Index of content:
Volume 8, Issue 7, July 1996

Anomalous behavior during leveling of thin coating layers with surfactant
View Description Hide DescriptionOur recently‐published linear analysis [Schwartz et al., Langmuir 11, 3690 (1995)] demonstrated that an initially rippled thin layer of Newtonian liquid with uniformly distributed surfactant may level in unexpected ways. While the presence of surfactant will, in general, slow the rate of leveling compared to that of a perfectly clean system, there was shown to exist a realistic parameter range where increasing, rather than reducing, the amount of surfactant present will hasten leveling. Here, for the two‐dimensional problem, we investigate the importance of nonlinearity though numerical solution of (i) the unsteady lubrication form of the evolution equations with surfactant, and (ii) finite‐element solution of the exact governing equations for slow viscousflow. Confirmation of the linear results is demonstrated and quantitative discrepancy only appears for large‐amplitude and short‐wavelength ripples. Surface tension gradient driven flow explains the anomalies; for moderate surfactants, the surface quickly ‘hardens,’ leading to a decay rate of one‐quarter of the clean‐surface rate, while for weak surfactants, leveling proceeds to a plateau level which decays much slower than the hard‐surface result.

Explosive breakup of a liquid jet by a swirling coaxial gas jet
View Description Hide DescriptionThe breakup of a round water jet by a swirling coaxial annular air jet, issuing from convergent jet nozzles has been studied experimentally. The intensity of the swirl and the water to air mass flux ratio have been varied over a large range. It was found that the liquid jet is little affected by the swirl when the swirl number, here defined by the ratio of tangential to axial air jet nozzle velocities, is below a critical value (S _{cr}). Just above this value the liquid jet undergoes an explosive radial expansion and disintegration. A simple model shows that the physically relevant parameters are the gas to liquid momentum flux ratio M and the ratio of the nozzle’s diameters. For small momentum flux ratios, S _{cr} was found to depend on M ^{−1/2} until an asymptotic constant value is reached at large M. Surface tension has no effect on the breakup when air velocities are large, however, membrane‐type breakup is dominant at the lower air velocities when the aerodynamic Weber number is of order 100 or less.

Observations of a cavitation bubble interacting with a solid boundary as seen from below
View Description Hide DescriptionWe present a schlieren photographic sequence of a laser generated cavity in water interacting with a solid boundary as seen from the boundary’s perspective looking through the bubble. Previous studies have photographed the collapsing bubble from the side and it is unclear from these photographs once a bubble becomes toroidal how the central hole develops. Our schlieren sequence displays the jet first penetrating the lower bubble surface and then follows the evolution of the central channel as the bubble contracts and reexpands. It is felt that the bubble shapes recorded add further knowledge to the possible damage mechanisms of cavitation.

On the existence of a Generalized Langevin model representation for second‐moment closures
View Description Hide DescriptionThe Generalized Langevin model representations of two second‐moment closure models for the rapid pressure‐strain term, proposed by Fu and Launder and by Jones and Musonge, are obtained. This representation makes it possible to use these models in PDF calculations of turbulent flows. The implications of three realizability constraints for the relationship between Langevin models and these second‐moment closures are discussed. A Generalized Langevin model representation exists only if the rapid pressure‐strain model satisfies realizability at the 2D turbulence limit.

Thermocapillary mobility of a suspension of droplets in a fluid
View Description Hide DescriptionThe effective thermocapillary mobility of a suspension of viscous, heat conductingdroplets in a fluid of different shear viscosity and thermal conductivity is studied in the framework of the theory of linear transport coefficients of heterogeneous media. The fluid velocity and the temperature field are combined conveniently into a single four‐vector field. This allows a multiple scatteringanalysis and calculation of the thermocapillary mobility in terms of a renormalized cluster expansion. A mean field result is derived that reduces to the Clausius‐Mossotti formula in the case of bubbles. Correlation corrections are calculated in pair approximation.

Energy‐conserving truncations for convection with shear flow
View Description Hide DescriptionA method is presented for making finite Fourier mode truncations of the Rayleigh–Bénard convection system that preserve invariants of the full partial differential equations in the dissipationless limit. These truncations are shown to have no unbounded solutions and provide a description of the thermal flux that has the correct limiting behavior in a steady‐state. A particular low‐order truncation (containing 7 modes) is selected and compared with the 6‐mode truncation of Howard and Krishnamurti [J. Fluid Mech. 170, 385 (1986)], which does not conserve the total energy in the dissipationless limit. A numerical example is presented to compare the two truncations and study the effect of shear flow on thermal transport.

