To have a better understanding of the flow of blood in arteries a theoretical analysis of the pressure wave propagation through a viscous incompressible fluid contained in an initially stressed tube is considered. The fluid is assumed to be Newtonian. The tube is taken to be elastic and isotropic. The analysis is restricted to tubes with thin walls and to waves whose wavelengths are very large compared with the radius of the tube. It is further assumed that the amplitude of the pressure disturbance is sufficiently small so that nonlinear terms of the inertia of the fluid are negligible compared with linear ones. Both circumferential and longitudinal initial stresses are considered; however, their origins are not specified. Initial stresses enter equations as independent parameters. A frequency equation, which is quadratic in the square of the propagation velocity is obtained. Two out of four roots of this equation give the velocity of propagation of two distinct outgoing waves. The remaining two roots represent incoming waves corresponding to the first two waves. One of the waves propagates more slowly than the other. As the circumferential and/or longitudinal stress of the wall increases, the velocity of propagation and transmission per wavelength of the slower wave decreases. The response of the fast wave to a change in the initial stress is on the opposite direction.

An approximate numerical method for calculating flow profiles in arteries is developed. The theory takes into account the nonlinear terms of the Navier-Stokes equations as well as the nonlinear behaviour and large deformations of the arterial wall. Through the locally measured values of the pressure, pressure gradient and pressure–radius function the velocity distribution and wall shear at a given location along the artery can be determined. The computed results agree well with the corresponding experimental data.

To give a realistic representation of the pulse propagation in arteries a theoretical analysis of the wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube is considered. The tube is assumed to be orthotropic and its longitudinal motion is constrained by a uniformly distributed additional mass, a dashpot and a spring. The fluid is assumed to be Newtonian. The analysis is restricted to propagation of small amplitude harmonic waves whose wavelength is large compared to the radius of the vessel. Elimination of arbitrary constants from the general solutions of the equations of motion of the fluid and the wall gives a frequency equation to determine the velocity of propagation. Two roots of this equation give the velocity of propagation of two distinct outgoing waves. One of the waves propagates slower than the other. The propagation properties of s lower waves are very slightly affected by the degree of anisotropy of the wall. The velocity of propagation of faster waves decreases as the ratio of the longitudinal modulus of elasticity to the circumferential modulus decreases; transmission of these waves is very little affected. The influence of the tethering on the propagation velocity of slower waves is negligibly small; transmission of these waves is seriously affected. In tethered tubes faster waves are completely attenuated.

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A constant-temperature heated-film anemometer system has been adapted for the detailed study of in-vivo aortic velocity fields. Two types of sensing probes were developed: a velocity probe and a velocity-gradient or fluid shear stress probe. These probes were evaluated for steady and pulsatile flow in rigid circular tubes using both a glycerin-water mixture and blood. Measurements using both devices agreed closely with the values predicted by well established theory. Moreover, the integrated velocity profiles that were measured correlated well with the simultaneously recorded flow values using orifice meter and electromagnetic flowmeter techniques. In-vivo studies were made along the thoracic aortas of anesthetized dogs and pigs. Velocity measurements along the aorta indicated that the velocity profiles are blunt. The flow-pulse forms obtained by the heated-film technique in vivo were also similar in magnitude and contour to those obtained simultaneously from an electromagnetic flowmeter. Fully developed turbulent flow was not observed; however, occasional "eddy" turbulence occurred in the aortic arch of dogs weighing less than 30 kg. Preliminary measurements indicate that peak wall-shear stresses reach values that are approximately one-third that of the endothelial yield stress.