Nonlinear dynamics of liquid columns: A comparative study

Abstract

One‐dimensional models have been used extensively over the past decade to study liquid columns. While the range of validity of these models is well known in the linear limit, this is not the case when nonlinear effects are important. By comparing results of a number of one‐dimensional models with results based on a velocity‐potential model in which no approximations have been made, the present aim is to establish exactly when the one‐dimensional models are applicable. First of all it is shown how the Cosserat equations in the inviscid limit may be obtained formally from the Euler equations. Subsequently, a linearized form of the Cosserat equations is derived and results of this model are compared with results obtained by means of the well‐established inviscid‐slice model and results obtained by the velocity‐potential approach. It is found that the applicability of the inviscid‐slice model is limited by short‐wavelength effects rather than amplitude effects. The range of validity of the inviscid‐slice model can be extended by including radial momentum contributions. However, even with the inclusion of radial momentum effects the one‐dimensional models are not well suited to describe the behavior close to the bifurcation point.