Numerical Simulations of Bouncing Jets

Abstract

The Kaye effect is a fascinating phenomenon of a leaping shampoo stream which was first described by Alan Kaye in 1963 as a property of non-Newtonian fluid. It manifest itself when a thin stream of non-Newtonian fluid is poured into a dish of fluid. As pouring proceeds, a small stream of liquid occasionally leaps upward from the heap. We investigate numerically the impact of the experimental setting as well as the fluid rheology on the apparition of bouncing jets. In particular, we observe the importance of the creation of a thin lubricating layer of air between the jet and the rest of the liquid. The numerical method consists of a projection method coupled with a level set formulation for the interface representation. Adaptive finite element methods are advocated to capture the different length scales inherent to this context.

In addition, we design and study two modifications of the first order standard pressure correction projection scheme for the Stokes system. The first scheme improves the existing schemes in the case of open boundary condition by modifying the pressure increment boundary condition, thereby minimizing the pressure boundary layer and recovering the optimal first order decay. The second scheme allows for variable time stepping. It turns out that the straightforward modification to variable time stepping leads to unstable schemes. The proposed scheme is not only stable but also exhibits the optimal first order decay. Numerical computations illustrating the theoretical estimates are provided for both new schemes.