Viscosity models

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 $c$.

Input file

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
   }

Wall motion

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.

Input file

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.