John S. Lowengrub

Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration

This is the first paper in a two-part series in which we analyze two model systems to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell with arbitrary density and viscosity contrast between the components. The systems stem from a simplification of a general system of equations governing the motion of a binary fluid (NSCH model [Lowengrub and Truskinovsky, Proc. R. Soc. London, Ser. A 454, 2617 (1998)]) to flow in a Hele-Shaw cell. The system takes into account the chemical diffusivity between different components of a fluid mixture and the reactive stresses induced by inhomogeneity. In one of the systems we consider (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we motivate, present and calibrate the HSCH/BHSCH equations so as to yield the classical sharp interface model as a limiting case. We then analyze their equilibria, one dimensional evolution and linear stability. In the second paper [paper II, Phys. Fluids 14, 514 (2002)], we analyze the behavior of the models in the fully nonlinear regime. In the BHSCH system, the equilibrium concentration profile is obtained using the classical Maxwell construction [Rowlinson and Widom, Molecular Theory of Capillarity (Clarendon, Oxford, 1979)] and does not depend on the orientation of the gravitational field. We find that the equilibria in the HSCH model are somewhat surprising as the gravitational field actually affects the internal structure of an isolated interface by driving additional stratification of light and heavy fluids over that predicted in the Boussinesq case. A comparison of the linear growth rates indicates that the HSCH system is slightly more diffusive than the BHSCH system. In both, linear convergence to the sharp interface growth rates is observed in a parameter controlling the interface thickness. In addition, we identify the effect that each of the parameters, in the HSCH/BHSCH models, has on the linear growth rates. We then show how this analysis may be used to suggest a set of modified parameters which, when used in the HSCH/BHSCH systems, yield improved agreement with the sharp interface model at a finite interface thickness. Evidence of this improved agreement may be found in paper II.

Modeling pinchoff and reconnection in a Hele-Shaw cell. II. Analysis and simulation in the nonlinear regime

This is the second paper in a two part series in which we analyze two diffuse interface models to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell with arbitrary density and viscosity contrast between the components. Diffusion between the components is limited if the components are macroscopically immiscible. In one of the systems (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we focus on buoyancy driven flow and the Rayleigh–Taylor instability. In the fully nonlinear regime before pinchoff, results from the HSCH and BHSCH models are compared to highly accurate boundary-integral simulations of the classical sharp interface system. In this case, we find that the diffuse interface models yield nearly identical results and we demonstrate convergence to the boundary-integral solutions as the interface thickness vanishes. We find that the break-up of an unstabl...

Pinch off and reconnection in liquid/liquid flows: joint experimental and numerical studies

Liquid/liquid systems appear in applications involving transport, mixing, and separation of petroleum, chemical, and waste products. Breakup and coalescence transitions often determine flow regimes as well as reaction and separation rates. Because they occur over very small time and length scales compared with the larger scales that dominate the flow, they are difficult to quantify experimentally and simulate numerically. Thus far, no accurate models exist for engineers to predict these flows. Experiments and computations were performed so that accurate engineering models can be developed. Jet pinch off and drop coalescence were examined in mixtures of water/glycerin and silicone oil. Index matching, laser sheet illumination, and the PIV method were applied to obtain visualization and velocity field sequences through transitions. The computations used a novel, physically-based method that captures interface breakup and coalescence automatically without resorting to ad-hoc cut-and-connect methods. To achieve enhanced accuracy near transitions, new adaptive time and space meshes were developed. The computations were validated through direct comparison with the experiments. The detailed results should lead to improved understanding of transition behavior. This understanding is needed to develop engineering models of multiphase flows. Such predictive models will lead to extensive cost savings in device and process design.