The Flow of Liquids Through Capillaries

A simple theory of the flow of a liquid from a reservoir, through a capillary, into a second reservoir is developed, and the conclusions from it are shown to accord with the observations of Bond and of Poiseuille, and with a qualitative study of the flow by means of colored streams. It is shown that, for viscosimeters of this type, the common interpretation of that term (with coefficient $m$) in the viscosimeter equation which is frequently called the kinetic energy correction is entirely incorrect. That term is an inertia correction; it does not arise from any loss of head attendant upon the imparting of kinetic energy to the liquid, but solely from a progressive change in the distribution of the flow in the exit reservoir. It is not the correction which was considered by Hagenbach. For ideal conditions, $m$ is probably equal to unity. At very low velocities, the distribution of the flow in each reservoir is independent of the velocity, and consequently the inertia term vanishes; at this stage the Couette correction is $2e$, twice what it is when the distribution in the exit reservoir is changing. From Bond's data, it is found that $e=0.573r$. At a certain velocity, simply related to $e$, the distribution of the flow in the exit reservoir begins to change, the inertia term appears. Bond found that this occurs when the Reynolds number is 10. Under certain stated conditions, the initial distribution of flow can be retained to a much higher velocity; in these cases the inertia correction does not enter, and $\mathrm{pt}$ is independent of the velocity. There are indications that for very short tubes $\mathrm{pt}$ ceases to be linear in the velocity before the flow in the tube becomes turbulent. An explanation is offered. Changes in the terminal configurations affect both the Couette correction and the inertia correction, and the second may be markedly affected by changes in the size and form of the exit reservoir. As commonly used, the subdivided tube method for determining $e$ is entirely unreliable.