Steady two-dimensional viscous flow in a jet

Abstract

An idealized two-dimensional flow due to a point source ofxmomentum is discussed. In the far field the flow is modelled by a jet region of large vorticity outside which the flow is potential. After use of the transformation\[ \zeta^3 = (\xi + i\eta)^3 = x + iy, \]the equations suggest naively obvious asymptotic expansions for the stream function in these two regions, namely\[ \sum_{n=0}^{\infty}\xi^{1-n}f_n(\eta)\quad {\rm and}\quad\sum_{n=0}^{\infty}\xi^{1-n}F_n(\eta/\xi) \]respectively. Consistency in matching these expansions is achieved by including logarithmic terms associated with the occurrence of eigensolutions.Fnis easy to find andJncan be found in closed form so the inner and outer eigensolutions may be fully determined along with the complete structure of the expansions.