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: A theoretical and experimental investigation of drop motion in rotating fluids is presented. The theory describing the vertical on-axis translation of an axisymmetric rigid body through a rapidly rotating low-viscosity fluid is extended to the case of a buoyant deformable fluid drop of arbitrary viscosity. In the case that inertial and viscous effects are negligible within the bulk external flow, motions are constrained to be two-dimensional in compliance with the Taylor-Proudman theorem, and the rising drop is circumscribed by a Taylor column. Calculations for the drop shape and rise speed decouple, so that theoretical predictions for both are obtained analytically. Drop shapes are set by a balance between centrifugal and interfacial tension forces, and correspond to the family of prolate ellipsoids which would arise in the absence of drop translation. In the case of a drop rising through an unbounded fluid, the Taylor column is dissipated at a distance determined by the outer fluid viscosity, and the rise speed corresponds to that of an identically shaped rigid body. In the case of a drop rising through a sufficiently shallow plane layer of fluid, the Taylor column extends to the boundaries. In such bounded systems, the rise speed depends further on the fluid and drop viscosities, which together prescribe the efficiency of the Ekman transport over the drop and container surfaces. A set of complementary experiments is also presented, which illustrate the effects of drop viscosity on steady drop motion in bounded rotating systems. The experimental results provide qualitative agreement with the theoretical predictions; in particular, the poloidal circulation observed inside low-viscosity drops is consistent with the presence of a double Ekman layer at the interface, and is opposite to that expected to arise in non-rotating systems. The steady rise speeds observed are larger than those predicted theoretically owing to the persistence of finite inertial effects. © 1995, Cambridge University Press. All rights reserved.

Description: The motion of membrane-bound objects is important in many aspects of biology and physical chemistry. A hydrodynamic model for this Fconfiguration was proposed by Saffman & Delbrück (1975) and here it is extended to study the translation of a disk-shaped object in a viscous surface film overlying a fluid of finite depth H. A solution to the flow problem is obtained in the form of a system of dual integral equations that are solved numerically. Results for the friction coefficient of the object are given for a complete range of the two dimensionless parameters that describe the system: the ratio of the sublayer (η) to membrane (ηm) viscosities, Λ=ηR/ηmh (where R and h are the object radius and thickness of the surface film, respectively), and the sublayer thickness ratio, H/R. Scaling arguments are presented that predict the variation of the friction coefficient based upon a comparison of the different length scales that appear in the problem: the geometric length scales H and R, the naturally occurring length scale [lscr ]m=ηmh/η, and an intermediate length scale [lscr ]H= (ηmhH/η)1/2. Eight distinct asymptotic regimes are identified based upon the different possible orderings of these length scales for each of the two limits Λ[Lt ]1 and Λ[Gt ]1. Moreover, the domains of validity of available approximations are established. Finally, some representative surface velocity fields are given and the implication of these results for the characterization of hydrodynamic interactions among membrane-bound proteins adjacent to a finite-depth sublayer is discussed briefly.

Description: Oscillatory translational and rotational motions of small particles in viscous fluids are studied for two cases: (i) circular disks and (ii) nearly spherical particles. For circular disks, four motions are treated: broadside and edgewise oscillatory translations and out-of-plane and in-plane oscillatory rotations. In each case the unsteady Stokes equations are reduced to dual integral equations and solved exactly for all frequencies. Streamline portraits of the flow fields are used to understand the evolution of the velocity and pressure fields. The motions of nearly spherical particles are then studied using the reciprocal theorem. Asymptotic formulae for the hydrodynamic resistance tensors are derived and discussed.

Description: When two drops of radius R touch, surface tension drives an initially singular motion which joins them into a bigger drop with smaller surface area. This motion is always viscously dominated at early times. We focus on the early-time behaviour of the radius rm of the small bridge between the two drops. The flow is driven by a highly curved meniscus of length 2πrm and width Δ ≪ rm around the bridge, from which we conclude that the leading-order problem is asymptotically equivalent to its two-dimensional counterpart. For the case of inviscid surroundings, an exact two-dimensional solution (Hopper 1990) shows that Δ ∝ rm3 and rm ∼ (tγ/πη) ln [tγ/(ηR)]; and thus the same is true in three dimensions. We also study the case of coalescence with an external viscous fluid analytically and, for the case of equal viscosities, in detail numerically. A significantly different structure is found in which the outer-fluid forms a toroidal bubble of radius Δ ∝ rm3/2 at the meniscus and rm ∼ (tγ/4πη) ln [tγ/(ηR)]. This basic difference is due to the presence of the outer-fluid viscosity, however small. With lengths scaled by R a full description of the asymptotic flow for rm(t) ≪ 1 involves matching of lengthscales of order rm2 rm3/2, rm, 1 and probably rm7/4.

