ALBERT

Description: A boundary integral method is presented for analysing particle motion in a rotating fluid for flows where the Taylor number 2T is arbitrary and the Reynolds number is small. The method determines the surface traction and drag on a particle, and also the velocity field at any location in the fluid. Numerical results show that the dimensionless drag on a spherical particle translating along the rotation axis of an unbounded fluid is determined by the empirical formula D/6n = 1 +(4/7) ^”1/2 +(8/9TI)2T, which incorporates known results for the low and high Taylor number limits. Streamline portraits show that a critical Taylor number c « 50 exists at which the character of the flow changes. For 3 “ 〈 2Tcthe flow field appears as a perturbation of a Stokes flow with a superimposed swirling motion. For T 〈 2TCthe flow field develops two detached recirculating regions of trapped fluid located fore and aft of the particle. The recirculating regions grow in size and move farther from the particle with increasing Taylor number. This recirculation functions to deflect fluid away from the translating particle, thereby generating a columnar flow structure. The flow between the recirculating regions and the particle has a plug-like velocity profile, moving slightly slower than the particle and undergoing a uniform swirling motion. The flow in this region is matched to the particle velocity in a thin Ekman layer adjacent to the particle surface. A further study examines the translation of spheroidal particles. For large Taylor numbers, the drag is determined by the equatorial radius; details of the body shape are less important. © 1994, Cambridge University Press. All rights reserved.

Description: 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 = ∆ρga2/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 ≫ 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 = 0(10) for λ = 0(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 〉 0(d/a) where d is the separation distance between the drops. © 1993, Cambridge University Press. All rights reserved.

Description: When a small air bubble bursts from an equilibrium position at an air/water interface, a complex motion ensues resulting in the production of a high-speed liquid jet. This free-surface motion following the burst is modelled numerically using a boundary integral method. Jet formation and liquid entrainment rates from jet breakup into drops are calculated and compared with existing experimental evidence. In order to investigate viscous effects, a boundary layer is included in the calculations by employing a time-stepping technique which allows the boundary mesh to remain orthogonal to the surface. This allows an approximation of the vorticity development in the region of boundary-layer separation during jet formation. Calculated values of pressure and energy dissipation rates in the fluid indicate a violent motion, particularly for smaller bubbles. This has important implications for the biological industry where animal cells in bioreactors have been found to be killed by the presence of small bubbles. © 1993, Cambridge University Press. All rights reserved.

Description: We have modelled the wall region of a turbulent boundary layer by expanding the instantaneous field in so-called empirical eigenfunctions, as permitted by the proper orthogonal decomposition theorem (Lumley 1967, 1981). We truncate the representation to obtain low-dimensional sets of ordinary differential equations, from the Navier-Stokes equations, via Galerkin projection. The experimentally determined eigenfunctions of Herzog (1986) are used; these are in the form of streamwise rolls. Our model equations represent the dynamical behaviour of these rolls. We show that these equations exhibit intermittency, which we analyse using the methods of dynamical systems theory, as well as a chaotic regime. We argue that this behaviour captures major aspects of the ejection and bursting events associated with streamwise vortex pairs which have been observed in experimental work (Kline et al. 1967). We show that although this bursting behaviour is produced autonomously in the wall region, and the structure and duration of the bursts is determined there, the pressure signal from the outer part of the boundary layer triggers the bursts, and determines their average frequency. The analysis and conclusions drawn in this paper appear to be among the first to provide a reasonably coherent link between low-dimensional chaotic dynamics and a realistic turbulent open flow system. © 1988, Cambridge University Press. All rights reserved.

Description: In this paper we examine some general features of the time-dependent dynamics of drop deformation and breakup at low Reynolds numbers. The first aspect of our study is a detailed numerical investigation of the ‘ end-pinching’ behaviour reported in a previous experimental study. The numerics illustrate the effects of viscosity ratio and initial drop shape on the relaxation and/or breakup of highly elongated droplets in an otherwise quiescent fluid. In addition, the numerical procedure is used to study the simultaneous development of capillary-wave instabilities at the fluid-fluid interface of a very long, cylindrically shaped droplet with bulbous ends. Initially small disturbances evolve to finite amplitude and produce very regular drop breakup. The formation of satellite droplets, a nonlinear phenomenon, is also observed. © Copyright Cambridge University Press 1989. All rights reserved.

Description: The behaviour of concentric double emulsion droplets in linear flows is examined analytically, for the case when both fluid-fluid interfaces remain nearly spherical, and numerically, for the effect of finite interface deformation. The theoretical analysis is used to calculate the velocity fields interior and exterior to the particle, the first effects of flow-induced deformation, and the effective viscosity of a dilute emulsion of compound droplets. The numerical simulations allow for a complete investigation of the finite deformation of both the outer drop and the encapsulated particle. For concentric multiphase particles, there appear to be two distinct mechanisms of globule breakup: (i) continuous extension of the globule corresponding to non-existence of a steady particle shape or (ii) contact of the two interfaces at the globule centre, owing to incompatibility of the steady inner and outer interface shapes, even though the globule is only modestly deformed. Finally, the effect of different flow-types, i.e, uniaxial or biaxial extensional flows, is shown, in one example, to suggest breakup of the inner droplet even though the outer droplet maintains a steady shape. © 1990, Cambridge University Press. All rights reserved.

