Volume 13, Issue 8, August 2001
 ARTICLES


Capillarygravity wave drag
View Description Hide DescriptionDrag due to the production of capillarygravity waves is calculated for an object moving along the surface of a liquid. Both two and three dimensional objects, moving at large Froude and Weber numbers, are treated.

Morphological transition in compressible foam
View Description Hide DescriptionA theory is constructed to describe the morphological transition that occurs in a compressible foam when its volume is increased. The foam is observed to separate into two bubble populations or “phases,” one consisting of a large number of small bubbles, the “liquid” phase, the other consisting of a small number of large bubbles, the “gaseous” phase. First, working along lines similar to the van der Waals theory for a fluid system, approximate forms of the equation of state of the foam are derived and explored. These describe the weakly compressible range well, but fail to capture the nature of the transition. Taking a clue from the phenomenology, a theory of the “phaseseparated” regime is then formulated working with the approximation that the two phases into which the foam separates are each relatively homogeneous. The successful singlephase formulas are applied to each phase, introducing an additional “order parameter” which gives the ratio of the average size of bubbles in the two phases. Approximate expressions are written for the Helmholtz free energy and the equilibria of the foam derived from these. The theory is compared to numerical simulations using SURFACE EVOLVER with very satisfying results. The transition from a uniform to a nonuniform configuration consists of avalanches of reconnections with smooth evolution between them. Hysteresis is observed as the foam is first expanded and then compressed.

Scalings for fragments produced from drop breakup in shear flow with inertia
View Description Hide DescriptionWhen a drop is sheared in a matrix liquid, the largest daughter drops are produced by elongative end pinching. The daughter drop size is found to scale with the critical drop size that would occur under the same flow conditions and fluid properties. Daughter drop volumes saturate to just below 60% of the critical volume as the mother drop size increases. For large Reynolds number, the daughter drop radius scales with the case power of the capillary number when the Reynolds number is fixed.

Bubbleinduced agitation and microstructure in uniform bubbly flows at small to moderate particle Reynolds numbers
View Description Hide DescriptionThe bubbleinduced agitation has been quantified in dilute and homogeneous bubbly flows. To characterize the microstructure, experimental techniques have been developed that provide the average perturbed liquid velocity and the pair density. For particulate Reynolds numbers about unity and contaminated bubbles, the liquid agitation is shown to increase as where α is the local void fraction. Although in partial agreement with the scaling proposed by Koch (Phys. Fluids, 1993), the pairdensity distribution does not exhibit any deficit. On the opposite, experiments at and clean bubbles reveal a strong deficit in the rear of test bubbles as well as a moderate accumulation both in a horizontal plane and all along the deficit zone. The extent of the deficit slowly decreases with the void fraction, roughly as It is shown that the agitation originates from this near wake region, the structure of which strongly differs from the one due to a single inclusion both in terms of its lateral extent and of the perturbed velocity decay rate. These features are related to the extra momentum sources induced by the microstructure surrounding the test bubble. Accordingly, the resulting velocity variance, that has been found independent of the vessel dimension and of the base flow velocity, is significantly higher than Koch’s predictions. Finally, a comparison is achieved with available data and models, and an explanation is suggested for the evolution of the bubbleinduced agitation at moderate

Lowsymmetric bubbles in Rayleigh–Taylor instability
View Description Hide DescriptionWe report a multimode analysis of the 3D–2D dimensional crossover for the nonlinear structure, which occurs in the nonlinear regime of the Rayleigh–Taylor instability(RTI). This structure is an array of bubbles and spikes periodic in the plane normal to the direction of gravity. The flow is assumed to be anisotropic in the plane and to have low rectangular symmetry. For regular bubbles, there is a twoparameter family of steady solutions, and we analyze stability of these nonlinear solutions. It is shown that 3D bubbles in RTI conserve a nearcircular contour, and cannot be transformed into 2D bubbles continuously. We discuss the mechanism of secondary instabilities of anisotropic RT flow.

Fractal dewetting of a viscous adhesive film between separating parallel plates
View Description Hide DescriptionA dewetting cylindrical film between two separating plates is observed to produce fractal finger patterns that penetrate radially inward. In lieu of the classical tip splitting, this fractal is generated by successive shielding of alternating fingers. By analyzing this key shielding dynamics via a forced twodimensional Hele–Shaw model with global force balance, a selfsimilar scaling for the shielding distance and shielding time were obtained for every generation of shielding events. The theory predicts that the plate detach time is reduced by a factor of due to the fractal fingers. This is confirmed experimentally for large Bond numbers when there are sufficient fingers to justify a continuum approximation in the theory. A cumulative node number density, which is not a power law, is also predicted but not confirmed experimentally.

