Volume 13, Issue 12, December 2001
 LETTERS


Theory of suspension segregation in partially filled horizontal rotating cylinders
View Description Hide DescriptionIt is shown that a suspension of particles in a partiallyfilled, horizontal, rotating cylinder is linearly unstable towards axial segregation and an undulation of the free surface at large enough particle concentrations. Relying on the shearinduced diffusion of particles, concentrationdependent viscosity, and the existence of a free surface, our theory provides an explanation of the experiments of Tirumkudulu et al. [Phys. Fluids 11, 507 (1999); 12, 1615 (2000)].
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 ARTICLES


Electrohydrodynamically driven chaotic mixing in a translating drop
View Description Hide DescriptionWhen a liquid drop of given dielectric constant, resistivity and viscosity is translating in the presence of an electric field in a liquid of different dielectric constant, resistivity and viscosity, the resulting internal circulation is a superposition of the Hadamard–Rybcynski and Taylor (quadrapole) circulations. For sufficiently large field strengths, the quadrapole structure of the Taylor flow causes an internal stagnation disk to occur. When this flow field is modulated by an ac component of the electric field, at low Stokes numbers, the resulting Lagrangian particle trajectories can exhibit chaotic behavior. This Lagrangian chaos is examined via Poincaré maps and both the degree and rates of mixing are studied for a wide range of field strengths and modulation frequencies.

Dynamics of droplet rebound from a weakly deformable gas–liquid interface
View Description Hide DescriptionA droplet of radius moves with kinetic energy in an incompressible, continuum ambient gas and rebounds from an initially planar, weakly deformable interface bounding a halfspace of liquid. We study the rebound process in the limit where σ is the surface tension of the two liquid–gasinterfaces and is the density of the fluid comprising the drop and the planar halfspace. When viscous dissipation in both the ambient gas and in the drop is negligible, the flow inside the drop is inviscid and driven by the deformation of the near contact dimple region. The dimple region is approximated as a section of a sphere of radius Analysis of the motion induced in the liquid phase, coupled with a physically appropriate description of the interfaces, provides a theoretical description of the deformation and flow modes that ensue. Analytical predictions based on a singular perturbation analysis valid in the limit indicate that the time taken by the drop to complete a bounce is and the angular extent of the near contact region which undergoes strong deformation is The asymptotes are compared to numerical solutions to the full governing equations.

Testing for scaling behavior dependence on geometrical and fluid parameters in the two fluid drop snapoff problem
View Description Hide DescriptionWe present experimental results on the snapoff dynamics of a drop with viscosity λη dripping through a fluid of viscosity η. This paper focuses on the Stokes regime where both the inner and outer fluid viscous stresses are balanced by the pressure gradients arising from the interfacial curvature. We track the time dependence of the drop profiles near snapoff and find that successive profiles can be rescaled onto a single curve. We explore the dependence of this scaling on the nozzle diameter, surface tension, density mismatch, and viscosity ratio λ. We find that only λ affects the rescaled profile. Finally we investigate the dependence of the breaking rate on λ.

The surface energy of finite clusters of soap bubbles
View Description Hide DescriptionA simple derivation is presented of a general integral equation relating the surface energy of a free cluster of bubbles to the pressure excesses in the bubbles and a new differential equation for the change in energy in reversible transformations of a cluster is derived. It is shown that the integral equation has a simple geometrical interpretation. A general form of the equation is written which can be applied to wet clusters with finite Plateau borders. Other consequences of the energy equation are drawn.

