Stability and Breakup of Liquid Threads and Annular Layers in a Corrugated Tube with Zero Base Flow

Abstract

We study the stability and breakup behavior of an axisymmetric liquid thread surrounded by another viscous fluid layer through a long wave approximation. The two fluids are immiscible and confined in a concentrically placed cylindrical tube and there is no base flow. The effect of the tube wall corrugation is taken into account in the model, which allows the access of the interaction between the wall shape and the thread interfacial dynamics. The linearized system is studied by Floquet theory due to the presence of nonconstant coefficients in the equation and the spectrum is computed numerically via the Fourier--Floquet--Hill method. The resulting features agree qualitatively with those obtained based on a lubrication model in the thin annulus limit, where the short wave disturbances that would be stabilizing due to the capillarity in the absence of wall corrugation, can excite some unstable long waves. Those results from the linear theory are also confirmed numerically by direct numerical simulation on the evolution equations. Meanwhile, a transition on the dominant modes from the one for a tube without corrugation to the one with a wall shape included is found. Finally the drop formation is shown in the nonlinear regime when the pinch off of the core thread is obtained. The annular film drainage regime is also approached slowly along with pinching when the tube wall is close to the thread interface. In addition, our results demonstrate the possibility of suppressing pinching, depending on the averaged annular layer thickness and the variation in tube radius.