Electrospinning and electrically forced jets. I. Stability theory
View Description Hide DescriptionElectrospinning is a process in which solid fibers are produced from a polymeric fluid stream (solution or melt) delivered through a millimeterscale nozzle. The solid fibers are notable for their very small diameters (<1 μm). Recent experiments demonstrate that an essential mechanism of electrospinning is a rapidly whipping fluid jet. This series of papers analyzes the mechanics of this whipping jet by studying the instability of an electrically forced fluid jet with increasing field strength. An asymptotic approximation of the equations of electrohydrodynamics is developed so that quantitative comparisons with experiments can be carried out. The approximation governs both long wavelength axisymmetric distortions of the jet, as well as long wavelength oscillations of the centerline of the jet. Three different instabilities are identified: the classical (axisymmetric) Rayleigh instability, and electric field induced axisymmetric and whipping instabilities. At increasing field strengths, the electrical instabilities are enhanced whereas the Rayleigh instability is suppressed. Which instability dominates depends strongly on the surface charge density and radius of the jet. The physical mechanisms for the instability are discussed in the various possible limits.

Electrospinning and electrically forced jets. II. Applications
View Description Hide DescriptionElectrospinning is a process in which solid fibers are produced from a polymeric fluid stream (solution or melt) delivered through a millimeterscale nozzle. This article uses the stability theory described in the previous article to develop a quantitative method for predicting when electrospinning occurs. First a method for calculating the shape and charge density of a steady jet as it thins from the nozzle is presented and is shown to capture quantitative features of the experiments. Then, this information is combined with the stability analysis to predict scaling laws for the jet behavior and to produce operating diagrams for when electrospinning occurs, both as a function of experimental parameters. Predictions for how the regime of electrospinning changes as a function of the fluid conductivity and viscosity are presented.

A generalized compressible Reynolds lubrication equation with bounded contact pressure
View Description Hide DescriptionA new modified Reynolds equation is derived based on physical principles for rarefied gas for compressible and extremely thin layer gas lubrication. For the onedimensional problem, theoreticalanalysis and numerical simulation are employed to show that the new equation does not predict an unphysical unbounded pressuresingularity in the limit of contact between the bearing surface and the moving surface. We also show the same is true for other existing models with higher than first order slippage correction, which introduce additional diffusion terms that are functions of the spacing with similar order to that of the convection terms. These developments remove the ambiguity of some previously published analyses and correct prior erroneous statements that all existing generalized Reynolds equationmodels predict nonintegrable singular pressure fields in the limit of contact. The asymptotic analysis also supplies a means for deriving the needed additional boundary condition at the boundary of a contact region. For the twodimensional problem, we show by numerical analysis that there are also no unbounded contact pressuresingularities for the new model and other models with corrections higher than first order, and that the singularity is weaker than in the onedimensional case for these lower order correction models due to the cross diffusion effect introduced by the additional dimension.

Stability of miscible displacements across stratified porous media
View Description Hide DescriptionWe consider the stability of miscible displacements across stratified porous media, where the heterogeneity is along the direction of displacement. Asymptotic results for long and short wavelengths are derived. It is found that heterogeneity has a longwave effect on the instability, which, in the absence of gravity, becomes nontrivial when the viscosity profiles are nonmonotonic. In the latter case, profiles with endpoint viscosities, predicted to be stable using the Saffman–Taylor criterion, can become unstable, if the permeability contrast in the direction of displacement is sufficiently large. Conversely, profiles with endpoint viscosities predicted to be unstable, can become stable, if the permeability decrease in the direction of displacement is sufficiently large. Analogous results are found in the presence of gravity, but without the nonmonotonic restriction on the viscosity profile. The increase or decrease in the propensity for instability as the permeability increases or decreases, respectively, reflects the variation of the two different components of the effective fluid mobility. While permeability remains frozen in space, viscosity varies following the concentration field. Thus, the condition for instability does not solely depend on the overall fluid mobility, as in the case of displacements in homogeneous media, but it is additionally dependent on the permeability variation.

Hydrodynamic modes of a sheared granular flow from the Boltzmann and Navier–Stokes equations
View Description Hide DescriptionThe initial growth rates for the hydrodynamic modes of the shear flow of a threedimensional collection of inelastic spheres is analyzed using two models. The first is the generalized Navier–Stokes equations, derived for the shear flow of inelastic spheres using the Chapman–Enskog procedure, where the energy equation has an additional dissipation term due to inelastic collisions. The second is the solution of the linearized Boltzmann equation, where the distribution function in the base state is determined using a Hermite polynomial expansion in the velocity moments. For perturbations with variations in the velocity and gradient directions, it is found that the solutions obtained by two procedures are qualitatively similar, though there are quantitative differences. For perturbations with variations in the vorticity direction, it is found that there are qualitative differences in the predictions for the initial growth rate of the perturbations.

