Pre-Analysis & Start-Up

Pre-Analysis

In the Pre-Analysis step, we'll review the following:

• Governing Equations: We will review the governing equations that need to be solved in this problem.
• Boundary Conditions: We will go into more details about the boundary conditions that are applied in this problem.

Governing Equations   

Before starting a CFD simulation, it is always good to take a look at the governing equations underlying the physics. In this case, although we have additional complexities such as pulsatile flow and non-newtonian fluids, the governing equations are the same as any other fluids problem. The most fundamental governing equations are the continuity equation and the Navier-Stokes equations. Here, let's have a quick review of the equations.
Continuity Equation:

However, as blood can be regarded as an incompressible fluid, the rate of density change is zero, thus the continuity equation above can be further simplified in the form below:

The Navier-Stokes Equation:

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

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:

In the equations above,

is the effective viscosity.

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

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.

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:

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.

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