Two-dimensional Stokes and Hele-Shaw flows with free surfaces

We discuss the application of complex variable methods to Hele-Shaw flows a nd two-dimensional Stokes flows, both with free boundaries. We outline the theory for the former, in the case where surface tension effects at the mov ing boundary are ignored. We review the application of complex variable met hods to Stokes flows both with and without surface tension, and we explore the parallels between the two problems. We give a detailed discussion of co nserved quantities for Stokes flows, and relate them to the Schwarz functio n of the moving boundary and to the Baiocchi transform of the Airy stress f unction. We compare the results with the corresponding results for Hele-Sha w hows, the principal consequence being that for Hele-Shaw flows the singul arities of the Schwarz function are controlled in the physical plane, while for Stokes flow they are controlled in an auxiliary mapping plane. We illu strate the results with the explicit solutions to specific initial value pr oblems. The results shed light on the construction of solutions to Stokes f lows with more than one driving singularity, and on the closely related iss ue of momentum conservation, which is important in Stokes flows, although i t does not arise in Hele-Shaw flows. We also discuss blow-up of zero-surfac e-tension Stokes flows, and consider a class of weak solutions, valid beyon d blow-up, which are obtained as the zero-surface-tension limit of flows wi th positive surface tension.