Volume 7, Issue 6, June 1995
Index of content:

Hysteresis in forced oscillations of pendant drops
View Description Hide DescriptionA hysteresis phenomenon has been revealed through experiments conducted with large‐amplitude forced oscillations of pendant drops in air. Under strong excitation, the frequency response of a drop forced at constant amplitude exhibits jump behavior; a larger peak response amplitude ε_{↓} appears at a lower frequency ω_{↓} during a downward (↓) variation of forcing frequency than during an upward (↑) variation, viz. ε_{↓}≳ε_{↑} and ω_{↓}<ω_{↑}. Similar results are obtained when forcing amplitude is varied at constant frequency. This behavior is characteristic of a system with a soft nonlinearity. These findings indicate that oscillating pendant drops constitute a convenient system for studying nonlinear dynamics.

An experimental study of dynamics of drop formation
View Description Hide DescriptionA liquid being ejected from a nozzle emanates from it as discrete, uniformly sized drops when the flow rate is sufficiently low. In this paper, an experimental study is presented of the dynamics of a viscous liquid drop that is being formed directly at the tip of a vertical tube into ambient air. The evolution in time of the drop shape and volume is monitored with a time resolution of 1/12 to 1 ms. Following the detachment of the previous drop, the profile of the new growing drop at first changes from spherical to pear‐shaped. As time advances, the throat of the pear‐shaped drop takes on the appearance of a liquid thread that connects the bottom portion of the drop that is about to detach to the rest of the liquid that is pendant from the tube. The focus here is on probing the effects of physical and geometric parameters on the universal features of drop formation, paying special attention to the development, extension, and breakup of the liquid thread and the satellite drops that are formed subsequent to its breakup. The role of surfactants in modifying the dynamics of drop formation is also studied. The effects of finite inertial, capillary, viscous, and gravitational forces are all accounted for to classify drastically different formationdynamics and to elucidate the fate of satellite drops following thread rupture.

Equilibrium shapes of nonaxisymmetric liquid bridges of arbitrary volume in gravitational fields and their potential energy
View Description Hide DescriptionBifurcation diagrams of nonaxisymmetric liquid bridges subject to a lateral gravitational force and to both lateral and axial gravitational forces are found by solving the Young–Laplace equation for the interface by a finite difference method. The potential energy of the equilibrium shapes is also calculated. The results obtained show that the slenderness of the bridge determines whether the breaking of the liquid bridge subject to a lateral gravitational force leads to equal or unequal drops. The stability limits calculated are compared with the ones obtained using asymptotic techniques around the cylinder, the agreement being extremely good for a wide range of the parameters.

The wetting of a plane surface by a fluid
View Description Hide DescriptionThe spreading of a thin, planar drop of fluid that completely wets a solid surface is described, from an initial state in which the drop has a compact shape through its final approach to an infinitesimally thin film extending to infinity. Because the slope of the dropsurface is everywhere small, the lubrication approximation can be used, and the effects of capillarity, viscosity, and intermolecular forces are all included. It is shown that a simple model for the intermolecular forces allows a mathematically acceptable solution to be found, without the need to invoke a large‐distance cutoff and without violating the small‐slope requirement of lubrication theory.

Film entrained by a fiber quickly drawn out of a liquid bath
View Description Hide DescriptionThe thickness of the film entrained by a fiber quickly withdrawn out of a bath of wetting liquid is of interest. For velocities larger than a threshold (usually of the order of 1 m/s), inertia can no longer be neglected and must be incorporated in a generalized form of the classical Landau–Levich–Deryaguin model. It is shown here that the effect of inertia is to make the film thicker, which agrees with recent observations.

