case, in most engineering applications transition to turbulence occurs in a region where there is a pressure gradient. This investigation, so far as we know, is the first to examine what effect this has on the eigenfunctions. Our results show that when there is a pressure gradient the magnitude of the eigenfunctions near the edge of the boundary layer can be much larger than it is for Blasius flow. This is particularly true when the pressure gradient is adverse, but such is the case even when it is favorable. We also investigate the effect of Reynolds number and frequency on the penetration depth; the latter term refers to one of the properties of these modes that distinguishes them from Tollmien–Schlichting waves, namely, that their magnitude is largest near the edge of the boundary layer, but much smaller inside.