Description: A numerical technique, based on the boundary integral method, is developed to allow the modelling of unsteady free-surface flows at large Reynolds numbers in cases where the surface is contaminated by some surface-active compound. This requires the method to take account of the tangential stress condition at the interface and is achieved through a boundary-layer analysis. The constitutive relation that forms the surface stress condition is assumed to be of the Boussinesq type and allows the incorporation of surface shear and dilatational viscous forces as well as Marangoni effects due to gradients in surface tension. Sorption kinetics can be included in the model, allowing calculations for both soluble and insoluble surfactants. Application of the numerical model to the problem of bursting gas bubbles at a free surface shows the greatest effect to be due to surface dilatational viscosity which drastically reduces the amount of surface compression and can slow and even prevent the formation of a liquid jet. Surface tension gradients give dilatational elasticity to the surface and thus also significantly prevent surface compression. Surface shear viscosity has a smaller effect on the interface motion but results in initially increased surface concentrations due to the sweeping up of surface particles ahead of the inward-moving surface wave. © 1995, Cambridge University Press. All rights reserved.

Description: A thin rigid disk translates edgewise perpendicular to the rotation axis of an unbounded fluid undergoing solid-body rotation with angular velocity Q. The disk face, with radius a, is perpendicular to the rotation axis. For arbitrary values of the Taylor number, ℱ = Ωa2/v, and in the limit of zero Reynolds number ℛe, the linearized viscous equations reduce to a complex-valued set of dual integral equations. The solution of these dual equations yields an exact representation for the velocity and pressure fields senerated by the translating disk. For large rotation rates. ℱ ≫ 1, the O(1) disturbance velocity field is confined to a thin O(ℱ-1/2) boundary layer adjacent to the disk. Within this boundary layer, the flow field near the disk centre undergoes an Ekman spiral similar to that created by a nearly geostrophic flow adjacent to an infinite rigid plate. Additionally, flow within the boundary layer drives a weak O(ℱ-1/2) secondary flow which extends parallel to the rotation axis and into the far field. This flow consists of two counter-rotating columnar eddies, centred over the edge of the disk, which create a net in-plane flow at an angle of 45° to the translation direction of the disk. Fluid is transported axially toward/away from the disk within the core of these eddies. The hydrodynamic force (drag and lift) varies as O(ℱ1/2) for ℱ ≫ 1; this scaling is consistent with the viscous stresses created in the Ekman boundary layer. Additionally, an approximate expression, suitable for all Taylor numbers, is given for the hydrodynamic force on a disk translating broadside along the rotation axis and edgewise transverse to the rotation axis. © 1995, Cambridge University Press. All rights reserved.

Description: Deformation due to hydrodynamic interactions between two deformable buoyant drops may result in the alignment and coalescence of horizontally offset drops. Three-dimensional boundary integral calculations are presented for systems containing two, three or four drops and it is argued that the interactions which occur between three drops or four drops may be characterized qualitatively by the two-drop interactions. In a dilute monodisperse suspension, the rate of coalescence of deformable drops is calculated using far-field analytical results and is found to be proportional to the Bond number. The rate of coalescence in a dilute polydisperse suspension of bubbles in corn syrup is determined by performing a large number of laboratory experiments for Bond numbers based on the larger bubble radius 15 〈 B 〈 120. The rate of coalescence is enhanced (by a factor of 10 for B = 10), owing to the effects of deformation, compared to the predictions of models which include hydrodynamic interactions and van der Waals forces among spherical bubbles. The rate of coalescence is greater than the rate predicted by the Smoluchowski model which ignores all hydrodynamic interactions. The experimental results are used to calculate the evolution of the bubble size distribution in suspensions using a standard one-dimensional population dynamics model; deformation affects the size distribution in suspensions, resulting in a wider range of bubble sizes. © 1995, Cambridge University Press. All rights reserved.

Description: An analytic formula is derived for the Rayleigh growth rate of a fluid cylinder immersed in a fluid of equal viscosity, and an extension is given for concentric fluid threads.

Description: Viscous stretching of a cigar-shaped drop due to the centrifugal pressure field in a surrounding rapidly rotating denser fluid is analysed. Scaling arguments are used to examine the various contributions to the viscous stresses resisting deformation, and a number of asymptotic regimes are identified which are delineated by the relative magnitudes of the aspect ratio, the viscosity ratio and unity. These asymptotic regimes may usefully be described as the bubble, pipe, sliding-rod and toffee-strand limits. Detailed analysis based upon a slenderness assumption combined with an integral representation of Stokes equations is used to derive evolution equations for the shape of the drop as a function of time in the different regimes. In the limit that interfacialtension effects are negligible, similarity solutions are developed in which the length of the drop is found to increase as t2/5, t1/4, (tInt)1/4 and t. The analytical results are in good agreement with numerical simulations based upon a boundary-integral solution to the full viscous flow equations.