`Law of Hagen-Poiseuille`

*Module:* `tutorials.06_fluid.01_HagenPouseille_Stationary.run`

*Section author: Elias Karabelas <elias.karabelas@medunigraz.at>*

This example demonstrates a simple application for fluid dynamics in a straight cylindrical pipe.

The law of Hagen-Poiseuille is a physical law that gives the
pressure drop for an incompressible Newtonian fluid in the laminar regime flowing through a long cylindrical pipe of constant cross section. This geometric
setup is depicted in figure `fig-pouseille-setup`

.

The law states that

where

- is the pressure drop (in Pascal)
- is the length of the pipe (in meter)
- is the dynamic viscosity of the fluid (in Pascal seconds)
- is the volumetric flow rate (in cubic meter per seconds)
- is the pipe radius (in meter)

For the derivation we will use the **Navier-Stokes equations**

First we assume that and constant. To derive Hagen-Poiseuille’s law we will rewrite the Navier-Stokes equations in cylindrical coordinates

()

Additionally the following is assumed

- Steady flow, meaning
- Radial and swirl components of the fluid velocity are zero meaning
- The flow is assumed to be axisymmetric, meaning
- The flow is fully developed, meaning

First with this assumptions the continuity equation (fourth line of ()) is trivially fulfilled. Further it follows that

()

From this we can deduce that . Plugging this into the last line of () we can solve for by twice integrating and get

with some arbitrary integration constants . To get a closed representation we will use the following observations

- On the boundary of the cylinder () we want
- For we want to be finite

The first obervation yields and the second yields that . Putting all together we get a closed expression for

As a last assumption, let $p$ be decreasing linearly from to yielding . Now we can calculate the volumetric flux through the cross section of the cylinder as

which gives the Law of Hagen-Poiseuille.

For the experiments we choose a cylindrical pipe with . We will vary the pressure drop as well as the viscosity . For simplicity we assume that . We have three different discretizations (coarse, medium, fine). The coarse discretization has a maximal edge length of , the medium has , and the fine has .

[Pa] | [Pa s] | ||
---|---|---|---|

Experiment 1 | 1.0 | 0.01 | 0.625 |

Experiment 2 | 5.0 | 0.01 | 3.125 |

Experiment 3 | 10.0 | 0.01 | 6.25 |

Experiment 4 | 15.0 | 0.01 | 9.38 |

Experiment 5 | 1.0 | 0.001 | 62.5 |

Experiment 6 | 5.0 | 0.001 | 312.5 |

Experiment 7 | 10.0 | 0.001 | 625.0 |

Experiment 8 | 15.0 | 0.001 | 937.5 |

The results for the coarse discretization are depicted in Table `table-results-coarse`

, the results for the
medium discretization are depicted in Table `table-results-medium`

, and the results for the fine discretization
are depcited in Table `table-results-fine`

. One can clearly see the influence of the Reynolds number on the
accuracy of the numeric solution. For in a cylindrical pipe the assumptions for the law
of Hagen-Poiseuille are not longer valid.

To run your own validation experiment just type in the command

```
./run.py --discretization TYPE --mu VALM --rho VALR --pressureDrop VALP --np NP
```

Here **TYPE** can be either *coarse*, *medium* or *fine*. This relates to the meshes used. The value **VALM** is the
value for the dynamic viscosity of the fluid in Pascal seconds. Value **VALR** is the value for the fluid
density in kilogram per cubic meters. **VALP** denotes the value for the pressure drop
given in Pascal. At the inflow a value of will be prescribed and at the outflow a value of
. Last, **NP** stands for the number of processors.

The above code uses the standard meshes provided with this run script. This cylindrical mesh has the default length , and default radius . You can also generate your own mesh. This is achieved via the flags:

```
./run.py --discretization TYPE --mu VALM --rho VALR --pressureDrop VALP --np NP --generate 1 --radius R --length L
--radiusfactor HR
```

Here, **R** denotes the radius of your cylinder in centimeter, **L** denotes the length of your cylinder in meter and
**HR** gives the factor for calculating the average edgelength in the mesh. It should be between zero and one. The edgelength is calculated as
. To use this functionalty you need to have **meshtool** in your search path as well as **gmsh**.
The software package **gmsh** can be downloaded from here. Depending on the chosen parameters the
generation of the meshes can take some time.

Coarse

coarse (1) | 4.8157e-04 | 4.9087e-04 | 0.0190 |

coarse (2) | 2.4077e-03 | 2.4544e-03 | 0.0190 |

coarse (3) | 4.8157e-03 | 4.9087e-03 | 0.0190 |

coarse (4) | 7.2243e-03 | 7.3631e-03 | 0.0188 |

coarse (5) | 4.5198e-03 | 4.9087e-03 | 0.0792 |

coarse (6) | 2.0044e-02 | 2.4544e-02 | 0.1833 |

coarse (7) | 3.6429e-02 | 4.9087e-02 | 0.2579 |

coarse (8) | 5.0822e-02 | 7.3631e-02 | 0.3098 |

Medium

medium (1) | 4.8869e-04 | 4.9087e-04 | 0.0044 |

medium (2) | 2.4435e-03 | 2.4544e-03 | 0.0044 |

medium (3) | 4.8875e-03 | 4.9087e-03 | 0.0043 |

medium (4) | 7.3321e-03 | 7.3631e-03 | 0.0043 |

medium (5) | 4.8394e-03 | 4.9087e-03 | 0.0141 |

medium (6) | 2.3317e-02 | 2.4544e-02 | 0.05 |

medium (7) | 4.5841e-02 | 4.9087e-02 | 0.0661 |

medium (8) | 6.7741e-02 | 7.3631e-02 | 0.08 |

Fine

fine (1) | 4.9044e-04 | 4.9087e-04 | 0.0009 |

fine (2) | 2.4523e-03 | 2.4544e-03 | 0.0009 |

fine (3) | 4.9046e-03 | 4.9087e-03 | 0.0008 |

fine (4) | 7.3573e-03 | 7.3631e-03 | 0.0008 |

fine (5) | 4.9081e-03 | 4.9087e-03 | 0.0001 |

fine (6) | 2.3535e-02 | 2.4544e-02 | 0.0411 |

fine (7) | 4.6091e-02 | 4.9087e-02 | 0.0610 |

fine (8) | 6.9728e-02 | 7.3631e-02 | 0.0530 |