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Application of the Biot–Savart law to parabolic vortex segments using elliptic integrals

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Abstract

The Biot–Savart law is used in aerodynamic theory to calculate the velocity induced by curved vortex lines. Explicit formulas are developed, using multivariate Appell hypergeometric functions, for the velocity induced by a general parabolic vortex segment. The formulas are derived by constructing a particular pencil of elliptic curves whose period integrals provide the solution to the induced velocity. We use numerical integration and a perturbation expansion to evaluate the validity of our formulas.

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Acknowledgements

Funding was provided by Simons Foundation (Grant No. 202367).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84322, USA

    Andreas Malmendier

  2. Department of Mechanical and Aerospace Engineering, Utah State University, Logan, UT, 84321, USA

    Jackson T. Reid

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  1. Andreas Malmendier
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Correspondence to Andreas Malmendier.

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Malmendier, A., Reid, J.T. Application of the Biot–Savart law to parabolic vortex segments using elliptic integrals. Z. Angew. Math. Phys. 71, 104 (2020). https://doi.org/10.1007/s00033-020-01319-3

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  • DOI: https://doi.org/10.1007/s00033-020-01319-3

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