The effect of an external magnetic field on oscillatory instability of convective flows in a rectangular cavity
View Description Hide DescriptionThe present study is devoted to the problem of onset of oscillatory instability in convective flow of an electrically conducting fluid under an externally imposed timeindependent uniform magnetic field.Convection of a lowPrandtlnumber fluid in a laterally heated twodimensional horizontal cavity is considered. Fixed values of the aspect ratio (length/height=4) and Prandtl number (Pr=0.015), which are associated with the horizontal Bridgman crystal growth process and are commonly used for benchmarking purposes, are considered. The effect of a uniform magnetic field with different magnitudes and orientations on the stability of the two distinct branches (with a singlecell or a twocell pattern) of the steady state flows is investigated. Stability diagrams showing the dependence of the critical Grashof number on the Hartmann number are presented. It is shown that a vertical magnetic field provides the strongest stabilization effect, and also that multiplicity of steady states is suppressed by the electromagnetic effect, so that at a certain field level only the singlecell flows remain stable. An analysis of the most dangerous flow perturbations shows that starting with a certain value of the Hartmann number, singlecell flows are destabilized inside thin Hartmann boundary layers. This can lead to destabilization of the flow with an increase of the field magnitude, as is seen from the stability diagrams obtained. Contrary to the expected monotonicity of the stabilization process with an increase of the field strength, the marginal stability curves show nonmonotonic behavior and may contain hysteresis loops.

Eddies induced in cylindrical containers by a rotating end wall
View Description Hide DescriptionThe flow generated in a viscous liquid contained in a cylindrical geometry by a rotating end wall is considered. Recent numerical and experimental work has established several distinct phases of the motion when fluid inertia plays a significant role. The current paper, however, establishes the nature of the flow in the thus far neglected low Reynolds number regime. Explicitly, by employing biorthogonality relations appropriate to the current geometry, it is shown that a sequence of exponentially decaying eddies extends outward from the rotating end wall. The cellular structure is a manifestation of the dominance of complex eigensolutions to the homogeneous problem and arises as the result of nonlinear forcing associated with an inertial correction to the Stokes flow.

Visual analysis of twodimensional magnetohydrodynamics
View Description Hide DescriptionMagnetohydrodynamics(MHD) offers a unique opportunity to study the behavior of twodimensional turbulent flows. A strong external magnetic fieldB perpendicular to the flow direction of an electrically conducting fluid will suppress velocity gradients in the direction of B. The resulting approximation is known as quasitwodimensional MHD. An experimental configuration is presented which meets this requirement, along with a spatially extended probe used to visualize the twodimensional flowkinematics inside the opaque liquid metalflow. As a prototypical example, the wake behind a circular cylinder is investigated for Reynolds numbers up to New and unexpected vortex patterns are observed that deviate significantly from usual hydrodynamic flows. Also, stability limits for the transition from stationary to nonstationary flow patterns are experimentally determined for the cylinder wake and another type of shear flow profile. These results confirm existing theoretical predictions and thus validate the quasitwodimensional approach.

Analysis of centrifugal convection in rotating pipes
View Description Hide DescriptionNew exact solutions, obtained for centrifugal convection of a compressible fluid in pipes and annular pipes, explain axially elongated counterflow and energy separation—poorly understood phenomena occurring in vortex devices, e.g., hydrocyclones and Ranque tubes. Centrifugal acceleration (which can be up to times gravity in practical vortex tubes), combined with an axial gradient of temperature (even small), induces an intense flow from the cold end to the hot end along the pipe wall and a backflow near the axis. To account for large density variations in vortex devices, we use the axial temperature gradient as a small parameter instead of the Boussinesq approximation. For weak pipe rotation, the swirl is of solidbody type and solutions are compact: and where the subscripts w and a denote values of axial velocity temperature T, and radial distance r, at the wall and on the axis. The axial gradient of pressure, being proportional to has opposite directions near the wall, and near the axis, this explains the counterflow. With increasing pipe rotation, the flow starts to converge to the axis. This causes important new effects: (i) the density and swirl velocity maxima occur away from the wall (vortex core formation), (ii) the temperature near the axis becomes lower than near the wall (the Ranque effect), (iii) the axial gradient of temperature drops from the wall to the axis, and (iv) the total axial heat flux (Nu) reaches its maximum and then decreases as swirl increases. These features can be exploited for the development of a microheatexchanger, e.g., for cooling computer chips.

