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By Biringen S., Chow C.-Y.

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Results at later time steps show that the paths of some vortices in that region are contorted, and certain parts of the vortex sheet may become crossed. The computation should actually be terminated before this unrealistic situation has developed. The chaotic motion of the spiral has been found by several authors. A brief description of some previous research works is referred to in Moore (1974). To eliminate the chaotic motion, Moore successfully used a concentrated tip vortex to represent the tightly rolled portion of the vortex sheet.

12 using a numerical scheme constructed for solving nonlinear hyperbolic equations. 13, we present a formulation of blood flow 50 An Introduction to Computational Fluid Mechanics by Example Sedat Biringen and Chuen-Yen Chow Copyright © 2011 John Wiley & Sons, Inc. INCOMPRESSIBLE POTENTIAL FLOWS 51 in elastic arteries modeled by a system of nonlinear hyperbolic partial differential equations. We also introduce multistep methods for numerically integrating hyperbolic partial differential equations.

1 Program Symbol Definition (Main program) A 1+ρ 2 B (1 − ρ)g C 3ρ 4d D Diameter of sphere, d , m D1V, D2V, etc. Velocity increments, DIZ, D2Z, etc. Displacement increments, G Gravitational acceleration, g, m/s H Time increment, h, s NU Kinematic viscosity of fluid, ν, m2 /s PI π RE Reynolds number vd ν RHO Density of body, ρ, kg/m3 RHOBAR Density ratio, ρ, or ρf ρ 1 vi , 2 vi , 1 zi , etc. 2 zi , 2 etc. 2 Motion of a Simple Pendulum (continued) Program Symbol Definition Y, Y0 Circumferential displacement, y or l θ, and its initial value, y0 , respectively, m YV Circumferential displacement in vacuum, yv , m (Subprogram RUNGE) D1P, D2P, etc.

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