A mathematical model for nonlinear analysis of flow pulses utilizing an integral technique

An existing one-dimensional mathematical model, for the arterial tree was extended to include the effects of radial variation of axial fluid velocity by the application of an integral technique. The resulting formulation reduced to a system of characteristics equations similar, in form to the equations for the onedimensional model and the computer program was modified to accommodate the integral formulation. The need for a kinematic boundary condition on the axial component of wall velocity was demonstrated. Results were obtained for a variety of velocity profiles. It was found that the slope of the front and back of the waves as well as the wave, amplitude are sensitive to changes in the velocity profile and the axial component of wall velocity. The velocity of the waves is also effected but not significantly.

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Zusammenfassung Die eindimensionale Theorie von Anliker et al. (ZAMP22, 217 (1971)) wird in dieser Arbeit dahingehend erweitert, dass der Einfluss des Geschwindigkeitsprofiles mitberücksichtigt wird. Die Navier-Stokes-Gleichungen und die Kontinuitätsgleichung werden mit Hilfe einer Integraltechnik auf ähnliche, für die Rechnung mit dem Computer geeignete Gleichungen zurückgeführt, wie sie von Anliker et al. verwendet wurden. Die charakteristischen Grössen des Geschwindigkeitsprofiles sowie die Geschwindigkeit der Gefässwand gehen als Parameter in die Theorie ein, so dass parametrische Studien durchgeführt werden können.

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Nomenclature a local internal radius of the vessel - a _f , A _f constants in the cosine profile - b defined by equation (22) - local normal and tangential unit vectors (see Figure 1) - f(r/a), g (z,t) defined by equation (9) - f _R friction factor - local mass flux into the vessel - p local pressure - p _c capillary pressure - p _o pressure at the terminal end - r, z radial and axial coordinates - S local cross sectional area - t time - flow velocity at the wall interface - u, v, w radial, circumferential and axial components of flow velocity - u _w , v _w , w _w radial, circumferential and axial components of flow velocity at the wall interface - wall velocity at the interface - U _w , W _w radial and axial components of wall velocity at the interface - W mass average flow velocity defined by equation (13) - w _o maximum flow velocity - ₀, ₁, ₂, ₃ parameters defined by equation (10) - ₄ W _w /w _w - _A , _B , _C parameters defined by equation (18) - outflow parameter - wave length - _L Lagrangian multiplier - viscosity coefficient for the fluid - density of the fluid - kinematic viscosity - ()' nondimensional quantity of order one - ()₊, ()_– values of () associated with roots of equation (23) This analysis was initiated during the authors appointment as a NASA-ASEE Summer Faculty Fellow to the Stanford-Ames Program and completed through the facilities of the Computer Science Center at the University of Maryland.