Oscillatory motion of a fluid along a circular tube

Abstract

1. This paper deals with oscillatory motions of a viscous incompressible fluid in a uniform circular tube. The oscillations may either be free or forced by a pulsating pressure, control coming in both instances from the "gravity head" which results from the passage of fluid. They are closely analogous with mechanical and electrical oscillations, but the theory is complicated by the circumstance that the fluid does not oscillate as a rigid body, its velocity ranging between a zero value at the wall and a maximum value at the axis of the tube. Our interest in the problem originated in a notion that oscillatory motion, either "free" or "forced", might be made the basis of a new method of viscometry. Of existing methods, all which aim at high accuracy appear (following the pioneer methods of Poiseuille 1840-I) to take advantage of the known solution for steady flow along a straight and uniform tube. Commercial viscometers employ the same prinicple, but the tube (for convenience) is made relatively short, and the motion is not steady, since the "head" of the fluid is changing throughout the tests. On this account their measurements are comparative rather than absolute; for since the motion is not calculable by theory they must be calibrated with the aid of fluids whose viscosity is known. They have the advantage (compared with Poiseuille's method) of not requiring a large amount of the fluid under test; but it is difficult to be certain of the exact temperature of the oil while it is passing through the tubular orifice, and this is a serious disadvantage in view of the rapidity with which the viscosity of oils decreases with increase of temperature. Our notion was that a method involving pulsating instead of steady flow on the one hand would have some claim to be regarded as an absolute method. This is not the place for a complete account of the resulting "oscillation viscometer", but such features as have a bearing on the fundamental problem of this paper receive brief notice in part II; Part I deals with theory and Part III with experimental results.