Unsteady laminar flow between a pair of disks corotating in a fixed cylindrical enclosure
View Description Hide DescriptionThe unsteady streamlined motion of a constant property fluid in the unobstructed space between a pair of disks corotating at angular velocity Ω in a fixed cylindrical enclosure is investigated numerically. Two‐dimensional (axisymmetric) and three‐dimensional calculations are performed using a second‐order accurate time‐explicit algorithm. The flow configuration corresponds to that investigated experimentally by Schuler et al. [Phys. Fluids A 2, 1760 (1990)]. The steady flow solutions are characterized by a symmetrical pair of counter‐rotating toroidal vortices in the cross‐stream (r‐z) plane. This secondary motion is driven by the radial imbalance between the outward‐directed centrifugal force and the inward‐directed pressure gradient force. Axisymmetric calculations predict a flow that is steady for Re<22 200, where Re is the Reynolds number based on the disk radius, the tip speed of the disks, and the kinematicviscosity of the fluid. Above this value the motion is unsteady periodic and, while the features of the cross‐stream flow pattern are broadly preserved, the symmetry of the motion about the midplane is broken by alternating periodic crossings of the toroidal vortices.
This instability is maintained through an interaction that arises between outward‐directed fluid in the disk Ekman layers and inward‐directed fluid in the return core flow. Three‐dimensional calculations at Re=22 200 and 44 400 show that the toroidal vortices acquire a time‐varying sinuous shape in the circumferential direction. These calculations reveal circumferentially periodic reversals of the axial velocity component in the cross‐stream plane, including the detached shear layer separating the region of motion in solid‐body rotation near the hub from the potential core, in agreement with the flow visualization observations of Humphrey and Gor [Phys. Fluids A 5, 2438 (1993)]. The wavelength of this oscillation is shown to be twice that of the circumferential velocity component which is responsible for the nodal distribution of axial vorticity. When plotted on the interdisk midplane, the axial component of vorticity manifests itself as an even integer number, 2n (n=1,2,...), of circumferentially periodic foci. Experiments show that the number of foci decreases in a stepwise manner with increasing Reynolds number. For the conditions of this study, the calculated dimensionless angular velocity of the foci, Ω_{ F }/Ω, ranges from 0.55 at Re=22 200 to 0.44 at Re=44 400. These values are close to the present experimental estimate Ω_{ F }/Ω=0.5.

The accumulation and dispersion of heavy particles in forced two‐dimensional mixing layers. Part 2: The effect of gravity
View Description Hide DescriptionThe dispersion and settling of small, heavy, spherical particles in a temporally evolving two‐dimensional mixing layer under gravity is investigated. The dilute limit is assumed, in which both the effect of the particles on the fluid flow and the interaction among the particles is negligible. The particle dynamics is quantified as a function of the dimensionless Stokes and Froude numbers, St and Fr, which express the ratios of the three time scales related to (i) the fluid flow, (ii) the particles’ inertia, and (iii) their settling velocity, respectively. For horizontal flow in which the upper stream is the seeded one, the mixing layer accelerates the settling of particles with small St, whereas particles with large St are slowed down in their settling motion. At intermediate St and for moderate settling velocities, root‐mean‐square (RMS) data for the particle concentration field demonstrate the generation of strong inhomogeneities by the mixing layer. These regions of high particle concentration have the form of bands in the initially unseeded stream. Scaling laws for their angles and the distance between them are given. Furthermore, analytical results for linearized flow fields are derived that demonstrate the optimal efficiency of the dispersion and settling process at intermediate St. The numerical simulations show the existence of different parameter regimes, in which the particle motion is dominated by the coherent vortices and by gravity, respectively. Scaling laws are derived for the particle dispersion and settling for both of these regimes, which show reasonable quantitative agreement with the simulation data. Flows that exhibit a vortex pairing process show a reduced tendency of the particles toward suspension. For vertically upward flow in which the faster stream is seeded, is observed a sharp maximum in the particle dispersion measures for intermediate St and settling velocities equal to one‐half the difference between the free‐stream velocities. Under these conditions, the cross‐stream fluid velocity components become optimally efficient in ejecting particles into the unseeded stream.

