ANALYSIS OF AXISYMMETRIC RADIALLY EXPANDING FREE-LIQUID SHEETS BY MEANS OF LUBRICATION THEORY

A nonlinear lubrication model for the analysis of inviscid radially expanding modulated free-liquid sheets is presented. Starting from the dimensionless continuity and momentum equations for incompressible, inviscid axisymmetric liquid sheets without surrounding gas-phase effects governing equations for sheet thickness, sheet centerline location in transverse direction as well as mean radial and transverse velocity components are derived. The resulting system of partial differential equation together with proper boundary conditions is solved numerically using central differences and a high-order Runge-Kutta ordinary differential equation (ODE)-solver. The model was validated with experimental data for the case of a stationary water bell and by comparing numerical results for modulated radially expanding sheets and small disturbance amplitudes with results obtained from linear theories (i.e., in terms of observed wavelengths) and experimental data (in terms of film disturbance envelope) for the same flow cases. For cases with larger disturbance amplitudes, nonlinear mode coupling between sinuous and varicose modes leads to film rupture, similar to that observed for planar sheets. Predicted film break-up lengths in the nonlinear regime show good agreement with experimental data and with highly resolved volume of fluid (VOF) simulations.