## Definitions

We consider the flow of an **incompressible Newtonian fluid**
interacting with an **elastic solid**. We denote by
the domain occupied by the fluid and by the solid at the
time .
Let be the part of the boundary where the elastic solid
interacts with the fluid.

### Fluid properties

The fluid is considered to be **Newtonian**,
**incompressible** and its state is described by the velocity and
pressure fields . The balance equations are

The material constitutive equation is

The constant density of the fluid is and the viscosity is denoted by . The Reynolds number is defined by , with the mean velocity , radius of the cylinder and height of the channel (see Figure 1).

### Structure properties

The structure is assumed to be **elastic** and
**compressible**. Its configuration is described by the
displacement , with velocity field
. The balance equations are

Written in the more common Lagrangian description, i.e. with respect to some fixed reference (initial) state , we have

The material is specified by giving the Cauchy stress tensor
(the 2nd Piola-Kirchhoff stress tensor is then given
by ) by the
following constitutive law for the **St. Venant-Kirchhoff**
material ()

The density of the structure in the undeformed configuration is . The elasticity of the material is characterized by the Poisson ratio ( for a compressible structure) and by the Young modulus . The alternative characterization is described by the Lam� coefficients and (the shear modulus):

### Interaction conditions

The boundary conditions on the fluid solid interface are assumed to be

### Domain definition

The domain is based on the 2D version of the well-known CFD benchmark in
[TurekSchaefer1996] and shown here in Figure 1. By omitting the elastic bar
behind the cylinder one can exactly recover the setup of the *flow around cylinder*
configuration which allows for validation of the flow part by comparing the results with
the older flow benchmark.

- The domain dimensions are: length , height .
- The circle center is positioned at (measured from the left bottom corner of the channel) and the radius is .
- The elastic structure bar has length and height , the right bottom corner is positioned at , and the left end is fully attached to the fixed cylinder.
- The control points are , fixed with the structure with , and .

The setting is intentionally non-symmetric (see [TurekSchaefer1996]) to prevent the dependence of the onset of any possible oscillation on the precision of the computation.

geometry parameters | value [m] | |
---|---|---|

channel length | L | 2.5 |

channel width | H | 0.41 |

cylinder center position | C | (0.2,0.2) |

cylinder radius | r | 0.05 |

elastic structure length | l | 0.35 |

elastic structure thickness | h | 0.02 |

reference point (at t=0) | A | (0.6,0.2) |

reference point | B | (0.2,0.2) |

### Boundary conditions

- A parabolic velocity profile is prescribed at the left channel
inflow
- The outflow condition can be chosen by the user, for example
*stress free*or*do nothing*conditions. The outflow condition effectively prescribes some reference value for the pressure variable . While this value could be arbitrarily set in the incompressible case, in the case of compressible structure this will have influence on the stress and consequently the deformation of the solid. In this proposal, we set the reference pressure at the outflow to have*zero mean value*. - The
*no-slip*condition is prescribed for the fluid on the other boundary parts. i.e. top and bottom wall, circle and fluid-structure interface .

### Initial conditions

Suggested starting procedure for the non-steady tests is to use a smooth increase of the velocity profile in time as

### Material parameters

material | [] | E [] | [] | |
---|---|---|---|---|

polybutadiene | 910 | 0.50 | 1.6 | 0.53 |

polyurethane | 1200 | 0.50 | 25 | 8.3 |

polypropylene | 1100 | 0.42 | 900 | 317 |

PVC | 1400 | 0.42 | 1500 | 528 |

steel | 7800 | 0.29 | 210000 | 81400 |

cork | 180 | 0.25 | 32 | 12.8 |

material | [] | E [] | [] | |
---|---|---|---|---|

air | 1.23 | 0.015 | 0.018 | |

aceton | 790 | 0.405 | 0.32 | |

ethyl alcohol | 790 | 1.4 | 1.1 | |

oil, vegetable | 920 | 76.1 | 70 | |

water | 1000 | 1.14 | 1.14 | |

blood | 1035 | 3 -- 4 | 3 -- 4 | |

glycerine | 1260 | 1127 | 1420 | |

honey | 1420 | 7042 | 10000 | |

mercury | 13594 | 0.0114 | 1.55 |

An overview of certain material properties for some relevant fluids and elastic materials is shown in the Table. The choice of the parameters for the benchmark is guided by several requirements:

First, we would like the flow to be in the laminar regime, which implies "small" Reynolds numbers. On the other hand, the flow should be capable of deforming the elastic structure. A typical fluid candidate for such experiments is glycerine.

In order not to introduce additional numerical complications connected with high aspect ratios in the geometry, the deformable structure has a certain thickness which requires that the stiffness of the material should be low enough to allow significant deformations. Certain rubber-like materials fit into such a setting, namely polybutadiene (for a future incompressible configuration) and polypropylene.

In the table the material parameters are presented for 2 combinations of glycerine and selected rubber-like material.

parameter | polybutadiene & glycerine | polypropylene & glycerine |
---|---|---|

[] | 0.91 | 1.1 |

0.5 | 0.42 | |

[] | 0.53 | 317 |

[] | 1.26 | 1.26 |

[] | 1.13 | 1.13 |