The structure of the axisymmetric high‐Reynolds number flow around an ellipsoidal bubble of fixed shape
View Description Hide DescriptionThe structure of the flow around an oblate ellipsoidal bubble of fixed shape is studied by means of direct numerical simulation for Reynolds numbers Re up to 10^{3}. In agreement with a previous study by Dandy and Leal [Phys. Fluids 29, 1360 (1986)] the computations demonstrate that if the bubble aspect ratio χ is high enough a standing eddy can exist at the rear of the bubble in an intermediate range of Re. This eddy disappears beyond a certain Reynolds number and it is shown that its existence is governed by the competition between accumulation and evacuation of the vorticity in the flow. The range of Re where the eddy exists increases very rapidly with χ meaning that this structure is certainly present in many experimental situations. The evolution of the drag coefficient with Re reveals that the oblateness has a dramatic influence on the minimum value of Re beyond which Moore’s theory [J. Fluid Mech. 23, 749 (1965)] can be used to predict the rise velocity of a bubble of fixed shape. In contrast, owing to the shape of the vorticity distribution at the surface of the bubble, no noticeable influence of the standing eddy on the drag is found. A quantitative comparison between the present results and those of previous authors shows that the computational description of the boundary layer around curved free surfaces is not a trivial matter since a strong influence of the numerical method is observed.

Comparative measurements in the canonical boundary layer at Re_{δ2 }≤6×10^{4} on the wall of the German–Dutch windtunnel
View Description Hide DescriptionMean velocity and Reynolds‐stress profiles were measured in the incompressible turbulent boundary layer with zero pressure gradient on the aerodynamically smooth sidewall of the German–Dutch windtunnel. Data were taken at Reynolds numbers Re_{δ2 }, based on momentum thickness δ_{2} of 2×10^{4}, 4×10^{4}, and 6×10^{4} by means of four different types of hot‐wire probes (three‐wire probes, X wire, and normal‐wire probes). There are also measurements of skin friction and of spectra. The data compare well with the few available other measurements, and all profiles show independence of Reynolds number in the outer region of the boundary layer when plotted against y/Δ, where Δ is the Rotta–Clauser length.

Homoclinic bifurcation in Blasius boundary‐layer flow
View Description Hide DescriptionIn an attempt to elucidate the laminar/turbulent transition mechanism in a Blasius boundary‐layer flow, a nonsemisimple resonance of phase‐locked secondary instability modes is investigated. The local nonlinear behavior is described by means of a center manifold reduction. The numerically computed normal form is of the symmetric Takens–Bogdanov type and predicts a homoclinic orbit which is possibly related to a physical bursting process. A global continuation procedure for equilibrated three‐dimensional (3‐D) waves in the full Navier–Stokes system validates some of the local predictions and very closely outlines the experimentally observed skin friction domain including subcritical transition.

The stability of a two‐dimensional rising bubble
View Description Hide DescriptionThe stability of an inviscid two‐dimensional bubble subject to two‐dimensional disturbances is considered and the bubbles are found to be linearly stable for all Weber numbers, for which a steady solution is known. Certain aspects of the nonlinear initial value problem are also studied. An initial condition that consists of a superposition of a suitable symmetric eigenmode (of the linear stability operator) on a steady state is found to result in pinching of the bubble neck as it tends to oscillate about the steady state. An estimate of the threshold amplitude of such a disturbance needed to cause breakup of a large aspect ratio bubble is obtained. The presence of gravity appears to inhibit this pinching process.

On the velocity field and tracer patterns in a twisted duct flow
View Description Hide DescriptionThe laminar flow in a twisted square duct is studied numerically to analyze the formation and reorganization of Dean roll cells under abrupt curvature change. The flow geometry consists of four 90° bends, the curvature plane of each making a 90° angle with that of its neighbor. Straight ducts are inserted between successive bends. The flow geometry studied here is a basic element of a twisted duct flow in which chaotic advection has been observed [Jones, Thomas, and Aref, J. Fluid Mech. 209, 335 (1989)]. Steady incompressible full Navier–Stokes equations have been solved by an approximate factorization technique. Vector plots of the secondary flow (Dean roll cells) show upstream diffusion of curvature effects. The upstream propagation is weak, however, and the Dean roll cells adopt the rotation protocol of the bend immediately at its entrance. Farther downstream, roll cells continue to develop and reach an axially invariant state before leaving the bend. A passive tracer was also introduced at the center of the entrance to the twisted duct. Tracer distribution was followed in the downstream direction and was visualized every 30°. Comparison of the secondary flow velocity vector plots and the corresponding tracer patterns shows that the tracer distribution lags behind the velocity field in adjusting to changes in curvature. Tracer patterns show strong stretching and folding, which lead to mixing enhancement.

