Peristaltic motion

Abstract

The flow of a viscous fluid through axially symmetric pipes and symmetrical channels is investigated under the assumption that the Reynolds number is small enough for the Stokes flow approximations to be made. It is assumed that the cross-section of the pipe or channel varies sinusoidally along the length. The flow is produced by a prescribed pressure gradient and by the variation in cross-section that occurs during the passage of a prescribed sinusoidal peristaltic wave along the walls. The theory is applied in particular to two extreme cases, peristaltic motion with no pressure gradient and flow under pressure along a pipe or channel with fixed walls and sinusoidally varying cross-section. Perturbation solutions are found for the stream function in powers of the ratio of the amplitude of the variation in the pipe radius or channel breadth to the mean radius or breadth respectively. These solutions are used to calculate, in particular, the flux through the pipe or channel for a given wave velocity in the first case and for a given pressure gradient in the second case. With a suitable notation it is possible to combine the analysis required for the two cases of pipe and channel flow.

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“…The results of the experiments were found to be in good agreement with the theoretical results of Shapiro [2]. Based on this experimental work, Burns and Parkes [3] studied the peristaltic motion of a viscous fluid through a pipe and a channel by considering sinusoidal variations at the walls. Shapiro et al [4], in 1969 analyzed peristaltic pumping with long wavelengths at low Reynolds number.…”

Transverse magnetic field driven modification in unsteady peristaltic transport with electrical double layer effects

“…Important mathematical works in this regard have been communicated by Burns and Parkes [8], Fung and Yih [9], Shapiro et al [10], Jaffrin and Shapiro [11], Shukla et al [12], Takabatake, and Ayukawa [13], Pozrikidis [14] and Li and Brasseur [15]. More recent studies which have extended the purely Newtonian fluid models described in [8]- [15] include Tripathi and Bég [16] who examined viscoelastic peristaltic propulsion, Tripathi and Bég [17] who studied peristaltic flows of magnetized couple stress fluids, Blanchette [18] who considered the influence of suspended drops on peristaltic pumping, Ellahi et al [19] who computed heat transfer in peristaltic flow. Further recent studies exploring other areas of peristaltic transport are the articles of Tripathi and Bég [20] concerning nanofluids, Kothandapani and Prakash [21] on magnetic nanofluids and Akbar et al…”

An Engineering Study of the Peristaltic Drive of Axonal Flow

“…(3n + 1) 2KcAXa I)dz -(J)dz 1l.0 + (-I)ndz, [7] Z2 where Q/1 a2c-42/2 + 24 sin (2Trz) + 4)2 sin2 (2rz) [8] (1 + 4 sin (2rz))(,n+1)/n and z1 and Z2 are the roots of the transcendental equation -Q/Tra2C + 4)2/2 = 24 sin (2rz) + 4)2 sin2 (27rz) [9] A pump is a device for transferring energy to a fluid; some of this energy is spent on raising the pressure of the fluid and some on increasing the kinetic energy of the fluid, which yields flow. Every pump is characterized by a specific relation between that pressure developed by it and the flow rate ("pump characteristic").…”

“…Eq. [7] specifies that characteristic relation. In general, the higher the flow rate at which the pump is operated, the lower is the pressure at the pump outlet.…”