Variational approach to a turbulent swirling pipe flow with the aid of helicity
View Description Hide DescriptionA turbulent swirling flow in a circular pipe is studied with the aid of a variational method. A prominent feature of the flow is the occurrence of a retarded or reversed meanflow profile near the central axis. Diffusioneffect is generally enhanced in a turbulent flow. A mechanism under which these two properties coexist is sought with resort to a variational approach combined with helicity. The meanflow field is expressed by the mean vorticity proportional to its curl. On this basis, the centerline axial vorticity is shown to play a central role in the occurrence of axialvelocity retardation or reversal in a turbulent swirling pipe flow. Suggestions on modeling the Reynolds stress are obtained from the meanvorticity equation.

Direct numerical simulation of particleladen rotating turbulent channel flow
View Description Hide DescriptionDirect numerical simulations (DNS) of particleladen rotating turbulent channel flows at the Reynolds number 194 and the rotation number 0.075 (both based on the friction velocity and the channel half width) on a grid were performed. Particles were traced using the deterministic method combined with the direct particle interactions via hardsphere collisions and twoway coupling. The particle gravity parallel to the spanwise direction and the particle volume fractions from to were considered. It is found that the presence of the heavier particles (Stokes number and their interparticle collisions near the pressuresurface significantly changes the turbulence properties, particularly in the nearwall regions. The DNS results showed that the interparticle collisions make the lighter particle distribution more even and then attenuate the turbulence intensity of the fluid, and make the heavier particle distribution more uneven and then enhance the turbulence intensity of the fluid. It is also found that the slightly heavier particles form strong streaky structures added “hooks” with a certain skew angle to the streamwise direction near the pressuresurface. When the particles become larger and heavier, and the particle volume fraction becomes higher the wellknown low/high speed streaks near the pressuresurface are destroyed by the particles due to rotation.

Frequency shifts of Rossby waves in the inertial subranges of βplane turbulence
View Description Hide DescriptionNonlinear interactions between waves and turbulence cause systematic frequency shifts in Rossby waves. The frequency shifts in the inertial subranges of statistically steady βplane turbulence were examined theoretically and numerically. The theoreticalanalysis is based on the Lagrangian closure called the Lagrangian renormalized approximation and predicts that when the β effect is small, the frequency shifts of Rossby waves are proportional to in the inverse energy transfer range, while they are proportional to with or without a logcorrection term in the enstrophy transfer range, depending on the flow conditions, where is the wave number in the eastward direction. Numerical simulations using points of forced βplane turbulence that exhibit the inertial subranges, show fairly good agreement with theoretical predictions.

The performance of dynamic subgridscale models in the large eddy simulation of rotating homogeneous turbulence
View Description Hide DescriptionThe performance of dynamic subgridscale (SGS) models is numerically examined in the large eddy simulation of rotating homogeneous turbulences in comparison with the corresponding filtered data of the direct numerical simulation (DNS). The examined dynamic SGS models are: the dynamic Smagorinsky model (DSM), the dynamic mixed model (DMM), the dynamic Clark model (DCM), and the dynamic twoparameter Clark model (DTCM). All models are mathematically reformulated in a rotating frame from the corresponding expressions in an inertial frame. It is shown that the DSM and the DMM are not consistent with the constraint of asymptotic material frame indifference, but the DCM and the DTCM are consistent. All models except the DSM show similar decays of the gridscale turbulent energies both in nonrotating and in rotating frames; they agree well with the DNS in the nonrotating case, but they are slightly less dissipative than the DNS in the rotating case. The DSM underestimated the gridscale energy dissipation in the nonrotating case, though there is no major difference from other models in the rotating case. However, the DSM in a rotating frame, which takes a different form from that in an inertial frame, leads to an unphysical fluctuating decay for a homogeneous turbulence suddenly submitted to a rotation.

Anisotropy in turbulence profiles of stratified wakes
View Description Hide DescriptionAt sufficiently high values of the Reynolds number and internal Froude number initially turbulent bluff body wakes evolve in the presence of a stable background density gradient with wakeaveraged mean and turbulence length and velocity scales that are independent of and F for at least two orders of magnitude extension in both parameters. The way in which the initially threedimensional motions transition to the characteristic (and  and Findependent) late wakes (where vertical velocities, is both of great practical interest, and complex, hence somewhat unclear. Here, digital particle imaging velocimetry type measurements on towedsphere wakes are described, so that the development of anisotropy can be measured by the time development of turbulence profiles in horizontal and vertical centerplanes. The observed anisotropies can be associated with energy transfer to internal wave modes, and suppression of other vertical displacements, that contrasts with sphere wakes at similar in a homogeneous fluid. Maximum Reynolds stresses occur at the boundary of a sinuous undulation of the wake, which increases in amplitude up to is the buoyancy frequency that characterizes the strength of the stratification). Although an intrinsic wake profile instability cannot be excluded, the observed wake element spacings can be accounted for by known spiral and Kelvin–Helmholtz instabilities in the near wake.