Hindered diffusion of spherical macromolecules through dilute fibrous media
View Description Hide DescriptionResults are presented for the effect of solute–fiber hydrodynamicinteractions on the hindered diffusion of a spherical macromolecule in random media comprised of cylindrical fibers. Hydrodynamicinteractions are calculated by representing the sphere as a collection of point singularities and accounting for the fibers by using a numerical version of slender‐body theory.Electrostatic and other nonhydrodynamic interactions are neglected. The calculations show that the hydrodynamic mobility of the solute decreases in an exponential‐like fashion as the fiber volume fraction is increased. Also, at a given volume fraction, a medium of thinner fibers hinders solute transport more than a medium of thicker fibers. The results compare well with experimental data, both for proteindiffusion in solutions of the polysaccharide Dextran and for proteindiffusion in cross‐linked agarose gels.

Dynamics of drops in cavity flows: Aggregation of high viscosity ratio drops
View Description Hide DescriptionThe interactions of deformable drops in cavity flows is studied numerically in the limit of low Reynolds numbers for a two‐dimensional model.Flow in a square cavity is driven by the steady motion of one of the walls. Deformable drops will migrate across streamlines until they reach an equilibrium trajectory or equilibrium position; the rate and direction of migration depend on both the viscosity ratio and capillary number. High viscosity ratio deformable drops have a tendency to aggregate and form clusters. The presence of a deformable dispersed phase results in an elastic behavior of the suspension.

Marangoni effects on drop deformation in an extensional flow: The role of surfactant physical chemistry. I. Insoluble surfactants
View Description Hide DescriptionThe shape of a drop centered in an axisymmetric extensional flow is determined by the viscous stresses that deform the drop and surface tension γ′ that resists the deformation. The ratio of these stresses is given by the capillary number, Ca. When Ca is small enough, the drop attains a steady shape. However, above a threshold value, Ca^{cr}, the drop elongates continuously, and no steady shape is attained. When surfactants are present on the drop interface, the surface tension is determined by the surface concentration profile, which varies throughout the deformation process. Initially, the surface tension is given by γ_{eq} ^{′}, in equilibrium with the uniform surface concentration Γ_{eq} ^{′}. When the flow is initiated, surfactant is swept toward the drop tips, reducing the surface tension there, and altering the interfacial stress balance tangentially through Marangoni stresses and normally through the Laplace pressure. In this paper, the effects of an insoluble surfactant on drop deformation are studied. In previous work, either a surfaceequation of state for the surface tension γ′ that is linear in the surface concentration Γ′ was used, an approximation that is valid only for dilute Γ′, or Γ′ sufficiently dilute for the linear approximation to be valid were studied. In this paper, a nonlinear surfaceequation of state that accounts for surface saturation and nonideal interactions among the surfactant molecules is adopted. The linear framework results are recovered for Γ′ that are sufficiently dilute. As Γ′ is increased, the effects of saturation and surfactantinteractions are probed at constant initial Γ_{eq} ^{′} and at constant initial γ_{eq} ^{′}. Finally, the case of strong intersurfactant cohesion is treated with a first‐order surface phase transformation model. At moderate surface concentrations, these nonlinear phenomena strongly alter the steady drop deformations and Ca^{cr} relative to the uniform surface tension and linear equation of state results.

The single‐particle distribution function for rapid granular shear flows of smooth inelastic disks
View Description Hide DescriptionThe velocity distribution function,f _{1}, for a (linear) shear flow of a system of rigid inelastically colliding disks in a plane is measured by applying a novel algorithm to results of (MD) simulations involving 200 000 particles. The need to consider such a relatively large system is explained. It is found that f _{1} is well fitted by an exponent of a second‐order polynomial in the norm of the fluctuating velocities with angle‐dependent coefficients (which also depend on the density and the granular temperature). Other characterizations of the system studied in this paper are presented as background material. A hitherto unnoticed property of systems with Lees–Edwards boundary conditions has been discovered and its origin is briefly explained.

Qualitative analysis of the Navier–Stokes equations for evaporation–condensation problems
View Description Hide DescriptionThe methods of qualitative theory of dynamical systems are used to provide new information of the Navier–Stokes solutions for gas flows driven by evaporation and condensation at interphase surfaces. It is shown that these solutions correspond to separatrixes of the saddle point in the (u,T) plane. The classification of solutions is given, and some special cases are studied in detail. The qualitative methods are applied to the problem of evaporation/condensation between two plates. It is shown that the topology of the saddle point implies the following situation: one of the functions u(x) (velocity) or T(x) (temperature) is always non‐monotone for sufficiently small Knudsen number. This explains some previously published numerical results. The case of finite Reynolds number is considered separately, and relationship with the kinetic‐theory results is discussed throughout.

