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Pre-Analysis & Start-Up
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{\large \begin{eqnarray*} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \textbf{v})=0 \end{eqnarray*} } |
However, as blood can be regarded as an incompressible fluid, the rate of density change is zero, thus since we are considering only the steady case, the time-dependent term is zero. Thus, the continuity equation above can be further simplified in the form below:
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{\large \begin{eqnarray*} \nabla \cdot \textbf{v}=0 \end{eqnarray*} } |
The Navier-Stokes Equation is written as follows:
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{\large \begin{eqnarray*} \rho (\frac{d \textbf{v}}{dt}+\textbf{v} \cdot \nabla \textbf{v})=- \nabla p + \mu \nabla ^2 \textbf{v} + \textbf{f} \end{eqnarray*} } |
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The easiest boundary condition to determine is the artery wall. We simply need to define the wall regions of this model and set it to “wall”. From a physical viewpoint, the “wall” condition dictates that the velocity at the wall is zero due to the no-slip condition.
Inlet:
As we know, mammalian blood flow is pulsatile and cyclic in nature. Thus the velocity at the inlet is not set to be a constant, but instead, in this case, it is a time-varying periodic profile. The pulsatile profile within each period is considered to be a combination of two phases. During the systolic phase, the velocity at the inlet varies in a sinusoidal pattern. The sine wave during the systolic phase has a peak velocity of 0.5m/s and a minimum velocity of 0.1m/s. Assuming a heartbeat rate of 120 per minute, the duration of each period is 0.5s. This model for pulsatile blood flow is proposed by Sinnott et, al. [3] A figure of the profile within two periods is given below:
To describe the profile more clearly, a mathematical description is also given below:
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Here our inlet velocity will be a constant 0.315 m/s. This was chosen to give us a Reynolds number of 600.
Outlets:
The systolic pressure of a healthy human is around 120 mmHg and the diastolic pressure of a healthy human is around 80 mmHg. Thus taking the average pressure of the two phases, we use 100 mmHg (around 13332 Pascal) as the static gauge pressure at the outlets.