zolabar / conformalmappingsympy

This repository hosts code that illustrates the content of the presentation and proceeding Conformal Mappings with Sympy: Towards Python-driven Analytical Modeling in Physics a collaborative work of Z. Lauer-Baré and E. Gaertig presented on THE 20th PYTHON IN SCIENCE CONF. (SCIPY 2021).

Conformal Mappings with SymPy

This repository hosts code that illustrates the content of the presentation and conference proceeding Conformal Mappings with SymPy: Towards Python-driven Analytical Modeling in Physics, a collaborative work of Zoufiné Lauer-Baré and Erich Gaertig presented on THE 20th PYTHON IN SCIENCE CONF. (SCIPY 2021).

Python scripts and Jupyter notebooks are provided.

Please refer to

Lauer-Baré Z. and Gaertig E., Conformal Mappings with SymPy: Towards Python-driven Analytical Modeling in Physics. Lauer-Baré, Z. & Gaertig, E. In Agarwal, M., Calloway, C., Niederhut, D., & Shupe, D., editors, Proceedings of the 20th Python in Science Conference, pages 85 - 93, 2021.

when using formulae, code, figures or animations from this repository. The conference talk can be seen on the Enthought YouTube channel.

The theoretical methods used here are conformal mappings, inspired by PHW33 and BC09 and Taylor-expansions, following LGK21 and LGKS23. These methods are used to solve the Stokes problem in an eccentric annular domain for Couette-Poiseuille flow and to calculate the corresponding flow force in a postprocessing step, as well as analyzing the limits for small gaps. The context of this work is the modelling of viscous fluid power systems (see LGK21 for more details).

Applications of conformal mappings with SymPy in the context of inviscid irrotational flow can be found on Plotting streamlines with Matplotlib and SymPy (T. S. Yu). Further applications of conformal mappings with SymPy in the context of inviscid irrotational flow applied to naval engineering are described in G21 with an open Python code repository in naval Python and SymPy.

Further, a Python package with cloud computing possibility via binder, for visualizing conformal mappings interactively, based on SymPy, NumPy and Plotly can be found at the page conformalMaps; see LA21.

Transformation of eccentric annulus to concentric annulus

code_block_moebius.py and moebius.ipynb with a Möbius transform of the type

The following animation was created with an adapted version of the code from the interactive Python code for conformal mappings mentioned above, where the results from LG21 were implemented.

Transformation of eccentric annulus to rectangle

code_block_bipolar.py and bipolar.ipynb with a conformal mapping related to bipolar coordinates

The following animation was created with an adapted version of the code from the interactive Python code for conformal mappings mentioned above, where the results from LG21 were implemented.

Postprocessing

The postprocessing is shown in the file moebius.ipynb , due to LaTeX rendering of web browser based Jupyter notebook.

Flow force calculation with diff and Taylor expansion of force in the gap

with series.

Literature

[BC09] Brown J.W., Churchill R.V., Complex variables and applications, Eighth edition, McGraw-Hill Book Company; 2009

[G21] Grm, A., Ships Added Mass Effect on a Flexible Mooring Dolphin in Berthing Manoeuvre. J. Mar. Sci. Eng. 2021, 9, 108. https://doi.org/10.3390/jmse9020108

[LGK21] Lauer-Baré Z., Gaertig E., Krebs J., Arndt C., Sleziona C., Gensel A., A note on leakage jet forces: Application in the modelling of digital twins of hydraulic valves, International Journal of Fluid Power, 2021, Vol. 22 (1), 113–146

[LGKS23] Lauer-Baré Z., Gaertig E., Krebs J., Sleziona C. Analytical formulae in fluid power, quo vadis in times of CFD and I4.0? 13th International Fluid Power Conference, p. 866-879, 2023

[LG21] Lauer-Baré Z. and Gaertig E., Conformal Mappings with SymPy: Towards Python-driven Analytical Modeling in Physics. Lauer-Baré, Z. & Gaertig, E. In Agarwal, M., Calloway, C., Niederhut, D., & Shupe, D., editors, Proceedings of the 20th Python in Science Conference, pages 85 - 93, 2021.

[LA21] Lauer-Baré Z. and Aditya, Conformal-Maps: Code for interactive conformal mapping with python and jupyter notebook (v1.0.1). Zenodo. 2021 https://doi.org/10.5281/zenodo.5717868.

[PHW33] Piercy N.A.V., Hooper M.S., Winny H.F., LIII. Viscous flow through pipes with cores, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1933