bluuebunny/crossref_metadata_2025_split huggingface dataset:
Buoyancy-driven interactions between two deformable viscous drops
Authors: Manga Michael , Stone H. A. 
Published: April 2006

Time-dependent interactions between two buoyancy-driven deformable drops are studied in the low Reynolds number flow limit for sufficiently large Bond numbers that the drops become significantly deformed. The first part of this paper considers the interaction and deformation of drops in axisymmetric configurations. Boundary integral calculations are presented for Bond numbers ℬ = Δρga2/σ in the range 0.25 ≤ ℬ < ∞ and viscosity ratios λ in the range 0.2 ≤ λ ≤ 20. Specifically, the case of a large drop following a smaller drop is considered, which typically leads to the smaller drop coating the larger drop for ℬ [Gt ] 1. Three distinct drainage modes of the thin film of fluid between the drops characterize axisymmetric two-drop interactions: (i) rapid drainage for which the thinnest region of the film is on the axis of symmetry, (ii) uniform drainage for which the film has a nearly constant thickness, and (iii) dimple formation. The initial mode of film drainage is always rapid drainage. As the separation distance decreases, film flow may change to uniform drainage and eventually to dimpled drainage. Moderate Bond numbers, typically ℬ = O(10) for λ = O(1), enhance dimple formation compared to either much larger or smaller Bond numbers. The numerical calculations also illustrate the extent to which lubrication theory and analytical solutions in bipolar coordinates (which assume spherical drop shapes) are applicable to deformable drops.The second part of this investigation considers the 'stability’ of axisymmetric drop configurations. Laboratory experiments and two-dimensional boundary integral simulations are used to study the interactions between two horizontally offset drops. For sufficiently deformable drops, alignment occurs so that the small drop may still coat the large drop, whereas for large enough drop viscosities or high enough interfacial tension, the small drop will be swept around the larger drop. If the large drop is sufficiently deformable, the small drop may then be ‘sucked’ into the larger drop as it is being swept around the larger drop. In order to explain the alignment process, the shape and translation velocities of widely separated, nearly spherical drops are calculated using the method of reflections and a perturbation analysis for the deformed shapes. The perturbation analysis demonstrates explicitly that drops will tend to be aligned for ℬ > O(d/a) where d is the separation distance between the drops.
On the irreversibility of internal-wave dynamics due to wave trapping by mean flow inhomogeneities. Part 1. Local analysis
Authors: Badulin Sergei I. , Shrira Victor I. 
Published: April 2006

The propagation of guided internal waves on non-uniform large-scale flows of arbitrary geometry is studied within the framework of linear inviscid theory in the WKB-approximation. Our study is based on a set of Hamiltonian ray equations, with the Hamiltonian being determined from the Taylor-Goldstein boundary-value problem for a stratified shear flow. Attention is focused on the fundamental fact that the generic smooth non-uniformities of the large-scale flow result in specific singularities of the Hamiltonian. Interpreting wave packets as particles with momenta equal to their wave vectors moving in a certain force field, one can consider these singularities as infinitely deep potential holes acting quite similarly to the ‘black holes’ of astrophysics. It is shown that the particles fall for infinitely long time, each into its own ‘black hole‘. In terms of a particular wave packet this falling implies infinite growth with time of the wavenumber and the amplitude, as well as wave motion focusing at a certain depth. For internal-wave-field dynamics this provides a robust mechanism of a very specific conservative and moreover Hamiltonian irreversibility.This phenomenon was previously studied for the simplest model of the flow non-uniformity, parallel shear flow (Badulin, Shrira & Tsimring 1985), where the term ‘trapping’ for it was introduced and the basic features were established. In the present paper we study the case of arbitrary flow geometry. Our main conclusion is that although the wave dynamics in the general case is incomparably more complicated, the phenomenon persists and retains its most fundamental features. Qualitatively new features appear as well, namely, the possibility of three-dimensional wave focusing and of ‘non-dispersive’ focusing. In terms of the particle analogy, the latter means that a certain group of particles fall into the same hole.These results indicate a robust tendency of the wave field towards an irreversible transformation into small spatial scales, due to the presence of large-scale flows and towards considerable wave energy concentration in narrow spatial zones.
The mean rise velocity of pairwise-interacting bubbles in liquid
Authors: Wijngaarden L. van 
Published: April 2006