Dynamics and rupture of planar electrified liquid sheets
View Description Hide DescriptionWe investigate the stability of a thin twodimensional liquid film when a uniform electric field is applied in a direction parallel to the initially flat bounding fluidinterfaces. We consider the distinct physical effects of surface tension and electrically induced forces for an inviscid, incompressible nonconducting fluid. The film is assumed to be thin enough and the surface forces large enough that gravity can be ignored to leading order. Our aim is to analyze the nonlinear stability of the flow. We achieve this by deriving a set of nonlinear evolution equations for the local film thickness and local horizontal velocity. The equations are valid for waves which are long compared to the average film thickness and for symmetrical interfacial perturbations. The electric field effects enter nonlocally and the resulting system contains a combination of terms which are reminiscent of the Kortweg–deVries and the Benjamin–Ono equations. Periodic traveling waves are calculated and their behavior studied as the electric field increases. Classes of multimodal solutions of arbitrarily small period are constructed numerically and it is shown that these are unstable to long wave modulational instabilities. The instabilities are found to lead to film rupture. We present extensive simulations that show that the presence of the electric field causes a nonlinear stabilization of the flow in that it delays singularity (rupture) formation.

Linear oscillations and stability of a liquid bridge in an axial electric field
View Description Hide DescriptionSmall amplitude oscillations of viscous, capillary bridges are studied in the presence of an electric dc field. The electric field is proposed as a means to maintain bridges longer than their perimeter and of uniform cylindrical shape. This is desired in the fabrication of semiconductor crystals. The material of the bridge and the surrounding medium is modeled either as a perfect or as a leaky dielectric. The frequency and the damping rate of the oscillations are calculated numerically by solving a generalized eigenvalue problem. It is shown that they depend on the ratios of the dielectric constants, and conductivities, of the two materials, the aspect ratio of the bridge, the ratio of viscous to the capillary force, which can also be viewed as the inverse Reynolds number of the flow, and, finally, the electrical Bond number, which is the ratio of the electric stresses to the capillary force. The stability limit of an initially cylindrical bridge is determined with respect to varicose disturbances. In agreement with previous studies it is shown that, if both materials are perfect dielectrics, application of an electric field has a stabilizing effect on the bridge, in the sense that the minimum value, of the aspect ratio for the bridge to remain stable drops below 0.5, irrespective of the specific value of the ratio ε. If both materials are leaky dielectrics, bridge stability is determined by the sign of and with the positive sign indicating bridge stabilization. The factor arises due to the appearance of a tangential electric stress in the perturbed state for leaky dielectrics. For both cases of leaky and perfect dielectrics, the most unstable mode is the one leading to amphora shaped bridges. It was also found that, when application of an electric field stabilizes the bridge, leaky dielectrics require a lower field than perfect dielectrics and that a large enough field tends to stabilize the bridge for almost the entire range of values of the aspect ratio Λ. These findings concur with earlier analytical results for the stability of jets in longitudinal electric fields and, in conjunction with certain experimental observations, point to the usefulness of the leaky dielectric model pertaining to the stability of bridges.

Sound–ultrasound interaction in bubbly fluids: Theory and possible applications
View Description Hide DescriptionThe interaction between sound and ultrasoundwaves in a weakly compressible viscousliquid with gas bubbles is considered. Using the method of multiple scales one and twodimensional nonlinear interactionequations are derived. The degeneracy of the interaction is found in bubbly fluids. This phenomenon lies in the fact that the interaction coefficients vanish at a certain frequency of ultrasound. We demonstrate that the integrable Davey–Stewartson I (DSI) system of equation can describe the twodimensional soundultrasound evolution. The DSI equations are remarkable by their solutions referred to as dromions. In bubbly fluids the dromion represents the localized focused ultrasoundwave which can alter the direction of its motion under changes in the boundary conditions for the sound wave. The condition of singular focusing of ultrasound in bubbly fluids is obtained. By numerical analysis of the interactionmodels, we reveal such processes as intensification of ultrasound by sound, nonlinear instability of a sound profile, and prove the validity of the singular focusing condition. Finally, possible applications of the results are outlined.

