University of Alberta
Edmonton, Alberta, Canada
Vortex dynamics at fluid interfaces are discussed. Special emphasis is placed on the dynamics near a free-surface. This work was motivated by earlier work on the creation of vortex rings by water drops impacting a pool [1]. These results are also relevant to those studying ocean surface dynamics and and other interfacial phenomena. The kinematics required at a fluid interface where tangential components of surface traction vanish are presented [2]. The vanishing tangential surface tractions force one principal axis of strain to remain aligned with the surface normal. From the rotation rate of the surface normal the level of vorticity required at the interface is calculated. Previous results restricted to special geometries are generalized and the role of the surface geometry is discussed. Effects of geometry on the flux of vorticity from a free-surface are also discussed. Special attention is paid to situations where curvature-dependent contributions to the vorticity flux may be neglected. The topology of vortex lines embedded in the surface is discussed in this context. These results show that vortex lines may be straight and geometry-induced vorticity flux is produced; conversely vortex lines may be curved and no geometry-induced vorticity flux is produced. A convenient method for assessing vorticity flux from a steady surface based on Gaussian curvature is derived. Finally, the creation of vorticity at fluid interfaces is discussed in the context of the vorticity transport equation. Here, an interface is modelled as a thin region across which density and viscosity vary rapidly but smoothly. Viscosity gradients in this region add an extra term to the usual Navier Stokes equation: %% \begin{equation} \rho\mathbf{a}= -{\rm grad} p + \mu {\rm div} {\rm grad}\ \mathbf{u} + \mathbf{D}\,{\rm grad} \mu \end{equation} %% Here, $\rho$ is the density $\mu$ is the dynamic viscosity and $\mathbf{D}$ is the rate-of-strain tensor. $\mathbf{u}$ is the velocity and $\mathbf{a}$ is the acceleration. Calculating the curl of this equation reveals curvature-dependent vorticity production terms. These new terms show vorticity is created not only by baroclinic torques but by torques due to viscosity gradients and a coupling of density gradients with viscosity gradients. These results are compared to those predicting the level of vorticity required at a free-surface discussed earlier.
References:
[1] Bill Peck and Lorenz Sigurdson. The three-dimensional vortex structure of an impacting water drop. {\em Phys. Fluids}, 6\penalty0 (2):\penalty0 564--576, February 1994.
[2] Bill Peck and Lorenz Sigurdson. On the kinematics at a free surface. {\em To appear IMA Journal of Applied Mathematics}, 1998.