This paper is concerned with calculations regarding a collection of small gas bubbles rising under buoyancy in a clear liquid. For dilute mixtures interactions can be restricted to those between two bubbles. In the analysis of binary interactions it is assumed that the Reynolds number for relative motion between bubbles and liquid is large, that surfactants, if present, do not modify the condition of zero tangential stress at the bubble-liquid interface, and that bubbles bounce at an encounter.A two-bubble probability density is derived from the analysis, valid on a short timescale associated with the interaction. It is shown that on a long timescale, based on viscous dissipation, clustering together of pairs takes place, most likely even when triple encounters are allowed for. An analysis is given of the vertical motion of pairs, followed by a calculation of the mean vertical bubble velocity with help of the (short timescale) probability density function. The result is compared with experimental data.
Turbulent shear flow over slowly moving waves
Authors: Belcher S. E. , Hunt J. C. R. 
Published: April 2006

We investigate the changes to a fully developed turbulent boundary layer caused by the presence of a two-dimensional moving wave of wavelength L = 2π/k and amplitude a. Attention is focused on small slopes, ak, and small wave speeds, c, so that the linear perturbations are calculated as asymptotic sequences in the limit (u* + c)/UB(L) → 0 (u* is the unperturbed friction velocity and UB(L) is the approach-flow mean velocity at height L). The perturbations can then be described by an extension of the four-layer asymptotic structure developed by Hunt, Leibovich & Richards (1988) to calculate the changes to a boundary layer passing over a low hill.When (u* + c)/UB(L) is small, the matched height, zm (the height where UB equals c), lies within an inner surface layer, where the perturbation Reynolds shear stress varies only slowly. Solutions across the matched height are then constructed by considering an equation for the shear stress. The importance of the shear-stress perturbation at the matched height implies that the inviscid theory of Miles (1957) is inappropriate in this parameter range. The perturbations above the inner surface layer are not directly influenced by the matched height and the region of reversed flow below zm: they are similar to the perturbations due to a static undulation, but the ‘effective roughness length’ that determines the shape of the unperturbed velocity profile is modified to zm = z0 exp (kc/u*).The solutions for the perturbations to the boundary layer are used to calculate the growth rate of waves, which is determined at leading order by the asymmetric pressure perturbation induced by the thickening of the perturbed boundary layer on the leeside of the wave crest. At first order in (u* + c)/UB(L), however, there are three new effects which, numerically, contribute significantly to the growth rate, namely: the asymmetries in both the normal and shear Reynolds stresses associated with the leeside thickening of the boundary layer, and asymmetric perturbations induced by the varying surface velocity associated with the fluid motion in the wave; further asymmetries induced by the variation in the surface roughness along the wave may also be important.
First-order resonance in the reflection of baroclinic Rossby waves
Authors: Graef Federico 
Published: April 2006

