J.M. Rallison

A numerical study of the deformation and burst of a viscous drop in an extensional flow

We study the deformation and conditions for breakup of a liquid drop of viscosity λμ freely suspended in another liquid of viscosity μ with which it is immiscible and which is being sheared. The problem at zero Reynolds number is formulated exactly as an integral equation for the unknown surface velocity, which is shown to reduce to a particularly simple form when Δ = 1. This equation is then solved numerically, for the case in which the impressed shear is a radially symmetric extensional flow, by an improved version of the technique used, for Δ = 0, by Youngren & Acrivos (1976) so that we model the time-dependent distortion of an initially spherical drop. It is shown that, for a given Δ, a steady shape is attained only if the dimensionless group Ω ≡4π G μ a /γ lies below a critical value Ω c (Δ), where G refers to the strength of the shear field, a is the radius of the initial spherical drop and γ is the interfacial tension. On the other hand, when Ω > Ω c the drop extends indefinitely along its long axis. The numerical results for Δ = 0·3, 0·5, 1, 2, 10 and 100 are in good agreement with the predictions of the small deformation analysis by Taylor (1932) and Barthes-Biesel & Acrivos (1973) and, at the smaller Δ, with those of slender-body theory (Taylor 1964; Acrivos & Lo 1978).

Creeping flow of dilute polymer solutions past cylinders and spheres

Abstract A dilute polymer solution is modelled as a suspension of dumbbells with finite extensibility. Time-dependent numerical calculations are performed of flow part cylindrical and spherical surfaces at low Reynolds number. A finite-difference scheme is employed in which the evolution in time of the dumbbells is followed from an initially unstretched equilibrium. Results are calculated with (i) a no-slip, and (ii) a zero-tangential-stress boundary condition at the body surface. At large Deborah number, D , the polymer is most highly stretched in thin regions of fluid close to and downstream of stagnation points of the flow. The most important region dynamically is found to be at the rear of the obstacle. Numerical refinements in space and time are included in order properly to resolve this fine-scale structure. Numerically stable results are obtained for values of D up to 16, and show that the flow field and drag force on the obstacle tend toward finite values at large D . Experimental measurements of the drag on a falling rigid sphere, and the velocity distribution around it, are compared with the numerical results for the no-slip boundary. Observations of bubble behaviour are discussed in the light of the results for the slip boundary.

The time-dependent deformation of a capsule freely suspended in a linear shear flow

An analysis is presented of the dynamics of a small deformable capsule freely suspended in a viscous fluid undergoing shear. The capsule consists of an elastic membrane which encloses another viscous fluid, and it deforms in response to the applied external stresses and the elastic forces generated within the membrane. Equations are derived which give its time-dependent deformation in the limit that the departure of the shape from sphericity is small. The form of the shear flow is arbitrary and a general (two-dimensional) elastic material is considered. Limiting forms are obtained for highly viscous capsules and for membranes which are area-preserving, and earlier results for surface tension droplets and incompressible isotropic membranes are derived as particular cases. Results for the viscosity of a dilute suspension of capsules are also given. The theoretical prediction for the relaxation rate of the shape is derived for an interface which has elastic properties appropriate for a red-blood-cell membrane, and is compared with experimental observations of erythrocytes.

The lubrication force between two viscous drops

The hydrodynamic force resisting the relative motion of two unequal drops moving along their line of centers is determined for Stokes flow conditions. The drops are assumed to be in near‐contact and to have sufficiently high interfacial tension that they remain spherical. The squeeze flow in the narrow gap between the drops is analyzed using lubrication theory, and the flow within the drops near the axis of symmetry is analyzed using a boundary integral technique. The two flows are coupled through the nonzero tangential stress and velocity at the interface. Depending on the ratio of drop viscosity to that of the continuous phase, and also on the ratio of the distance between the drops to their reduced radius, three possible flow situations arise, corresponding to nearly rigid drops, drops with partially mobile interfaces, and drops with fully mobile interfaces. The results for the resistance functions are in good agreement with an earlier series solution using bispherical coordinates. These results have i...

A numerical study of the deformation and burst of a viscous drop in general shear flows

The time-dependent deformation and burst of a viscous drop in an arbitrary shear flow at zero Reynolds number is studied. The viscosities of the drop and the suspending fluid are assumed to be equal. A numerical scheme to track the (non-axisymmetric) drop shape in time is presented, and used to investigate the deformation induced by two-dimensional shear and orthogonal rheometer flows. Steady deformations, critical flow rates and burst modes are determined, and compared with asymptotic (small and large) deformation theories, and with experiment.

