L. Gary Leal

Buoyancy-driven motion of a deformable drop through a quiescent liquid at intermediate Reynolds numbers

Numerical solutions have been obtained for steady streaming flow past an axisymmetric drop over a wide range of Reynolds numbers (0.005 [les ] Re [les ] 250), Weber numbers (0.005 [les ] We [les ] 14), viscosity ratios (0.001 [les ] λ [les ] 1000), and density ratios (0.001 [les ] ζ [les ] 1000). Our results indicate that at lower Reynolds numbers the shape of the drop tends toward a spherical cap with increasing We , but at higher Re the body becomes more disk shaped with increasing We . Unlike the recirculating wake behind an inviscid bubble or solid particle, the eddy behind a drop is detached from the interface. The size of the eddy and the separation distance from the drop depend on the four dimensionless parameters of the problem. The motion of the fluid inside the drop appears to control the behaviour of the external flow near the body, and even for cases when λ and ζ [Lt ] 1 (a ‘real’ bubble), a recirculating wake remains unattached.

Existence of solutions for all Deborah numbers for a non-newtonian model modified to include diffusion

Abstract We consider the existence of solutions for a non-Newtonian fluid that is based upon a nonlinear dumb-bell model. It is shown that a rigorous existence proof can be obtained for solutions on a bounded domain at arbitrary values of Deborah number provided the model includes a spatial diffusion term that is usually neglected in the derivation of the model by assuming that the structure is spatially homogeneous. Although this diffusion term is critical to the existence proof, it is expected to be numerically small compared to other terms in the constitutive model, except possibly in the vicinity of very large stress gradients, which it will tend to smooth out. The proof also requires that the stress always remain bounded. Although it is likely that this will be true for a model with a nonlinear (FENE) spring, it is difficult to prove rigorously. Hence, in the existence proof we resort to an ad hoc assumption that is equivalent to asserting that the polymer breaks (degrades) if the end-to-end distance exceeds some prescribed values that is less than the full cotour length but is otherwise arbitrary.

Smoothed particle hydrodynamics techniques for the solution of kinetic theory problems Part 2. The effect of flow perturbations on the simple shear behavior of LCPs

Abstract The effect of slight perturbations to simple shear flow of liquid-crystalline polymers (LCPs) is explored by using the SPH technique to solve the unapproximated orientation distribution function equation arising from the Doi Theory. First, the case of simple shear flow is outlined, and it is shown that skewed distributions play an important role in the transition from periodic to steady behavior as the shear rate is increased. Next, we consider perturbations to flows that are slightly more extensional than simple shear, parametrized by the flow type parameter α . They are shown to eliminate all periodic director behavior (tumbling and wagging), even when the relative increment in flow type is small. At lower shear rates (or more properly, lower Peclet number Pe based upon the rotational diffusivity), the elimination occurs through a homoclinic bifurcation, the transition being rather abrupt as the flow type is changed. At higher Pe, periodic behavior is suppressed more gradually through a Hopf bifurcation, with tumbling being replaced by wagging and negative θ flow-aligning, where θ is the angle of the director in the shear plane. The effect of these perturbations on rheological behavior is also explored. As the flow is made slightly more extensional, the zero-shear rate limiting value of the generalized viscosity η decreases dramatically, due to the slowing down of tumbling as the system approaches a homoclinic orbit; as Pe is increased, the viscosity rises again before falling, due to the induction of wagging behavior where tumbling would normally prevail in simple shear. Finally, it is found when the flow type is changed sufficiently, the interesting, non-monotonic behavior of rheological functions seen in simple shear of LCPs is replaced by monotonic behavior, even though the flow is still relatively close to simple shear.

Conservation and constitutive equations for adsorbed species undergoing surface diffusion and convection at a fluid-fluid interface

Abstract A rigorous mathematical theory is presented for mass transfer of a solute by diffusion and convection at and near an interface between two immiscible solvent liquids. The objective of this study is to determine, for a simple model system, the conditions under which a detailed resolution of the concentration and mass flux profiles near the interface can be replaced (insofar as macroscopically observable profiles are concerned) with jump conditions that relate the bulk or macroscopic concentration and flux values on the two sides of the interface. The “statistical—mechanical” model underlying this theoretical development consists of spherical Brownian particles, either wholly immersed in one of the two contiguous fluids, or else straddling the interface. Adsorption forces tending to cause accumulation of the Brownian “surfactant” particles at the interface are regarded as deriving from a position-dependent potential energy function. This system is characterized by three independent length scales: a macroscale L , which is relevant to solute concentration gradients in the bulk fluids away from the interface; and two smaller scales: I , which is characteristic of the potential energy gradients normal to the interface, and I 1 , which provides a proper scale for the hydrodynamic “wall effects” of the fluid interface upon the particle motion. It is the macroscale L which is characteristic of the normal continuum description of solute mass transfer. Relative to L , the microscales l and l 1 are vanishingly small. Singular perturbation techniques are employed to provide a complete spatial resolution of solute concentration and flux profiles, right down to the scales l 1 and/or l , with δ = l/L and δϵ = l 1 /L is independent small parameters. Three distinct situations emerge for δ ⪡ 1, depending upon the value of ϵ and the rate of decrease of any hydrodynamic wall effects with increasing distance from the undeformed interface. First, when ϵ ⪡ 1 and/or the wall effects fall off faster than linearly with increasing distance, a set of local interface jump conditions can be derived which includes both the effects of the potential energy of attraction/repulsion and hydrodynamic wall effects, while transport in the contiguous bulk-phase fluids occurs at a rate that is consistent with the advection velocity and diffusivity in an infinite, unbounded fluid. Second, when δ ⪡ 1, but ϵδ = O(1), and “wall effects” again fall off faster than linearly, a set of local interface jump conditions can still be derived, but these must be supplemented by “hindered” transport corrections for advection and diffusion in the bulk-phase fluids. Finally, in other circumstances, we show that a local theory is not possible; here, one cannot ignore the detailed features near the interface. In those cases where local jump conditions can be derived rigorously, the qualitative features of these interfacial jump conditions are discussed in order to obtain a physical understanding of interfacial transport processes for our model system. It is shown that rigorously derived interface conditions for these cases resemble the normally assumed macroscopic jump conditions for interfacial mass transfer. We consider particularly those conditions for which interface transport effects are macroscopically significant.