The nonlinear interaction between an incident and a reflected Rossby wave produces a steady flow parallel to the (non-zonal) reflecting wall and a transient flow oscillating at twice the frequency of the incident-reflected pair. If the transient forcing is resonant, i.e. a free Rossby wave, the resonant response must have zero amplitude at the wall in order to fulfil the boundary condition there; a straightforward expansion predicts a linear growth of its amplitude in the offshore direction y. Resonance is possible only if 0 < |sin α| ≤ $\frac13$, where α is the angle between the wall and the easterly direction. This requirement is met by several boundaries in the ocean. A simple graphical method to find a resonant triad is described.Using the method of multiple scales, it is shown that the wave amplitudes of the triad are slowly varying periodic functions of y, such that the energy flux of the triad through any plane parallel to the wall vanishes, as required by energy conservation. The waves participating in the resonant triad become wave packets. The three waves do not exchange energy in time due to the additional constraint on the motion imposed by the boundary condition at the wall. It is shown that the wave amplitudes cannot be slowly varying functions of v and time.As a possible oceanic application of the theoretical findings, the distance from the wall where one would expect to find large semi-annual amplitudes if annual Rossby waves are impinging on the boundary is of the order of 100 km. Motivated by similar studies (Plumb 1977; Mysak 1978), there are speculations on what would happen if three incident-reflected Rossby wave pairs (or modes) are taken, allowing each mode amplitude to be slowly varying in time.
The vaporization of a liquid front moving through a hot porous rock
Authors: Woods Andrew W. , Fitzgerald Shaun D. 
Published: April 2006

We develop an analytical model to describe the generation of vapour as water moves through a hot porous rock, as occurs in hot, geothermal reservoirs. Typically the isotherms in the liquid lag behind the water-vapour interface and so water is supplied to the interface at the interface temperature. This temperature is lower than that in the rock far ahead of the interface. Therefore, as the hot porous rock is invaded with water, it cools and the heat released is used to vaporize some of the water. At low injection rates, vapour formed from the injected liquid may readily move ahead of the advancing liquid-vapour interface and so the interfacial pressure remains close to that in the far field ahead of the interface. The mass fraction that vaporizes is then limited by the superheat of the rock. For larger injection rates, the interfacial vapour pressure becomes considerably greater than that in the far field in order to drive the vapour ahead of the moving interface. As a result, the interfacial temperature increases. The associated reduction in the thermal energy available for vaporization results in a decrease in the mass fraction of vapour produced.Since the vapour is compressible, the motion of the vapour ahead of the interface is governed by a nonlinear diffusion equation. Therefore, the geometry of injection has an important effect upon the mass fraction of water that vaporizes. We show that with a constant supply of water from (i) a point source, the mass fraction of water which vaporizes increases towards the maximum permitted by the superheat of the rock; (ii) a line source, a similarity solution exists in which the mass fraction vaporizing is constant; and (iii) a planar source, the liquid-vapour interface steadily translates through the rock with a very small fraction of the injected water vaporizing.
Simple shear flow of a suspension of fibres in a dilute polymer solution at high Deborah number
Authors: Harlen O. G. , Koch Donald L. 
Published: April 2006

The behaviour of fibre suspensions in dilute polymer solutions at high Deborah numbers is analysed. In particular, we calculate the change to the extension of the polymers and the orientation of the fibres caused by hydrodynamic interactions between the polymers and the fibres. At a sufficiently high Deborah number the combined effect of the fibre velocity disturbances and the mean shear flow produce a dramatic increase in the extension of the polymers, similar to the coil-stretch transition observed in extensional flow.The non-Newtonian stresses caused by the polymers produce a perturbation to the angular velocity of the fibres, giving rise to a net drift across Jeffery orbits towards the vorticity axis. Unlike the second-order-fluid analysis of Leal (1975), this effect does not depend on the second-normal-stress difference.
Stretching and bending of line elements in random flows
Authors: Drummond I. T. 
Published: April 2006

We study the stretching and bending of line elements transported in random flows with known Eulerian statistics in two and three dimensions. By making use of a cumulant expansion for the log-size of material elements we are able to analyse the exponential stretching they exhibit in random flows and identify conditions under which it will and will not occur. The results are confirmed by our numerical simulation.We also examine the evolution of curvature in material elements and confirm by numerical simulation that it is governed by an appropriate version of the Pope equation. By modelling this equation as stochastic differential equation we are able to explain the appearance of a power-law tail in the probability distribution for large curvature observed by Pope, Yeung & Girimaji (1989) for surface elements. In two dimensions the appearance of the tail can indeed be attributed to the occurrence of events in which the material element undergoes contraction rather than stretching while subject to bending. In three dimensions the relationship between episodes of contraction and strong bending is less direct. This power-law tail allows us to reconcile the observed asymptotic stability, which we confirm here, of the powers and cumulants of the log-curvature with the unboundedness of powers of the curvature itself.
Surface waves on shear currents: solution of the boundary-value problem
Authors: Shrira Victor I. 
Published: April 2006

