A three-dimensional linear stability analysis for two-dimensional free boundary flows by the finite-element method

The stability of two-dimensional, steady flows of Newtonian liquid with free boundaries affected by surface tension to small, three-dimensional disturbances is calculated by the finite-element method. The only simplification made is the neglect of the time partial derivative in each component of the Navier-Stokes equation, which leads to a major reduction in the order of the resulting matrix eigenvalue problem. The disturbances introduced vary sinusoidally in the third dimension and exponentially in time, and as a result the two-dimensional finite-element grid used for the base flow calculations can be used also for the stability calculations. Stability predictions for a stationary layer and for air displacing liquid in a Hele-Shaw cell show that the method can be effective.