Morphological stability analysis of directional solidification into an oscillatory fluid layer
View Description Hide DescriptionWe study the stability of a planar solidmelt boundary during directional solidification of a binary alloy when the solid is being periodically vibrated in the direction parallel to the boundary (or equivalently, under a far field uniform and oscillatory flow parallel to the planar boundary). The analysis is motivated by directional solidification experiments under the low level residual acceleration field characteristic of a microgravity environment, and possible effects on crystal growth in space. It is known that periodic modulation of the solidmelt interface under the conditions stated induces second order stationary streaming flows within a boundary layer adjacent to the interface, the thickness of which is the same as the wavelength of the modulation. We derive an effective solute transport equation by averaging over the fast time scale of the oscillatory flow, and obtain the resulting dispersion relation for a small disturbance of a planar interface. We find both regions of stationary and oscillatory instability. For small ratios of the viscous to solutal layer thicknesses, the flow generally destabilizes the planar interface. For the flow stabilizes the stationary branch, but it can also excite an oscillatory instability. For large the effect of the flow is small.

Transient ship waves on an incompressible fluid of infinite depth
View Description Hide DescriptionIn this paper, we focus our attention on transient ship waves induced by a submerged moving body in an incompressible fluid assumed to be inviscid or viscous. The Oseen equations are extended to the transient case for the viscous fluid and solved by employing a combinative model. Transient ship waves from calmness to the generation of steadystate ship waves are described. When the viscosity vanishes, the above results are degenerated into those for the inviscid fluid.

Short wave phase shifts by large free surface solitary waves: Experiments and models
View Description Hide DescriptionIn this paper, we compare experiments on short gravity wave phase shifting by surface solitary waves to a Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) refraction theory. Both weak interactions (headon interaction) and strong interactions (overtaking interaction) are examined. We derive a dispersion relation and wave action conservation relation which are similar to the ones obtained for short waves refraction on slowly varying media. The model requires an exact solitary wavesolution. To this end, a steady wavesolution is numerically computed using the algorithm devised by ByattSmith [Proc. R. Soc. London, Ser. A 315, 405 (1970)]. However, two other solitary wavesolutions are incorporated in the model, namely the classical Korteweg and De Vries (KdV) [Phil. Mag. 39, 422 (1895)] solution (weakly nonlinear/small amplitude solitary wave) and the Rayleigh [Phil. Mag. 1, 257 (1876)] solution (strongly nonlinear/large amplitude solitary wave). Measurements of the short wave phase shift show better agreement with the theoretical predictions based on the ByattSmith numerical solution and the Rayleigh solution rather than the Korteweg and De Vries one for large amplitude solitary waves.Theoretical phase shifts predictions based on Rayleigh and ByattSmith numerical solutions agree with the experiments for A new heuristic formula for the phase shift allowing for large amplitude solitary waves is proposed as a limiting case when the short wave wave number increases.

Influence of inertia on the transient axisymmetric freesurface flow inside thin cavities of arbitrary shape
View Description Hide DescriptionThe influence of inertia is examined for transient axisymmetric free surface flow inside a thin cavity of arbitrary shape. The flow field is obtained by solving the lubrication equations, which are averaged over the cavity gap by expanding the velocity in terms of Chandrasekhar functions and using the Galerkin projection method. The formulation accounts for the transverse flow, as well as nonlinearities stemming from inertia and front location. Both flows under an imposed flow rate, and an imposed pressure at the cavity entrance are examined. The influence of inertia, aspect ratio, gravity, and cavity geometry on the evolution of the front, flow rate, and pressure is assessed particularly in the early stage of flow. Comparison with existing results shows full qualitative agreement for cavities of various geometries and flow conditions. Inertia is found to have a significant influence on early transient behavior, leading to the development of a flow of the “boundarylayer” type upon inception. The effect of inertia is further explored by developing a multiplescale analysis to obtain an approximate solution at small Reynolds number, Re. Comparison with the exact (numerical) solution indicates a wide range of validity for the multiplescale approach, even in the moderately small Re range.