We consider a classic boundary-value problem for deep-water gravity-capillary waves in a shear flow, composed of the Rayleigh equation and the standard linearized kinematic and dynamic inviscid boundary conditions at the free surface. We derived the exact solution for this problem in terms of an infinite series in powers of a certain parameter e, which characterizes the smallness of the deviation of the wave motion from the potential motion. For the existence and absolute convergence of the solution it is sufficient that e be less than unity.The truncated sums of the series provide approximate solutions with a priori prescribed accuracy. In particular, for the short-wave instability, which can be interpreted as the Miles critical-layer-type instability, the explicit approximate expressions for the growth rates are derived. The growth rates in a certain (very narrow) range of scales can exceed the Miles increments caused by the wind.The effect of thin boundary layers on the dispersion relation was also investigated using an asymptotic procedure based on the smallness of the product of the layer thickness and wavenumber. The criterion specifying when and with what accuracy the boundary-layer influence can be neglected has been derived.
The role of coherent structures in bubble transport by turbulent shear flows
Authors: Sene K. J. , Hunt J. C. R. , Thomas N. H. 
Published: April 2006

Using Auton's force law for the unsteady motion of a spherical bubble in inhomogeneous unsteady flow, two key dimensionless groups are deduced which determine whether isolated vortices or shear-layer vortices can trap bubbles. These groups represent the ratio of inertial to buoyancy forces as a relaxation parameter [tcy ] = ΔU2/2gx and a trapping parameter [Gcy ] = ΔU/VT where ΔU is the velocity difference across the vortex or the shear layer, x is streamwise distance measured from the effective origin of the mixing layer and VT is the terminal slip speed of the bubble or particle. It is shown here that whilst buoyancy and drag forces can lead to bubbles moving in closed orbits in the vortex flows (either free or forced), only inertial forces result in convergent trajectories. Bubbles converge on the downflow side of the vortex at a location that depends on the inertial and lift forces. It is important to note that the latter have been omitted from many earlier studies.A discrete-vortex model is used to simulate the large-scale unsteady flows within horizontal and vertical mixing layers between streams with velocity difference ΔU. Trajectories of non-interacting small bubbles are computed using the general force law. In the horizontal mixing layer it is found that Γ needs to have a value of about 3 to trap about 50% of the bubbles if Π is about 0.5 and greater if Π is less. The pairing of vortices actually enhances their trapping of bubbles. In the vertical mixing layer bubbles are trapped mainly within the growing vortices but bubbles are concentrated on the downflow side of the vortices as Γ and Π increase. In a companion paper we show that lateral dispersion of bubbles can be approximately described by an advective diffusion equation with the diffusivity about equal to the eddy viscosity, i.e. rather less than the diffusivity of heat or other passive scalars.
Scale-dependent merging of baroclinic vortices
Authors: Verron J. , Valcke S. 
Published: April 2006

The influence of stratification on the merging of like-sign vortices of equal intensity and shape is investigated by numerical simulations in a quasi-geostrophic, two-layer stratified model. Two different types of vortices are considered: vortices defined as circular patches of uniform potential vorticity in the upper layer but no PV anomaly in the lower layer (referred to as PVI vortices), and vortices defined as circular patches of uniform relative vorticity in the upper layer but no motion in the lower layer (referred to as RVI vortices). In particular, it is found that, in the RVI case, the merging behaviour depends strongly on the magnitude of the stratification (i.e. the ratio of internal Rossby radius and vortex radius). The critical point here appears to be whether or not the initial eddies have a deep flow signature in terms of PV.The specific phenomenon of scale-dependent merging observed is interpreted in terms of the competitive effects of hetonic interaction and vortex shape. In the case of weaker stratification, the baroclinic structure of the eddies can be seen as dominated by a mechanism of hetonic interaction in which bottom flow appears to counteract the tendency of surface eddies to merge. In the case of larger stratification, the eddy interaction mechanism is shown to be barotropically dominated, although interface deformation still determines the actual eddy vorticity profile during the initialization stage. Repulsion (hetonic) effect therefore oppose attraction (barotropic shape) effects in a competitive process dependent on the relationship between the original eddy lengthscale and the first internal Rossby radius.A concluding discussion considers the implications of such analysis for real situations, in the ocean or in the laboratory.
The spin-up from rest of a fluid-filled torus
Authors: Madden F. N. , Mullin T. 
Published: April 2006

