Exact Solutions for Flows in Periodic Domains

In this work it is shown that recent advances in conformal geometry can be leveraged to obtain analytic solutions for flows in periodic domains. The solutions are constructed in a parametric circular domain and then conformally mapped to the periodic physical domain. By expressing the problem in terms of the transcendental Schottky--Klein prime function, the ensuing solutions are valid for domains of arbitrary connectivity, i.e. any number of objects per period window. Moreover, the conformal mapping to the desired physical domain may be constructed using a new periodic Schwarz--Christoffel formula. The mathematical analysis is valid for conformally invariant equations and is therefore applicable to a range of scenarios in fluid mechanics including potential flows, advection-diffusion problems and interfacial dynamics. Accordingly, the solutions find relevance in areas such as vortex dynamics, transport phenomena, turbomachinery flows, mesh generation and fractal growth.