Publication Details

The kinematics of a potential vortex offers an interesting flow history for a rheologically complex material, andearlier work on that subject led us to consider the behavior of a Newtonian drop in three related time dependentflow fields [K. Sarkar, W.R. Schowalter, Deformation of a two-dimensional drop at non-zero Reynolds number intime-periodic extensional flows: numerical simulation, J. Fluid Mech., 2000, submitted for publication; K. Sarkar,W.R. Schowalter, Deformation of a two-dimensional viscous drop in time-periodic extensional flows: analyticaltreatment, J. Fluid Mech., 2000, submitted for publication]. In the work reported here the drop, characterized by anupper-convected Maxwell model (UCM), is suspended in an incompressible Newtonian fluid. Again, three relatedflows are considered. The first is that of a potential vortex, modeled by an extensional flow field near the drop withrotating axes of stretching. The second is a generalization of the first and is calledrotating extensional(RE) flow,in which the frequency of revolution of the flow is varied independently of the shear rate. Finally, we consideroscillating extensional(OE) flow.Calculations were performed at small but non-zero Reynolds numbers using an ADI front-tracking/finite differ-ence method. We have developed an analytic elastic-viscous stress splitting scheme obtained by an integration byparts of the constitutive equation. The scheme explicitly separates the diffusive part of the momentum equation fora wide range of differential constitutive relations. An ADI implementation is executed for the diffusive part. Weinvestigate the effects of periodicity, Reynolds number and relaxation time on the drop dynamics. For a vortex andan RE flow, the long-time deformation reaches a steady value, and the drop attains a revolving, steady elliptic shape.The long-time values of deformation show complex non-monotonic behavior with variation in Weissenberg number,an effect of the decreased damping and increased elasticity, as well as the presence of a shear wave triggered by theUCM constitutive relation. The first two effects are modeled successfully by a simple ODE presented in AppendixA. The wave effects are briefly discussed in Appendix B.

This paper presents applications of boundary element methods to electrical impe- dance tomography. An algorithm for imaging the interior of a domain that consists of regions of constant conductivity is developed, that makes use of a simpler parametrization of the shapes of the regions to achieve efficiency. Numerical results from tests of this algorithm on synthetic data are presented, and show that the method is quite promising.

Ensemble averaging combined with multiple scattering ideas is applied to the Stokes flow over a stochastic rough surface. The surface roughness is modelled by compact protrusions on an underlying smooth surface. It is established that the effect of the roughness on the flow far from the boundary may be represented by replacing the no- slip condition on the exact boundary by a partial slip condition on the smooth surface. An approximate analysis is presented for a sparse distribution of arbitrarily shaped protrusions and explicit numerical results are given for hemispheres. Analogous conclusions for the two-dimensional case are obtained. It is shown that in certain cases a traction force develops on the surface at an angle with the direction of the flow.

Sarkar K, Prosperetti A 1995 “Effective boundary conditions for Laplace Equation with a Rough Boundary,” Proceedings of the Royal Society of London A, 451, 425-453.

A substantial amount of research on acoustic scattering by underwater bubbles is based on the theory of incoherent scattering. More recent work, devoted to much denser bubble assemblies, has instead used effective-media formulations that presuppose coherent effects. Here the mutual relationship between the two approaches is elucidated. It is shown that, underlying the incoherent results, is a WKB approximate solution of the effective equations. As an application, the scattering by tenuous subsurface bubble layers and acoustical bubble counting techniques are examined. Significant differences with previous results are found.

Sarkar K, Meneveau C 1993 “Gradients of potential fields on rough surface: perturbative calculation of the singularity distribution function f(a) for small surface,” Physical Review E, 47, 957-966.

This paper is a continuation of an earlier one [Prosperetti et al., J. Acoust. Soc. Am. 93, 3117-3127 (1993) ] in which the low-frequency backscattering of sound by hemispherical bubble clouds at the ocean's surface was studied. Here, clouds of various geometrical shapes (spheroids, spherical segments, cones, cylinders, ellipsoids) are considered and results in substantial agreement with the earlier ones and with the experiments of Chapman and Harris [J. Acoust. Soc. Am. 34, 1592-1597 ( 1962)] are found. The implication is that the backscattering levels are not strongly dependent on the shape of the clouds, which strengthens the earlier conclusion that bubble clouds produced by breaking waves can very well be responsible for the unexpectedly high backscattering levels observed experimentally. The accuracy of the Born approximation used by others for similar problems is also examined in the light of the exact results. Significant differences are found for gas concentrations by volume of the order of 0.01% or higher. Finally, shallow nonaxisymmetric plumes are briefly considered