On the instability of a free viscous rim

This paper is devoted to the theoretical description of the dynamics of a rim formed by capillary forces at the edge of a free, thin liquid sheet. The rim dynamics are described using a quasi-one-dimensional approach accounting for the inertia of the liquid in the rim and for the liquid flow entering the rim from the sheet, surface tension and viscous stresses. The governing equations are derived from the mass, momentum and moment-of-momentum-balance equations of the rim. The theory provides a basis from which to analyse the linear stability of a straight line rim bounding a planar liquid sheet. The combined effect of the axisymmetric disturbances of the radius of the rim cross-section as well as of the transverse disturbances of the rim centreline is considered. The effect of the viscosity, relative film thickness and rim deceleration are investigated. The predicted wavelength of the most unstable mode is always very similar to the Rayleigh wavelength of the instability of an infinite cylindrical jet. This prediction is confirmed by various experimental data found in the literature. The maximum rate of growth of rim disturbances depends on all the parameters of the problem; however, the most pronounced effect can be attributed to the rim deceleration. This conclusion is confirmed by nonlinear simulations of rim deformation.