Description: Motivated by the recent work of Bajer & MOFFATT (1990), we investigate the kinematics of bounded steady Stokes flows. Specifically, we consider the streamlines inside a neutrally buoyant spherical drop immersed in a general linear flow. The Eulerian velocity field internal to the drop, known analytically, is a cubic function of position. For a wide range of parameters the internal streamlines, hence the fluid particle paths, may wander chaotically. Typical Poincare sections show both ordered and chaotic regions. The extent and existence of chaotic wandering is related to (i) the orientation of the vorticity vector relative to the principal axes of strain of the undisturbed flow and (ii) the magnitude of the vorticity relative to the magnitude of the rate-of-strain tensor. In the limit of small vorticity, we use the method of averaging to predict the size of the dominant island region. This yields the critical orientation of the vorticity vector at which this dominant island disappears so that particle paths fill almost the entire Poincare section. The problem studied here appears to be one of the simplest, physically realizable, bounded steady Stokes flows which produces chaotic streamlines. © 1991, Cambridge University Press. All rights reserved.

Description: A computer-controlled four-roll mill is used to examine two transient modes of deformation of a liquid drop: elongation in a steady flow and interfacial-tension-driven motion which occurs after the flow is stopped abruptly. For modest extensions, drop breakup does not occur with the flow on, but may occur following cessation of the flow as a result of deterministic motions associated with internal pressure gradients established by capillary forces. These relaxation and breakup phenomena depend on the initial drop shape and the relative viscosities of the two fluids. Capillary-wave instabilities at the fluid-fluid interface are observed only for highly elongated drops. This study is a natural extension of G. I. Taylor's original studies of the deformation and burst of droplets in well-defined flow fields. © 1986, Cambridge University Press. All rights reserved.

Description: Transient effects associated with the deformation and breakup of a drop following a step change from critical to suberitical flow conditions are studied experimentally and numerically. In the experiments, we consider step changes in both the shear rate and flow type for two-dimensional linear flows generated in a four-roll mill. Numerically we consider step changes in shear rate only for a uniaxial extensional flow. Depending upon the degree of deformation prior to the change in flow conditions, the drop may either return to a steady deformed shape, or continue to stretch at a reduced rate, or, for intermediate cases, the drop may break without large-scale stretching. This behaviour is a consequence of the complicated interaction between changes of shape due to interfacial tension and changes of shape due to the motion of the suspending fluid. This mode of breakup is most pronounced for high viscosity ratios, because very large extensions are necessary to guarantee breakup if the flow is stopped abruptly. For drops that are not too deformed, the sudden addition of vorticity to the external flow is characterized by rapid rotation of the drop to a new steady orientation followed by deformation and/or breakup according to the effective flow conditions at the new orientation. Finally, for viscous drops in flows with vorticity, it is demonstrated experimentally that breakup can be achieved if the initial shape is sufficiently non-spherical even though the same drop could not be made to break in the same flow at any capillary number when beginning with a near-spherical shape.

Description: The effects of surface-active agents on drop deformation and breakup in extensional flows at low Reynolds numbers are described. In this free-boundary problem, determination of the interfacial velocity requires knowledge of the distribution of surfactant, which, in turn, requires knowledge of the interfacial velocity field. We account for this explicit coupling of the unknown drop shape and the evolving surfactant distribution. An analytical result valid for nearly spherical distortions is presented first. Finite drop deformation is studied numerically using the boundary-integral method in conjunction with the time-dependent convective-diffusion equation for surfactant transport. This procedure accurately follows interfacial tension variations, produced by non-uniform surfactant distribution, on the evolving interface. The numerical method allows for an arbitrary equation of state relating interfacial tension to the local concentration of surfactant, although calculations are presented only for the common linear equation of state. Also, only the case of insoluble surfactant is studied. The analytical and numerical results indicate that at low capillary numbers the presence of surfactant causes larger deformation than would occur for a drop with a constant interfacial tension equal to the initial equilibrium value. The increased deformation occurs owing to surfactant being swept to the end of the drop where it acts to locally lower the interfacial tension, which therefore requires increased deformation to satisfy the normal stress balance. However, at larger capillary numbers and finite deformations, this convective effect competes with ‘dilution5of the surfactant due to interfacial area increases. These two different effects of surface-active material are illustrated and discussed and their influence on the critical capillary number for breakup is presented. © 1990, Cambridge University Press. All rights reserved.