Modeling of cardiovascular flow is the main function of **SimVascular**. Currently, the default flow solver of **SimVasculr** is **svSolver**. All of the features in **svSolver** exists in **svFSI** including RCR boundary condition, Coupled Momentum Method, GenBC etc. Also, we have carefully tested the new solver to make sure it produce the same results as **svSolver**. In this section, we won’t dwell on the similarities between these two solvers, and instead will focus two new features in flow simulation using **svFSI**, i.e., non-Newtonian flow model and simulation of prescribed motion.

Blood is a complex mixture that consists of plasma, blood cells and platelets. The blood viscosity is a complicated subject. It is strongly dependent on several factors such as temperature, hematocrit and, especially, the shear rate. From experimental studies, it is determined that the blood behaves like Newtonian flow at high shear rate ($>100 s^{-1}$). In most large arteries such as aorta, coronary arteries, the shear rate is well above this number and blood can be treated as a Newtonian fluid, that is the viscosity is a constant. On the other hand, when the shear rate is below this threshold, blood presents strong shear thinning behavior, i.e. the viscosity decreases with increasing shear rate. Many viscosity models have been proposed to represent this non-Newtonian behavior [1].

Currently, **svFSI** supports three viscosity models: Newtonian, Carreau-Yasuda and Casson [2].

Carreau-Yassuda model is defined as

$$\eta=\eta_\infty + (\eta_0 - \eta_\infty) \left[ 1 + \left( \lambda\dot(\gamma)^a \right) \right]^{\frac{n-1}{a}}$$

Here:

- $\eta_\infty$: limiting high shear-rateviscosity;
- $\eta_0$: limiting low shear-rate viscosity;
- $\lambda$: shear-rate tensor multiplier;
- $\dot{\gamma}$: shear rate;
- $a$: shear-rate tensor exponent;
- $n$: power-law index.

Casson model is defined as

$$\eta=\frac{1}{\dot{\gamma}}\left[ k_0 ( c ) + k_1 ( c )\sqrt{\dot{\gamma}} \right]^2$$

Here, $k_0 ( c )$ and $k_1 ( c )$ are functions of the hematocrit $c$.

The input file mostly follows the master input file `svFSI_master.inp`

. Some specific input options are discussed below:

For Newtonian fluid:

```
Viscosity: Constant {
Vsalue: 0.04
}
```

For Casson fluid

```
Viscosity: Cassons {
Asymptotic viscosity parameter: 0.3953
Yield stress parameter: 0.22803
Low shear-rate threshold: 0.5
}
```

For Carreau-Yasuda fluid

```
Viscosity: Carreau-Yasuda {
Limiting high shear-rate viscosity: 0.022
Limiting low shear-rate viscosity: 0.22
Shear-rate tensor multiplier (lamda): 0.11
Shear-rate tensor exponent (a): 0.644
Power-law index (n): 0.392
}
```

In this section, we discuss how to set up a simulation with prescribed wall motion. For example, one may wish to extract the motion of the walls of the ventricle from CT/MR scans, and reconstruct the flow fields by solving the Navier-Stokes equations. This is accomplished by computing the displacement trajectory of the endocardial surface and prescribed it as the boundary condition. This must be done offline by the user for their specific problem. This is typically done for a small subset of the total times, and displacements between the specified times are prescribed by using a piece-wise linear interpolant. Some researchers refer to this method as the one-way coupled fluid-structure interaction modeling. But since only fluid equation is solved here, we categorize it as flow simulation.

There are many established pipelines to obtain the wall motion from CT/MR scans[3-5]. Here, we would recommend using the cardiac geometric modeling tool developed in the **SimCardio** project.

For the wall motion file, the file format is as follows. First, specify the dimension of the problem (three), the number of times at which to specify the displacements, and the number of vertices in the moving wall mesh. Then specify the times at which the displacements occur. Next, for each vertex, specify its index and then the prescribed displacements for each time. Note that in the case of multiple moving faces, these numbers may not start at one for any given face, as indexing is global. If a vertex is on an edge between two faces, it should have the same index and displacement fields, specified redundantly in both files.

```
Dimension n_times m_vertices
t_1
t_2
...
t_n
vertex_1_index
displacement_1_vertex_1
displacement_2_vertex_1
...
displacement_n_vertex_1
vertex_2_index
displacement_1_vertex_2
displacement_2_vertex_2
...
displacement_n_vertex_2
...
```

