Peristaltic Flow with Application to Ureteral Biomechanics

Peristalsis is produced by successive waves of contraction in elastic, tubular structures which push their fluid or fluid-like contents forward. The most common industrial use is in pumping. In the urinary system, peristalsis is due to involuntary muscular contractions of the ureteral wall which drives urine from the kidneys to the bladder through the ureters. Mathematical analyses of peristaltic flow with application to the ureter are presented. Also included are results from an experimental peristalsis simulator.The problem is introduced and a literature survey of relevant works on peristaltic motion provided. There is an overview of the anatomy and physiology of the urinary tract. Theoretical investigations of peristaltic transport of single-phase, two-phase and fluid-particle combinations are presented. In the mathematical analysis of two-dimensional peristaltic fluid flow, the fluid is assumed to be incompressible and Newtonian. The wavenumber is assumed to be small, and a regular perturbation expansion is used to solve the equations. Comparison is made with available experimental data. The next study is of streamline patterns and their local and global bifurcations in peristaltic flow. A simplified stream-function solution is obtained for long-wavelength and low-Reynolds number problems. This is used to set up a system of nonlinear autonomous differential equations for the particle paths. Local bifurcations and their topological changes are examined using methods of dynamical systems. Three different flow situations appear: backward, trapping, and augmented flow. Several stagnation points appear in the flow field as the parameters change. The bifurcations are summarized in a global diagram for the planar and axisymmetric problems. Theoretical results are compared with the experimental data available in the literature. Following this, a model of peristaltic flow of a fluid-particle suspension (a two-phase model) is studied. This is motivated by the possible presence of particles accompanying ureteral peristalsis. The flow is analyzed for a solitary traveling wave in an axisymmetric tube with an incompressible, Newtonian fluid in which identical, solid spherical particles are distributed. The equations are solved for a small radius to length ratio of the wave to obtain an approximate, closed-form solution of the flow and particle velocities. It is concluded that peristaltic pumping is affected as particle volume fraction is increased. In addition, the pressure drop diminishes as the amplitude ratio of the wave amplitude to its radius decreases, and the pressure in the contracted part of the ureter increases with the particle volume fraction. Following this, the particle motion in an unsteady peristaltic fluid flow is studied. The analytical solution for the fluid is used in a momentum equation for a small rigid sphere in non-uniform flow. Several forces on the particle are taken into account: Stokes drag, virtual mass, Faxen, Basset and gravity forces. For particles like stones or bacteriuria, parameters corresponding to calcium oxalates for calculosis and Escherichia coli for bacteria are considered. The results show that retrograde or reflux motion of the particles is possible and bacterial transport can occur in the upper urinary tract when there is a partial occlusion of the wave. Dilute particle mixing is also investigated, and it is established that some of the particles participate in the formation of a recirculating bolus, and some are delayed in transit and eventually reach the walls. Analytical modeling is limited, however, to two-dimensional configurations and with restrictions on particle size and distribution. A laboratory experiment, on the other hand, permits treatment of the three-dimensional problem with larger particles. A mechanical peristaltic pump is designed and constructed for this purpose. Ureteral peristalsis parameters are kept in mind to maintain geometrical and dynamical similarity. Experimental measurements of the mean flow rate indicate a linear relationship with the speed of the peristaltic wave. Particle transit time is also investigated, and it is found to increase with the number of particles. Finally, some conclusions of the research and recommendations for future work are discussed. The results obtained may be relevant to the transport of other physiological fluids and industrial applications in which peristaltic pumping is used.