Pre-Analysis & Start-Up

...

{\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:

{\large
\begin{eqnarray*}
\nabla \cdot \textbf{v}=0
\end{eqnarray*}
}

The Navier-Stokes Equation is written as follows:

{\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*}
}

One thing to notice in the Navier-Stokes equation is that the viscosity coefficient of

{\large\begin{equation*}\mu\end{equation*}}

is not a constant but rather a function of shear rate. Blood gets less viscous as the shear rate increases (shear thinning). Here, we model the blood viscosity using the Carreau fluids model. The mathematical formulation of the Carreau model is as follows:

{\large
\begin{eqnarray*}
\mu_{eff}(\dot{\gamma})=\mu_{inf}+(\mu_0-\mu_{inf})(1+(\lambda \dot{\gamma})^2)^\frac{n-1}{2}
\end{eqnarray*}
}

In the equations above,

{\large\begin{equation*}\mu_{eff}\end{equation*}}

is the effective viscosity.

{\large\begin{equation*}\mu_0, \mu_{inf}, \lambda \textrm{ and } n\end{equation*}}

are material coefficients.
For the case of blood [2],

{\large
\begin{align*}\\
&\mu_0=0.056(kg/m \cdot s)\\
&\mu_{inf}=0.0035(kg/m\cdot s)\\
&\lambda=3.313(s)\\
&n=0.3568
\end{align*}
}

Boundary Conditions

Wall:

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:

Image Removed
To describe the profile more clearly, a mathematical description is also given below:

...

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.

Go to Step 2: Geometry

Go to all FLUENT Learning Modules