Impulsive displacement of a liquid in a pipe at high Reynolds numbers

We consider the problem of an impulsive displacement of a liquid, originally at rest in a circular pipe, which is displaced by another liquid. The purpose of this analysis is to show that at a sufficiently high inertia the initial essentially inviscid motion can be extended to cover the entire displacement process, thus creating an inviscid window to which an inviscid analysis can be applied. We simplify the problem first, by considering a 1-liquid problem where the displacing liquid and displaced liquid are the same. We identify two characteristic times in this problem: the time it takes an inviscid liquid to be displaced, and the time it takes a viscous liquid to attain a steady state. Taking the ratio of the two defines the Reynolds number for the problem and we show that the motion becomes essentially inviscid once the Reynolds number is sufficiently high. We obtain the general solution of the 1-liquid problem which determines the nondimensional viscous displacement time as a function of the Reynolds number. We derive from the general solution: a critical Reynolds number above which the motion remains unsteady throughout the entire displacement process, and a formula which determines quantitatively whether applying an inviscid analysis to the 1-liquid viscous problem at a given Reynolds number is admissible within an acceptable error tolerance. We also show that at the limit the Reynolds number approaches infinity the viscous displacement time approaches the inviscid displacement time and that the velocity profile and the shape of the material surface separating the displacing from the displaced liquid approach their counterpart in the inviscid solution. Second, based on these results we propose that an inviscid solution is applicable to the 2-liquid viscous problem once the condition of a high Reynolds number is independently met by the two participating liquids. We obtain the solution to the inviscid 2-liquid displacement problem and calculate various examples. Finally, we present a stability analysis of the flat interface between the two inviscid liquids, which shows which of the examples is stable, neutrally stable, or unstable. The paucity of data for an impulsive displacement in the high Reynolds number range makes quantitative comparisons difficult. However, the excellent agreement obtained between the critical Reynolds number derived in this analysis and the result obtained in a numerical analysis of the viscous 2-liquid problem elsewhere constitutes at least a partial validation of the theory. Additional confirmation is obviously recommended.