For example,

```
3 21 14907
0.000000
3.800E-2
7.600E-2
1.140E-1
1.520E-1
1.900E-1
2.280E-1
2.660E-1
3.040E-1
3.419E-1
3.800E-1
4.180E-1
4.560E-1
4.940E-1
5.320E-1
5.699E-1
6.080E-1
6.460E-1
6.839E-1
7.220E-1
7.600E-1
1
0.000000 0.000000 0.000000
2.800E-1 -8.99E-2 -1.00E-2
8.799E-1 -2.09E-1 -8.10E-1
1.339999 -1.59E-1 -1.43000
1.509999 7.000E-2 -1.59000
1.310000 3.100E-1 -1.52000
1.069999 5.100E-1 -1.41000
1.009999 6.000E-1 -1.28000
9.699E-1 3.600E-1 -1.10000
1.049999 -1.09E-1 -8.80E-1
1.169999 -6.69E-1 -7.60E-1
1.169999 -1.17999 -7.90E-1
1.129999 -1.31999 -9.69E-1
1.049999 -1.25999 -1.34000
1.000000 -9.39E-1 -1.64000
9.899E-1 -4.29E-1 -1.96000
1.000000 3.000E-2 -2.15000
9.299E-1 3.600E-1 -1.90000
6.599E-1 4.200E-1 -1.14000
1.999E-1 1.500E-1 -3.50E-1
0.000000 0.000000 0.000000
2
0.000000 0.000000 0.000000
2.700E-1 -8.99E-2 0.000000
8.500E-1 -1.99E-1 -6.80E-1
1.280000 -1.59E-1 -1.16999
1.430000 5.000E-2 -1.28000
1.230000 2.599E-1 -1.19999
1.010000 4.499E-1 -1.09000
9.500E-1 5.099E-1 -9.69E-1
9.100E-1 2.400E-1 -8.19E-1
1.000000 -2.59E-1 -6.70E-1
1.120000 -8.29E-1 -5.69E-1
1.130000 -1.31999 -5.89E-1
1.090000 -1.44999 -7.50E-1
1.000000 -1.36000 -1.06999
9.400E-1 -1.00999 -1.31000
9.300E-1 -4.69E-1 -1.56999
9.500E-1 0.000000 -1.72999
8.900E-1 3.299E-1 -1.53999
6.200E-1 3.999E-1 -9.39E-1
1.900E-1 1.399E-1 -2.90E-1
0.000000 0.000000 0.000000
...
```

Note that in this example, we wish the mesh motion to be periodic in time, and thus the final displacement is zero.

To set up the input file, set the equation to be FSI to allow the mesh to move under the ALE framework, even though there is not necessarily a structure. For the moving wall, add the motion file when specifying the wall boundary condition, and turn on the option “Impose on state variable integral”.

```
Add BC: moving_wall {
Type: Dirichlet
Time dependence: General
Temporal and spatial values file path: wall_motion.dat
Profile: Flat
Zero out perimeter: 1
Impose flux: 0
#---------------- Add this line to the moving boundary face -----------
Impose on state variable integral: 1
}
```

It is recommended to include remeshing if the wall motion is such that the domain undergoes large changes. For example, set

```
Remesher: Tetgen {
Max edge size: lumen { val: 3.0 }
Min dihedral angle: 10.0
Max radius ratio: 1.1
Remesh frequency: 100
Frequency for copying data: 1
}
```

The max edge size should be consistent with the original mesh size.

Under the mesh equation, we similarly add the motion file.

```
Add equation: mesh {
Coupled: 1
Min iterations: 1
Max iterations: 8
Tolerance: 1e-3
Residual dB reduction: -20
Poisson ratio: 0
Output: Spatial {
Displacement: t
}
#---------- Add the BC for the moving_wall to the mesh equation as well ------
Add BC: moving_wall {
Type: Dirichlet
Time dependence: General
Temporal and spatial values file path: wall_motion.dat
Profile: Flat
Zero out perimeter: 1
Impose flux: 0
#---------------- Add this line to the moving boundary face -----------
Impose on state variable integral: 1
}
```

Follow this instruction if you need to restart your simulation after stoppage.

[2] Boyd, Joshua, James M. Buick, and Simon Green. **Analysis of the Casson and Carreau-Yasuda Non-Newtonian Blood Models in Steady and Oscillatory Flows Using the Lattice Boltzmann Method**. Physics of Fluids 19, no. 9 (September 2007): 093103.

[5] Kong, F., and Shadden, S. C. (2020). **Automating Model Generation for Image-based Cardiac Flow Simulation**. ASME. J Biomech Eng.