Slow flow across macroscopically rectangular fiber lattices and an open region: Visualization by magnetic resonance imaging
View Description Hide DescriptionCreeping flow of a Newtonian fluid across aligned and staggered rows of cylinders (fiber lattices) bounded by an open region is studied experimentally by magnetic resonance imaging(MRI)velocimetry. The model systems are formed by circular cylindrical rods, macroscopically arranged in rectangular fashion and confined inside a Hele–Shaw cell. The thus formed fiber arrays are bounded by the open region from one side and the wall of the cell on the other side, thus forming a heterogeneous fibrous medium of dual porosity. The influence of the fiber lattice volume fraction and lattice unitcell geometry on the local aspects of the flow in the interior of and exterior to the fiber arrays are investigated. The steadystate velocity maps of the longitudinal and, in particular, transverse velocity components are shown to be advantageous in studying the local aspects of the flow field in such a heterogeneous porous medium. The most important feature of local velocity distributions in the regions ahead of and behind the latticechannel arrangements is evidenced as substantial transverse velocities. This local flow aspect is termed edge effect and found to be dependent on lattice porosity. Local flow disturbances are present on either side of the open channel–fiber lattice interfaces, at the lengthscale corresponding to the size of unit cells of the fiber lattices. Regions with regular patterns of very low fluid velocities are identified throughout the fiber lattices. The local values for the velocity vector at the entrance/exit of the fiber lattices are considerably higher than the average values within the fiber arrangements. These local flow enhancements, which are caused by the proximity of velocity gradients in the adjoining free flow region, are termed entrance/exit effects.

Temperaturedependent viscous gravity currents with shear heating
View Description Hide DescriptionWe have studied the effects of viscous dissipation on gravity current in the Stokes flow regime for both constant volume and constant flux boundary conditions. We have also examined the influence of temperaturedependent viscosity, as well as the relative importance of thermal and chemical buoyant forces. For the constant volume case a threestage evolution was found. This aspect concerning the existence of the multiple stages is new and was not found previously. This threestage behavior comes as a result of the interaction between the two quasiisothermal regimes. The first regime corresponds to an early stage with a uniformly high temperature, whereas the third stage represents the final period, when most part of the current has cooled down to uniformly low temperature. This evolutionary process with threestages is characteristic of a temperaturedependent viscous fluid and does not depend too much on viscous dissipation, which induces a longer transient period. In contrast to constant volume case, there is only one stage of development for the constant flux current. Although temperaturedependent viscosity influences the current dynamics, the rate of expansion follows a asymptote which is close to the prediction for a constant viscosity model with a constant flux condition. Viscous dissipation exerts definitely a stronger influence in the constant flux gravity currents as compared to the constant volume case, because of the faster velocities produced by the constant flux condition.

Effect of unequal cylinder spacing on vortex streets behind three sidebyside cylinders
View Description Hide DescriptionThe turbulent nearwake of three sidebyside circular cylinders with equal or unequal spacing has been experimentally investigated using various techniques, including the hot wire, laser Doppler anemometer, and flow visualization. The work aims to understand the effect of unequal cylinder spacing on the vortex streets behind the three cylinders. When the cylinder centertocenter spacing is identical, i.e., the flow is symmetric about the centerline, with one wide wake behind the central cylinder and one narrow wake on each side of the wide wake. The dominant frequency in the narrow wakes is about 5.4 times that in the wide wake. The observation is consistent with previous reports, thus lending credence to the present measurement. As is slightly increased to 1.6, a remarkable change occurs in the flow. A comparison is made between the cases of equally and unequally spaced cylinders in terms of the pressure around the cylinders, drag, lift, dominant frequencies, and vortex formation mechanisms. The flow topology(vortex patterns) and downstream evolution are also discussed in detail.