The effect of variable viscosity on the interfacial stability of two‐layer Poiseuille flow
View Description Hide DescriptionIn this paper the linear stability analysis of the interface between two Newtonian liquids with temperature‐dependent viscosity in plane Poiseuille flow is presented. A piecewise linear temperature profile is considered. The linearized equations describing the evolution of small, two‐dimensional disturbances are derived and the stability problem is formulated as an eigenvalue problem for a set of ordinary differential equations. The continuous eigenvalue problem is solved numerically by a pseudospectral method based on Chebyshev polynomial expansions. The method leads to a generalized matrix eigenvalue problem, which is solved by the QZ algorithm. Results on the onset of instability are presented in the form of stability maps for a range of thickness ratios, disturbance wave numbers, imposed temperature differences, constant‐temperature viscosity ratios, thermal conductivity ratios, and Reynolds numbers. Increasing the imposed temperature difference, constant‐temperature viscosity ratio, or Reynolds number can have a stabilizing or destabilizing effect, depending on the flow configuration (thickness ratio) and disturbance wavelength. Increasing the thermal conductivity ratio has a destabilizing effect on the interface for all configurations and disturbance wavelengths.

Translational instability of a bubble undergoing shape oscillations
View Description Hide DescriptionThis paper studies the translational instability of an oscillating bubble. It is shown that when a spherical bubble undergoing volume oscillation becomes unstable, giving rise to shape oscillations of two neighboring modes, the translational mode is intimately coupled with the two shape modes and this results in translational instability of the bubble. The main contribution is twofold. First, the integral relations for motions of bubbles in an infinite perfect liquid are not relied on, hence result is applicable to liquids with weak viscous effect. Second, the method of deriving the amplitude equations, which is similar to that of normal form calculations for ordinary differential equations, has not been applied to partial differential equations before.

Viscous structure of plane waves in spatially developing shear flows
View Description Hide DescriptionThis paper is concerned with the propagation of linear plane waves in incompressible, two‐dimensional weakly nonparallel shear flows for large Reynolds numbers. Waves are analyzed for arbitrary complex frequency ω and local wave number k when nonparallel effects are assumed to be due to weak viscousdiffusion. The inviscid approximation is shown to correctly describe, at leading order, the cross‐stream variations of local plane waves at all stations where they are locally amplified in a frame of reference moving at the local phase speed Reω/Rek, i.e., at stations where the temporal growth rate σ≡Fmω−FmkReω/Rek remains positive. This result also holds as long as the local phase speed lies outside the range of values reached by the basic velocity profile. By contrast, the inviscid approximation fails to represent cross‐stream variations in the critical layers when waves are locally neutral (σ=0), and in large viscous regions when they become damped (σ<0). Uniformly valid WKBJ approximations are derived in these regions and the results are applied to the description of forced spatial waves and self‐excited global modes. © 1995 American Institute of Physics.

Some considerations on the instabilities of nonpolar liquids subjected to charge injection
View Description Hide DescriptionThe electrohydrodynamic stability of a plane layer of nonpolar liquid is analyzed under certain basic assumptions. The liquid is supposed to be perfectly insulating, the space charge resulting from a unipolar injection of charge. The injection is considered to be nonautonomous, that is, following a field dependent law. The regions of linear stability under stationary conditions have been computed both in terms of nondimensional parameters and directly measurable variables, such as the applied voltage and the gap spacing between the electrodes. The results have shown that the onset of instability is strongly dependent on an injection strength parameter and can be approximated by the linear criterion corresponding to autonomous injection, provided that the injected charge is given by the injection law. Experiments have also been performed to determine the instability threshold under transient conditions. An interpretation of these experiments has been done by generalizing an approximated theoretical study for autonomous injection to the case of a nonautonomous injection.