Note on the time-dependent deformation of a viscous drop which is almost spherical

The theory of the shear-induced small deformation of viscous drops at zero Reynolds number is reviewed. The general result for arbitrary shear and surface tension is presented, and the asymptotic forms for weak flow and for high internal viscosity are derived from it. In the latter case, numerical solutions are compared with the experiments of Torza, Cox & Mason (1972).

DO WE UNDERSTAND THE PHYSICS IN THE CONSTITUTIVE EQUATION

Abstract The failure of some careful attemps to provide numerical solutions of the equations for non-Newtonian flow suggests to us some inadequacies of the constitutive equations. (After all no one would doubt the validity of the conservation of mass and momentum.) To understand the physics in the constitutive equation, and thence to correct its undesirable features, it is helpful to look at a micro-structural model which leads to the constitutive equation. The bead-and-spring dumbbell model for a dilute polymer solution leads to an Oldroyd-like equation. The simplest version of the bead-and-spring model has a linear spring and a constant friction coefficient for the beads. While this model is simple and usefully combines viscous and elastic behaviour, it has the very unphysical feature of blowing up in strong straining flows (i.e. at a Deborah number in excess of unity), with the spring lengthening indefinitely in time and the steady extensional viscosity becoming unbounded at a critical flow strength. The hope that the corresponding large stresses would not occur in a flow calculation seems to have been misguided: some simple examples show that the large stresses may not act through the momentum equation to inhibit the flow. To cure this unphysical behaviour one clearly needs to use a non-linear spring force which gives a finite limit to the extension. Incorporating this modification into the constitutive equation enables the numerical solution of (some) flow problems to proceed to large Deborah numbers. Care is of course still needed in the numerical calculations, for example in resolving thin layers of high stress. (A boundary layer theory needs to be developed for the nonlinearity introduced by the non-Newtonianness.) A further modification of the bead-and-spring model may be necessary if argument is sought between numerical calculations and experiments. Many flows of interest subject the fluid to a sudden strong strain. In such circumstances the polymer chains will not be in thermodynamic equilibrium and so will not give the standard entropic spring. It may be possible to model this behaviour by a large temporary internal viscosity.

A split Lagrangian-Eulerian method for simulating transient viscoelastic flows

A novel numerical method for simulating time-dependent flow of viscoelastic fluids derived from dumbbell models is described. The constitutive equation is solved in a co-deforming frame, where the natural time-derivative is the upper-convected derivative. Mesh reconnection is achieved using a variant of Delaunay triangulation. The velocity and pressure are found via a finite element solution of the momentum equations. The method is tested by applying it to the benchmark problem of a sphere falling along the axis of a cylindrical tube.

The instability mechanism for two elastic liquids being co-extruded

Chen has recently shown (J. Non-Newtonian Fluid Mech., 40 (1991) 155-175) that a jump in the first normal stress difference between two elastic fluids in core-annular flow leads to a longwave varicose instability when the core is more elastic and occupies less than 32% of the cross-section, or when the annulus is more elastic and occupies less than 68% of the cross-section. The purpose of this note is to present a simplified calculation which reveals the physical mechanism, to incorporate the effect of a jump in the second normal stress difference, and to demonstrate the existence of a non-axisymmetric sinuous longwave instability that grows more rapidly than the varicose mode when the annulus is more elastic than the core.

The effect of particle interactions on dynamic light scattering from a dilute suspension

The calculation of dynamic laser-light scattering by dilute suspensions of Brownian particles is reviewed. It is shown that present theories of diffusion can provide approximations for the autocorrelation of the intensity of the scattered light that are only uniformly accurate for correlation times up to order ( D 0 k 2 ) −l where D0 is the diffusivity of a single particle and k is the scattering wave vector. The meanings of, and connections between, down-gradient, self- and tracer diffusion for both short and long times are established and it is shown how these may be inferred from light-scattering experiments for optically monodisperse and polydisperse systems. For dilute systems, equations giving the time evolution of the intermediate and self-intermediate scattering functions, F ( k , t ) and F s ( k , t ), accurate to first order in the volume concentration of particles are constructed, and are solved for suspensions of hard spheres with and without hydrodynamic interaction. For short and long times (semi-) analytic solutions are given; for intermediate times numerical results are presented. The formal correspondence of the limiting values of the time-dependent solutions with the results of Batchelor (1976, 1983) and others for steady sedimentation in polydisperse systems is established.