Cauchy integral formula for generalized analytic functions in hydrodynamics

Abstract

It is shown that for several classes of generalized analytic functions arising in linearized equations of hydrodynamics and magnetohydrodynamics, the Cauchy integral formulae follow from the one for generalized holomorphic vectors in a uniform fashion. If hydrodynamic fields (velocity, pressure and vorticity) admit representations in terms of corresponding generalized analytic functions, those representations and the Cauchy integral formulae form two essential parts of the generalized analytic function approach, which readily yields either closed-form solutions or boundary integral equations. This approach is demonstrated for problems of axisymmetric and asymmetric Stokes flows, two-phase axisymmetric Stokes flows, two-dimensional and axisymmetric Oseen flows.