Weakly nonlinear stability analysis of thin viscoelastic film flowing down on the outer surface of a rotating vertical cylinder

Abstract

The paper presents both of the linear and nonlinear stability theories for characterization of viscoelastic film flows down on the outer surface of a rotating infinite vertical cylinder. After showing the insufficiency of the linear model in characterizing certain flow behaviors, a generalized nonlinear kinematic model is then derived to represent the physical system. The model is solved by the long wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behaviors. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. In the second step, an elaborated nonlinear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The modeling results indicate that by increasing the rotation speed, Ω, and decreasing the radius of cylinder, R, the film flow will generally make the flow system less stable. In this study, the interaction of the rotation and the radius of cylinder are taken into account. Generally, Reynolds number is divided into three regions, which are Re<3, 3< Re<8 and 8< Re, for discussion corresponding to the pre-selected Rossby number ( Ro=0.1) and viscoelastic parameter ( k=0.05).