Particle motion in Stokes flow near a plane fluid-fluid interface. Part 2. Linear shear and axisymmetric straining flows

We consider the motion of a sphere or a slender body in the presence of a plane fluid–fluid interface with an arbitrary viscosity ratio, when the fluids undergo a linear undisturbed flow. First, the hydrodynamic relationships for the force and torque on the particle at rest in the undisturbed flow field are determined, using the method of reflections, from the spatial distribution of Stokeslets, rotlets and higher-order singularities in Stokes flow. These fundamental relationships are then applied, in combination with the corresponding solutions obtained in earlier publications for the translation and rotation through a quiescent fluid, to determine the motion of a neutrally buoyant particle freely suspended in the flow. The theory yields general trajectory equations for an arbitrary viscosity ratio which are in good agreement with both exact-solution results and experimental data for sphere motions near a rigid plane wall. Among the most interesting results for motion of slender bodies is the generalization of the Jeffrey orbit equations for linear simple shear flow.

Particle motion in Stokes flow near a plane fluid-fluid interface. Part 1. Slender body in a quiescent fluid

The present study examines the motion of a slender body in the presence of a plane fluid–fluid interface with an arbitrary viscosity ratio. The fluids are assumed to be at rest at infinity, and the particle is assumed to have an arbitrary orientation relative to the interface. The method of analysis is slender-body theory for Stokes flow using the fundamental solutions for singularities (i.e. Stokeslets and potential doublets) near a flat interface. We consider translation and rotation, each in three mutually orthogonal directions, thus determining the components of the hydrodynamic resistance tensors which relate the total hydrodynamic force and torque on the particle to its translational and angular velocities for a completely arbitrary translational and angular motion. To illustrate the application of these basic results, we calculate trajectories for a freely rotating particle under the action of an applied force either normal or parallel to a flat interface, which are relevant to particle sedimentation near a flat interface or to the processes of particle capture via drop or bubble flotation.

Thermocapillary motion of a deformable drop toward a planar wall

Abstract The thermocapillary migration of a deformable drop moving normal to a planar wall is considered. The temperature is assumed to be constant at the wall and approaches a linear field far from the wall. Inertial effects as well as thermal convection are neglected. All physical parameters save surface tension are assumed to be constant and the motion is assumed to be quasi-steady and axisymmetric. A numerical solution procedure based on the boundary integral method is applied. In particular, Greens functions appropriate to the planar wall configuration are used for both the thermal and fluid dynamic portions of the problem. The effects of viscosity ratio, thermal conductivity ratio, and dimensionless rate of change of surface tension with temperature are discussed. Degree of deformation is found to increase with increasing effective capillary number.

Thin Fluid Film Squeezed With Inertia Between Two Parallel Plane Surfaces

The present study is concerned with estimating the inertial effects on the draining of thin fluid layer between two parallel plane boundaries. In particular, we consider the case in which an initially stationary object with a circular plane lower surface begins suddenly moving under the action of a constant applied force toward a parallel plane wall when the inertia of the object and that of the intervening fluid in the gap are not negligible. The method of solution is a matched asymptotic expansion involving characterization of the solution by different characteristic time scales in different parts of the solution domain in the limit of small but finite Reynolds number based on the gap height

Effect of Molecular Weight and Flow Type on Flow Birefringence of Dilute Polymer Solutions

Birefringence measurements can be used to study conformation changes in a polymer solution due to the presence of flow. In the dilute concentration range, such studies have been carried out for simple shear flow1 and uniaxial extensional flow,2 but only for a single molecular weight in each case and for concentrations above 300 ppm. Neither the effect of flow type nor molecular weight has been adequately investigated, though each may be extremely important in “applications” such as the well-known “drag reduction” phenomenon. The present paper summarizes the results of flow birefringence experiments for 50 and 100 ppm solutions of three different MW samples of polystyrene dissolved in a viscous poly-chlorinated biphenyl solvent. A range of two-dimensional flow types was generated using a four roll mill.