Solitary waves of permanent form in a deep fluid with weak shear
View Description Hide DescriptionThe Benjamin–Davis–Acrivos–Ono equation is generalized to account for finite, large amplitude solitary waves in a sheared deep fluid. It is shown how fine structure of stratification and weak noncritical shear in such geophysicalflows do affect length (shape), wave (phase) velocity, and even stability of finite amplitude solitary waves.

Drop‐formed vortex rings—The generation of vorticity
View Description Hide DescriptionVortex rings are seen to form when dyed water drops strike a water surface and their formation and structure depend on height of fall and surface tension. The assumption that a vortex sheet envelopes the penetrating drop, frequently stated without explanation in the literature, does not explain these factors and this paper shows why it is incorrect. Alternative mechanisms have been proposed in the literature but none explains adequately the vorticity generation or the restriction of vortex ringformation to low Weber numbers. This paper proposes a mechanism based on the generation of vorticity on relaxation of surface stresses at coalescence. The condition that the surfaceviscous stress be continuous across the water‐air interface leads to a boundary condition on vorticity and the total amount of vorticity generated depends on the quantity which can be diffused into the fluid interior from the boundary during coalescence. At low values of Weber number this condition appears to be sufficient to generate enough vorticity to allow flow separation at the surface, such separation being a necessary condition for vortex sheet roll‐up and ring production. The existence of a critical Weber number above which vortex rings do not form is the result of a balance between the rate at which the ring of contact moves outward associated with on the one hand, the action of surface tension forces and, on the other, the rate of surface destruction due to the coming together of surfaces. If surface destruction dominates then the fluid elements to which the surfaceviscous stress boundary condition applies will become part of the fluid interior before diffusion has carried significant vorticity away from the surface.

Experimental investigation of a salt water turbulent boundary layer modified by an applied streamwise magnetohydrodynamic body force
View Description Hide DescriptionSingle‐component velocity field measurements, mean and fluctuating wall shear stress measurements, and photographic flow visualizations have been made of a magnetohydrodynamic(MHD) body‐force modified turbulent boundary layer. The turbulent boundary layer flowed over a flat plate in salt water at zero pressure gradient; the MHD force was created by the interaction of a permanent magnetic field and an applied electric field from a magnet/electrode array integral to the surface of the plate. A MHD force, when applied to an electroconducting fluid and acting in a streamwise direction, can generate a near‐wall jet, decreasing the boundary layer thickness and suppressing the intensity of the turbulent fluctuations across the boundary layer. At very high interactions, the force causes an increase in mean wall shear and in turbulence; in the zero free‐stream velocity limit, the force acts as a pump. An increase in local skin friction, however, is offset by a grain in thrust due to the force. At moderate interactions, mean quantities are unaffected, but fluctuating wall shear stress and turbulence intensity are suppressed by up to 30% of their unperturbed values across the lower part of the boundary layer. At very low interactions, effects are seen only near the wall. An interaction parameter is derived that characterizes these regimes. The effects likely occur because the MHD force pumps high momentum fluid along the wall, disrupting the liftup of shear‐generated wall vorticity. This jet effect is associated with increased convection of turbulent kinetic energy by the mean flow. With the force directed axial upstream, turbulence amplification is seen, along with a reduced mean velocity.

Exponential decay rate of the power spectrum for solutions of the Navier–Stokes equations
View Description Hide DescriptionUsing a method developed by Foias and Temam [J. Funct. Anal. 87, 359 (1989)], exponential decay of the spatial Fourier power spectrum for solutions of the incompressible Navier–Stokes equations is established and explicit rigorous lower bounds on a small length scale defined by the exponential decay rate are obtained.