Effects of particle sedimentation and rotation on axisymmetric gravity currents
View Description Hide DescriptionThe axisymmetric propagation of a relatively dense gravity current of a given initial volume over a horizontal boundary is considered when the intruding fluid is a suspension of heavy particles and the ambient fluid is steadily rotating about a vertical axis. The investigation employs a shallowwater model of the motion. With the introduction of a strained temporal coordinate, it is possible to derive asymptotic expressions for the evolution of the radius and height of the current, the radial and temporal variation of the horizontal velocity, the volume fraction of particles, and the angular velocity. In this way it is possible to distinguish how the Coriolis force and the effects of particle sedimentation inhibit the radial spreading of the flow. The analytical relationships arise directly from the shallowwater equations and thus improve upon previous simple expressions which are based on an a priori prescription of the shape for the current. The analytical results compare favorably with both numerical integration of the full system of equations and experimental data.

The strain rate in evolutions of (elliptical) vortices in inviscid twodimensional flows
View Description Hide DescriptionIn this paper the strain rate in evolutions of elliptical vortices in inviscid twodimensional flows is considered. Previous work [Vosbeek et al., Phys. Fluids 9, 3315 (1997)] has revealed that there exists a relationship between the (in)stability of an elliptical vortex of uniform vorticity (Kirchhoff vortex) and the spatial distribution of the strain rate. It is examined here how the strain distribution evolves in time for unstable Kirchhoff vortices. Furthermore, it is shown that there also exists a relationship between strain and (in)stability of a vortex consisting of nested elliptical patches representing a more general vorticity distribution.

Interaction of Stokes boundary layer flow with a sound wave
View Description Hide DescriptionWe analyze the interaction of a standing sound wave with the flow generated by the oscillation of a plate in its own plane (Stokes second problem). The sound wave acts in the direction transverse to the plate and it is considered that the plate oscillation and the sound wave have the same frequency but a nonzero relative phase. The sound wave induces a modification of the axial velocity that consists of two parts, an oscillation with twice the frequency of the plate oscillation and a steady streaming that persists beyond the Stokes boundary layer, resulting in a double boundary layer structure. This mechanism for generating steady streaming differs from those studied previously in the literature. The relative phase of the two oscillatory motions determines the direction of the net flow. The direction of the steady streaming far away from the plate, coincides with the direction of the displacement of the plate at the moment of maximum compression and is proportional to the velocity of the plate at this moment.

Unsteady heat transfer from a sphere in a uniform crossflow
View Description Hide DescriptionThree different types of unsteady heat transfer problems are considered: the first considers the free thermal evolution of a spherical particle subjected to a uniform isothermal ambient flow, where the particle temperature evolves toward the ambient flow temperature; the second considers the unsteady heat transfer problem in response to a sudden jump in the particle temperature; and the third considers oscillatoryheat transfer due to oscillatory particle temperature in an isothermal ambient flow. In each case a range of Reynolds or Peclet number is considered. The first set of simulations show that unsteady heat transfer from a spherical particle under free thermal evolution can be described by an effective Nusselt number, at least over the range of parameters considered in this study. The effective Nusselt number under unsteady free thermal evolution deviates from the steadystate counterpart and the difference depends on the heat capacity ratio between the surrounding fluid to the particle. The existence of an effective Nusselt number indicates a faster decay of the thermal history kernel, which is confirmed with response to a step change in particle temperature in the second set of simulations. The final set of simulations considers the behavior of the thermal history kernel in the frequency space. We observe the low frequency response, or correspondingly the long time behavior, of the thermal history kernel at finite Peclet number to deviate significantly from the classical one over square root decay obtained for zero Peclet number limit.

Anisotropy of a thermal field at dissipative scales in the case of smallscale injection
View Description Hide DescriptionThe anisotropy of a thermal field at the level of dissipation has been studied experimentally and by means of modeling, downstream of a heated line source placed, successively, in a turbulent boundary layer and a turbulent plane jet. This situation represents anisotropic smallscale injection of a passive scalar in a turbulent medium. All three instantaneous temperature gradients have been measured. In the central region of the thermal sheet, experimental data reveal a high degree of anisotropy of temperature dissipation near the line source and return to isotropy further downstream. Comparison of measurements with modeling allows interpreting the data and estimating the returntoisotropy time scale.
