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SimVascular

This project also makes source code available.

Goals

1. Derive governing PDEs for 1D blood flow. Result will be equations for pressure P, flow rate Q, and cross-sectional area S as functions of one spatial coordinate z and time t:

Conservation of mass

$\frac{\partial S}{\partial t} + \frac{\partial Q}{\partial z} + \psi = 0$

Conservation of momentum

$\frac{\partial Q}{\partial t} + \frac{\partial}{\partial z}[(1 + \delta) \frac {Q^2}{S}] + \frac{S}{P} \frac{\partial P}{\partial z} = S f + N \frac{Q}{S} + \nu \frac{\partial^2 Q}{\partial z^2}$

Constitutive equation

$P = \hat P (S(z,t), z, t)$

where $\psi$ , $\delta$ , f, and N are to be defined and $\nu$ is the kinematic viscosity

2. Derive a linear approximation to these equations

3. Discuss how to solve the nonlinear & linear equations

4. Discuss how to apply 1D theory to solve blood flow problems.

Tools

Reynolds Transport Theorem

Material or convective derivative of a volume integral

Both integral $\zeta$ and volume $V(\bar x, t)$ depend on time.

$\frac{D}{Dt} \int\limits_{V} \zeta\, dV = \int\limits_{V} \frac{\partial \zeta}{\partial t} \, dV + \int\limits_{\partial V} \zeta \, u_n \, ds$

Divergence Theorem

$\int\limits_{V} \nabla \cdot (\bar F) \, dV = \int\limits_{\partial V} \bar F \cdot \bar n \, dS$

1D Nonlinear Theory (last edited 2007-08-01 18:32:14 by wtkatz)