Displacement of a liquid from a horizontal circular tube by another liquid

Abstract

We consider the axisymmetric problem of expulsion of one liquid from a horizontal circular pipe by another for the laminar flow regime.

Studies of Taylor [1], Cox [2, 3], and others have been devoted to the experimental investigation of the displacement of a liquid from a capillary. Experiments on the displacement of mutually immiscible liquids show that the length of the combined flow region and the amount of displaced liquid remaining in this region are determined primarily by the magnitude of the interphase tension forces at the leading edge of the interface. The equilibrium of these forces and the hydrodynamic differential, established some time after the beginning of displacement, give rise to the rigid bullet-shaped form of the interface leading edge. This portion of the interface, whose stiffness is achieved as a result of the force balance, is termed the “head” of the interface between the liquids. The radius of the head-the relatively small autonomous portion of the interface-defines the dimensions and deformations of the entire remaining interface.

The existence of a rigid autonomous interface head is the basis of the physical displacement model used in this article.