Chaotic advection in creeping flow of viscoelastic fluids between slowly modulated eccentric cylinders
View Description Hide DescriptionRecent experiments show that very low levels of elasticity can either enhance or diminish the area over which chaotic advection occurs in creeping flows [T. C. Niederkorn and J. M. Ottino, J. Fluid Mech. 256, 243 (1993)]. No mechanistic explanation of this phenomenon is currently available. This has motivated us to consider the problem of two‐dimensional flow between counter‐rotating eccentric cylinders where the angular velocities are subject to slow, continuous modulation. Regular perturbation theory for low levels of elasticity is used to semi‐analytically determine the viscoelastic correction to the Newtonian flow field based on the Oldroyd‐B constitutive model. The geometric theory of Kaper and Wiggins [J. Fluid Mech. 253, 211 (1993)] is then applied to make predictions about how elasticity affects chaotic advection in quasi‐steady flows. It is found that elasticity can act to either increase or decrease the area over which chaotic advection occurs, depending on the boundary motion. This is accomplished through three distinct mechanisms: (1) area changes of the maximum area over which chaotic advection can occur, the potential mixing zone (PMZ); (2) area changes of the region in the PMZ where fluid particles execute non‐chaotic trajectories below a critical modulation frequency; (3) area changes of the region between the extrema of the Newtonian stagnation streamlines which does not belong to the PMZ. The mechanism responsible for these area changes is a modified pressure gradient in the angular direction, which in turn appears to be due to first normal stress differences caused by shearing. Numerical calculations of fluid particle trajectories confirm the predictions of the geometric theory. For the boundary motions considered here, the calculations yield two additional results about the effect of low levels of elasticity on chaotic advection. First, the critical modulation frequency is decreased. Second, the rate of chaotic mixing, as measured by the largest Liapunov exponent, is increased for modulation fre‐ quencies greater than the critical Newtonian value.

Boundary conditions for the lattice Boltzmann method
View Description Hide DescriptionWhen the Lattice Boltzmann Method(LBM) is used for simulating continuum fluid flow, the discrete mass distribution must satisfy imposed constraints for density and momentum along the boundaries of the lattice. These constraints uniquely determine the three‐dimensional (3‐D) mass distribution for boundary nodes only when the number of external (inward‐pointing) lattice links does not exceed four. We propose supplementary rules for computing the boundary distribution where the number of external links does exceed four, which is the case for all except simple rectangular lattices. Results obtained with 3‐D body‐centered‐cubic lattices are presented for Poiseuille flow, porous‐plate Couette flow, pipe flow, and rectangular duct flow. The accuracy of the two‐dimensional (2‐D) Poiseuille and Couette flows persists even when the mean free path between collisions is large, but that of the 3‐D duct flow deteriorates markedly when the mean free path exceeds the lattice spacing. Accuracy in general decreases with Knudsen number and Mach number, and the product of these two quantities is a useful index for the applicability of LBM to 3‐D low‐Reynolds‐number flow.

Linear stability analysis of plane Couette flow with viscous heating
View Description Hide DescriptionThe linear stability of a plane Couette flow of Newtonian fluids with exponential dependence of viscosity on temperature according to the Arrhenius law and Nahme law is investigated. The modified Chebyshev polynomials of the second kind are used in Galerkin’s method to solve the eigenvalue problem. The combinations of Chebyshev polynomials to form the modified ones for satisfying the required boundary conditions, and their orthogonality conditions, are derived. It is found that an analysis that does not include enough terms of the trial functions may lead to erroneous results. The results show that instability occurs for fluids with highly temperature‐sensitive viscosity in a range of the Brinkman numbers; when the temperature‐sensitive viscosity is small enough, instability is not observed. Furthermore, only two kinds of modes, the viscous and inviscid ones, are found for the instability of the flow system. Also, the results indicate that the fluids obeying the Arrhenius‐type model are more stable than those of the Nahme‐type model if both are based on the same temperature‐sensitive viscosity, reference viscosity, and temperature.

Analytic expression for Taylor–Couette stability boundary
View Description Hide DescriptionWe analyze the mechanism that determines the boundary of stability in Taylor–Couette flow. By simple physical argument we derive an analytic expression to approximate the stability line for all radius ratios and all speed ratios, for co‐ and counterrotating cylinders. The expression includes viscosity and so generalizes Rayleigh’s criterion. We achieve agreement with linear stability theory and with experiments in the whole parameter space. Explicit formulae are given for limiting cases.

