I don't know if the application that I'm going to refer is well-known. I've read it from a book, Wikipedia has an article dedicated to the author E. O. Wilson, that refers in chapter 14 of the book [1] (I know the Spanish edition of the book Editorial Planeta, Crítica 2022); the chapter is potentially interesting for mathematicians.
Wikipedia has an article including the section In engineering applications from the Wikipedia article dedicated to Logarithmic spiral.
If I understad well whirls don't fit with your question/post, but I include the following references as general reference in case that you or the readers want to know these notes that I consider interesting, in fact these aren't applications, if I remember well [2] include notes about whirls, and [3] seems interesting for me (is a note in Spanish language).
Please let me to know (you or your colleagues) if my post isn't a good answer that I can to delete it in next few hours, many thanks.
References:
[1] Edward O. Wilson, Tales from the Ant World, Published by Liveright (2020).
[2] Leo Zippin, Uses of Infinity, Dover Publications (2000).
[3] Juan Luis Varona, $\zeta(2k)$ versus $zeta(k)$: una relación geométrica en una espiral, Miniaturas matemáticas, La Gaceta de la RSME Vol. 18 (2015), Núm. 2, Pág. 352.