We present the results of an experimental and numerical study of the spin-up from rest to solid-body rotation of a fluid-filled torus. In separate experiments, the rotation rate of the container is suddenly increased to a fixed value and the final rotation rate is used to define a non-dimensionalized control parameter,C. At low values ofC, the observed flows during the transient phase are axisymmetric and spin-up is achieved through viscous diffusion. This in turn is followed by significant secondary flow and the appearance of ‘fronts’ asCis increased. During the transient phase the fluid motion near the inner wall of the container is dynamically unstable according to Rayleigh's criterion. Thus at higher values ofCwave-like structures break the axisymmetry, non-uniqueness in the details of the process is found and finally, an innear wall instability is observed directly. A plot of the spin-up time versusCshows breaks in the slope at transition points between each of the above dynamical regimes but the overall trend is found to be insensitive to the details of the fluid motion. Further elucidation of the dynamical processes is provided by a novel variant of the now standard phase-space reconstruction techniques. The results show a systematic splitting of the phase paths asCis increased.Finally, in the complementary numerical study, the time-dependent Navier–Stokes equations are solved for axisymmetric flows. Here, the flow is computed using a velocity–streamfunction–vorticity formulation in a two-dimensional plane with a velocity component normal to this plane. The quantitative and qualitative agreement between the numerical and experimental results is excellent for moderate values of the dynamical control parameterC.
Asymptotic analysis of the radiation by volume sources in supersonic rotor acoustics
Authors: Ardavan H. 
Published: April 2006

The application of Lighthill's acoustic analogy to the generation of sound by rotating surfaces with supersonic speeds results in radiation integrals in which the stationary points of the phase function – that describes the sapce-time distance between each source point and a fixed observation point – have variable positions and coalesce at a caustic in the space of source points. Here, the leading term in the asymptotic expansion of the corresponding Green's function at this caustic is calculated, both in the time and the frequency domains, and it is shown that the radiation generated by volume sources, which are steady in the uniformly rotating blade-fixed frame, has an amplitude that does not obey the spherical spreading law, i.e. does not fall off like RP–1 with the radial distance RP away from the source. Within a finite solid angle, depending on the extent of the source distribution, the amplitude of this newly identified sound decays like RP–½, so that it is stronger in the far field than any previously studied element. That this is not incompatible with the conservation of energy is because the emission time intervals associated with the volume elements of the source distribution which contribute towards the non-spherically decaying component of the radiation are by a large (RP-dependent) factor greater than the time intervals during which the signals generated by these elements are received. The contributing source elements are those whose positions at the retarded time coincide witht the locus of singularities of the Green's function, i.e. with the one-dimensional locus of points, fixed in the rotating frame, which approach the observer with the wave speed and zero acceleration along the radiation direction. Because the signals received at two neighbouring instants in time arise from distinct, coherently radiating filamentary parts of the source which have both different extents and different strengths, the resulting overall waveform in the far zone consists of the superposition of a (continuous) set of narrow subpulses with uneven amplitudes. These subpulses are narrower the larger the distance from the source.The differences between the new results and those of the earlier works in the literature are shown to arise from the error terms in the far-field and high-frequency approximations, approximations that are inappropriate for volume sources moving supersonically.