Motion of a viscous fluid in a tube which is subjected to a series of pulses
Introduction. Consider a circular pipe which contains a viscous fluid. The flow due to a constant pressure gradient in a stationary pipe is the well-known Poiseuille flow. Now suppose that a pipe, containing a viscous fluid, is subjected to a series of pulses in an axial direction which do not appreciably move the pipe. The fluid inside will acquire a velocity in the direction of the pulses, relative to the pipe, due to the transfer of momentum from the boundary by viscous stresses. We determine the flow, relative to the pipe, which is obtained when a pressure gradient and a series of pulses act. Since the equation of motion is linear the two effects may be considered separately and the two results added together. In particular we are interested in the case when the pressure gradient and the pulses have opposite senses. In this case there arises the possibility that the discharge from the pipe could be completely stopped.** In Sec. 1 we consider the simpler case of flow in a two dimensional channel. The case of a pipe with circular cross-section is taken up in Sec. 3. 1. Flow in a straight walled two-dimensional channel. Let x be the coordinate measured along the channel length, 2a be the distance between the walls and the origin of coordinates be midway between the walls. For the steady state motion the velocity in the axial direction u(y) satisfies the differential equation d2u dp "d^^Tx (1)