Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes

Abstract

Some insight into the behaviour of tightly fitting solid pellets, which may be deformable, and are being forced by a pressure difference to move slowly along a distensible tube filled with viscous fluid, is sought by theoretical study of a simple axisymmetric model (§2). In this, the pellet's clearance in the tube is taken to be a small fraction of the tube radius; the fraction may, at a pressure characteristic of that ahead of the pellet, be either positive or negative. Even if it is positive, the tube may still be distended (or the pellet compressed, or both) as the pellet passes, because the thickness of lubricating film generated may exceed the clearance. Naturally, still greater elastic deformation can occur in the case of negative clearance.

Highly simplified elastic properties are assumed; with an eye on tubes occurring in physiological systems (with Poisson's ratio close to 0.5), the local distension of the tube is taken to vary linearly with the local excess pressure; as a still cruder approximation, a similar relation for local reduction of pellet radius is assumed. A parabolic approximation to the pellet's undistorted meridian section, in the region where the lubricating film is thin, is also assumed, leading to a simple relation between pressure and local film thickness which is used, together with Reynolds's lubrication equation, to evaluate both. An arbitrary constant, the rate of leakback of fluid past the pellet, is determined by the condition that the pressure difference forcing the pellet must just balance the skin-frictional resistance to its motion.

The problem is non-dimensionalized (§3) and reduced to that of finding a particular solution of a differential equation containing a certain parameter L. In addition to numerical solutions for particular values of L (§6), perturbation solutions for both small and large L are obtained (§§4 and 5), to give mathematical and physical insight; the perturbation for large L (corresponding to negative clearance) is highly singular, requiring the matching of approximate solutions different in each of six different layers.

A striking feature of the solutions is a necking of the gap between pellet and tube behind the pellet. This is so pronounced in the case of negative clearance (figure 2) that it might give the false impression that the pellet was being propelled by peristaltic contraction of the tube instead of by fluid pressure gradient. The physical reason for this is elucidated (§6).

In the case of positive clearance, rather small pressure differences suffice, on this theory, to propel the pellet, because different parts of the lubricating layer act on it with frictional resistances of different signs, which almost cancel out. By contrast, for negative clearance, the resistance becomes a large multiple of that found in a purely fluid-filled tube of length and mean velocity equal to that of the pellet. This multiple increases, and the film thickness correspondingly decreases (figure 7), as the pellet velocity decreases.

One physiological system on which the model may throw some light is the narrow capillary with red blood cells being squeezed through it in single file, lubricated by plasma (§1 and 8). At the higher flow speeds, around 0.1 mm/s, the lubricating film, predicted to be about 0·2μm thick, appears likely to play a significant role in mass transfer to and from the tissue spaces. At much lower speeds, predicted film thicknesses are so small that any of a number of mechanisms, including loss of fluid through the porous capillary wall due to the local excess pressure in the layer, might cause movement to ‘seize up’ altogether.