The deformation of an immiscible toroidal drop embedded in axisymmetric compressional Stokes flow is analysed via the boundary integral formulation in the case of equal viscosity. Numerical simulations are performed for the drop having initially the shape of a torus with circular cross-section. The quasi-stationary dynamic simulations reveal that, when the viscous forces, proportional to the intensity of the flow, are relatively weak compared with the surface tension (the ratio of these forces is characterized by the capillary number, Ca), three different scenarios of drop evolution are possible: indefinite expansion of the liquid torus, contraction to the centre and a stationary toroidal shape. When the intensity of the flow is low, the stationary shapes are shown to be close to circular tori. Once the outer flow strengthens, the cross-section of the stationary torus assumes first an elliptic and then an egg-like shape. For the capillary number greater than a critical value, Ca cr , toroidal stationary shapes were not found. Remarkably, Ca cr is close to the critical capillary number found previously for a simply connected drop flattened in compressional flow. Thus, a new example of non-uniqueness of stationary drop shape in viscous flow is obtained. Approximate stationary solutions in the form of tori with circular and elliptic cross-sections are obtained by minimizing the normal velocity over the drop interface. They are shown to be in good agreement with the stationary shapes from quasi-dynamic simulations for the corresponding intervals of the capillary number.

The Stokes equations describing the creeping motion of two arbitrary-sized touching spheres are solved exactly through the use of tangent-sphere coordinates. For the case of a linear shear field at infinity, explicit results covering the entire range of size ratios are presented for: (a) the forces and torques on the aggregate; (b) the hydrodynamic forces on the individual spheres comprising a freely suspended aggregate, which are in general non-zero; (c) the contribution of the pair to the bulk stress of a dilute suspension; and (d) under free suspension conditions, the velocity of any material point relative to that of the undisturbed flow.

The dynamics of the deformation of a drop in axisymmetric compressional viscous flow is addressed through analytical and numerical analyses for a variety of capillary numbers, $\mathit{Ca}$, and viscosity ratios, $\lambda $. For low $Ca$, the drop is approximated by an oblate spheroid, and an analytical solution is obtained in terms of spheroidal harmonics; whereas, for the case of equal viscosities ($\lambda = 1$), the velocity field within and outside a drop of a given shape admits an integral representation, and steady shapes are found in the form of Chebyshev series. For arbitrary $Ca$ and $\lambda $, exact steady shapes are evaluated numerically via an integral equation. The critical $\mathit{Ca}$, below which a steady drop shape exists, is established for various $\lambda $. Remarkably, in contrast to the extensional flow case, critical steady shapes, being flat discs with rounded rims, have similar degrees of deformation ($D\sim 0. 75$) for all $\lambda $ studied. It is also shown that for almost the entire range of $\mathit{Ca}$ and $\lambda $, the steady shapes have accurate two-parameter approximations. The validity and implications of spheroidal and two-parameter shape approximations are examined in comparison to the exact steady shapes.

The closed equations for the velocity correlation tensor and for the mean-squared displacement of a particle suspended in a stationary homogeneous turbulent flow, with an arbitrary linear law of fluid-particle interaction, are obtained using two assumptions suggested previously for the problem of turbulent self-diffusion: the ‘independence approximation’ and the Gaussian property of the functional distribution of particle velocities. The numerical solution of the derived equations is given for an isotropic system with a model turbulence spectrum. The following characteristics of the particle motion are obtained: (a) the mean kinetic energy, (b) diffusivity, (c) rate of energy dissipation, (d) velocity correlation function, and (e) the correlation function of the relative fluid-particle velocity. The impact of various spectral modes on the characteristics of the particle motion is discussed.

Abstract