Gerald G. Kleinsteinhttp://works.bepress.com/gkleinstein/All documents in Gerald G. Kleinsteinen-usCopyright (c) 2016 All rights reserved.Mon, 08 Feb 2016 12:24:04 -00003600Impulsive displacement of a liquid in a pipe at high Reynolds numbershttp://works.bepress.com/gkleinstein/2/<p>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.</p>
http://works.bepress.com/gkleinstein/2/ArticlesA note on Taylor's stability rulehttp://works.bepress.com/gkleinstein/3/<p>Analyzing the stability of the interface between two fluids of different densities which is accelerating normal to its surface, Taylor concluded a rule that "this surface is stable or unstable according to whether the acceleration is directed from the heavier to the lighter fluid or vice versa." We show by following Taylor's analysis that in the range where the acceleration of the interface is in the direction of the gravitational acceleration and its magnitude is smaller, this rule reverses direction. The interface is stable or unstable according to whether the acceleration is from the lighter to the heavier fluid or vice versa.</p>
http://works.bepress.com/gkleinstein/3/ArticlesOn the derivation of boundary conditions from the global principles of continuum mechanicshttp://works.bepress.com/gkleinstein/1/<p>We consider the motion of a fluid exterior to a moving rigid obstacle, or interior to a moving rigid shell. The boundary conditions, such as the no-slip condition and the condition of an isothermal wall, applied in the solution of the system of differential equations describing these motions, are currently assumed to be an approximation derived from experimental observation rather than an exact law. It is the purpose of this paper to show that the boundary conditions at a material interface between a fluid and a solid are derivable from the global principles of balance of continuum mechanics and the Clausius-Duhem inequality.</p>
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