Unsteady asymptotic solutions of the two-dimensional Euler equations

A technique is described for deducing a class of unsteady asymptotic solutions of the two-dimensional Euler equations. In contrast to previously known analytical results, the vorticity function $\left [ {\omega \left ( {x, y, t} \right )} \right ]$ for these solutions has a complicated dependence on the spatial coordinates $\left ( {x, y} \right )$ and time $\left ( t \right )$. The results obtained are in implicit form and are valid in those regions of space and time where $t\omega \to {o^ + }$ or $t\omega \to + \infty$. These asymptotic solutions may be split into an unsteady, two-dimensional and irrotational basic flow and a disturbance that is strongly nonlinear at appropriate locations within the domain of validity. The generality and complexity of these solutions make them theoretically interesting and possibly useful in applications.