Abstract The flow field of a vortex in a viscous shear flow is found by constructing a uniformly valid asymptotic expansion consisting of an inner solution field represented, to lowest order, by a two-dimensional, nonlinear, inviscid Stuart vortex and an outer solution field represented, to lowest order, by either a two-dimensional parallel or self-similar viscous flow. The technique involves scaling both the transverse and streamwise coordinates in the vicinity of the vortex as well as allowing for a ‘slow’ variation of the outer viscous flow. Criteria are established for both the size of the vortical structure and proximity to the boundary surfaces. The composite solution is a consistent mathematical picture of the flow field at a fixed streamwise location as the vortical structure evolves past this point. Such a formulation is also useful in the specification of boundary or initial conditions in numerical fluid dynamic calculations, where an inconsistent setting of these conditions leads to spurious results for rather long computation times.