REGENTS OF THE UNIVERSITY OF MINNESOTA
This project carries out mathematical and numerical studies of potential flows of viscous and viscoelastic fluids. The Helmholtz decomposition divides the velocity into a rotational part and an irrotational part. In general, the rotational and irrotational velocities are both required and they are tightly coupled at the boundary, especially when the no-slip condition is applied. The project's goal is to develop a framework for the analysis of both components in the Helmholtz decomposition. Previous NSF supported studies of this topic by the principal investigator's group showed that the common opinion that potential flows are irrotational motions of inviscid fluids in general is not correct: it is neither necessary nor useful to put the viscosity to zero. There are essentially two irrotational theories: viscous potential flow and the dissipation method. Viscous potential flow works best for gas-liquid flows where the viscous normal stress is computed from the irrotational component. The dissipation method (introduced by Stokes in 1851) is based on the self-equilibration of viscous stresses for irrotational flow which do not give rise to forces in the equations: they do work and give rise to energy and dissipation. Unfortunately, predicting which of these two irrotational theories will give the better result is not known a priori but is one of the main goals of this study.
Viscous potential flow can be used to study a variety of physical phenomena, including cavitation, capillary breakup and rupture, Rayleigh-Taylor and Kelvin-Helmholtz instabilities (drop and jet breakup), phase change problems involving heat and mass transfer (nuclear reactors), the viscous decay of capillary-gravity waves, waves and rupture of moving thin films, Hele-Shaw flows (oil recovery), etc. It is important to recognize that the theory of potential flows of viscous fluids is a viable topic with a rich physical content.