Marangoni instability of bi‐component droplet gasification
View Description Hide DescriptionSimplified analyses using a linear stability approach are developed to predict influences of thermal and solutal Marangoni effects on hydrodynamic stability of bi‐component droplets evaporating in a spherically‐symmetrical manner in hot environments. It is predicted that with zeotropic mixtures and for ∂σ/∂T<0 and ∂σ/∂y<0 (where σ is surface tension,T temperature, and y the surface mass fraction of the more volatile droplet component), the thermal and solutal Marangoni effects oppose each other in that the thermal effect is stabilizing and the concentration effect is destabilizing. The model is applied to alkane/alkane and alcohol/water mixture droplets. The alkane mixture droplets were predicted to be hydrodynamically stable. For alcohol/water mixtures, the results suggest that critical radii for marginal stability exist; when a droplet is initially pure methanol which subsequently absorbs water from the ambient, the critical radius is predicted to depend upon the relative humidity of the environment.

Electron vortex orbits and merger
View Description Hide DescriptionPure electron plasma columns are contained inside hollow conducting cylinders in an axial magnetic field. In the 2D E × B drift approximation, an electron column is a vortex evolving in (r,θ) according to the Euler equation. First the center‐of‐mass orbits of two vortices sufficiently well‐separated to be stable to merger are characterized. Equilibria are observed in which the vortices orbit about the center of the cylinder, with either oscillations about stable equilibria or exponential divergence away from unstable equilibria. The equilibrium positions, oscillation frequencies, and instability rates for these spatially extended vortices agree well with the predictions of point vortex theory, apparently because surface waves and shape distortions do not couple significantly to the center‐of‐mass motion. Next, the merger of two vortices with unequal radii is quantified. Merger is accompanied by the formation of filamentary arms, and results ultimately in an axisymmetric central core surrounded by a lower density halo. The self‐energy of the merged core is found to be roughly the sum of the self‐energy of the merging vortices. The fraction of the total circulation entrained into the core varies from 70% to 90% as the ratio of the initial vortex radii is varied from 1:1 to 2:1. The point‐like vortex dynamics and the circulation loss with merger are both consistent with the ‘‘punctuated Hamiltonian’’ models of decaying turbulence.

Axisymmetrization of an isolated vortex region by splitting and partial merging of satellite depletion perturbations
View Description Hide DescriptionIn numerical studies, we observe the essentially inviscid approach to axisymmetry of an isolated‐and‐perturbed monopolar and uniformly decreasing distributed vortex region in a two‐dimensional incompressible fluid. In particular, an initial small‐but‐finite amplitude, 3‐fold, ‘‘edge’’‐located hole or depletion perturbation of a monopole evolves into a 2‐fold state. This occurs through the nonlinear processes of hole stretching, splitting and partial merger. The initial growth rate for this downward cascade is proportional to the initial perturbation magnitude. Near‐inviscid simulations are made with surgery‐regularized contour dynamics codes (CDS), using from 5 to 11 contours. Pseudospectral (PS) simulations with varying Newtonian and hyperviscosities, that is considering continuum and dissipation effects, yield consistent results. Our numerical results are in agreement with small‐but‐finite amplitude perturbations that were used in recent laboratory experiments on magnetized electron columns. This process may also occur in late time evolutions associated with the ‘‘bump‐on‐tail’’ initial condition in Vlasov plasmas.

Scaling properties of vortex ring formation at a circular tube opening
View Description Hide DescriptionA vortex sheet model is applied to study vortex ring formation at the edge of a circular tube, for accelerating piston velocities U _{ p }∼t ^{ m }. We determine properties of the vortex ring as a function of the piston motion and investigate the extent to which similarity theory for planar vortex sheet separation applies. For piston strokes up to half the tube diameter, we find that the ring diameter, core size and circulation are well predicted by the planar similarity theory. The axial ring translation is a superposition of an upstream component predicted by the theory and a downstream component which is linear in the piston stroke. The front of the fluid volume exiting the tube is also linear in the piston stroke and travels with 75% of the piston velocity. The core size decreases and the distribution of fluid near the core becomes more asymmetric as the parameter m increases.

On predicting the turbulence‐induced secondary flows using nonlinear k‐ε models
View Description Hide DescriptionLow turbulent Reynolds number direct simulation data are used to calculate the invariants of the Reynolds stress and the turbulent dissipation rate in a square duct. The results show that, depending on the region where the analysis is carried out, the turbulent flow field comes close to one‐, two‐, and three‐component states. Modeling such flows—even at higher Reynolds numbers—will require models that can approach all three states. A number of related nonlinear k‐ε models are tested a priori using the direct simulation data. The numerical simulation using Reynolds averaged Navier–Stokes equations with these models was performed. Their ability to predict the secondary flows, with a low‐Reynolds k‐ε model, cannot be gauged from realizability.