Steady, spatial, algebraically growing eigenfunctions are now known to occur in several important classes of boundary-layer flow, including two-dimensional hypersonic boundary layers and more recently in Blasius boundary layers subject to three-dimensional linearized disturbances, and in more general three-dimensional boundary layers. These spatial eigensolutions are particularly important and intriguing, given that they exist within the broad limits of the classical steady boundary-layer approximation, and as such are independent of Reynolds number.

In this paper we make the natural extension to these previous (stability) analyses by incorporating the effects of unsteadiness into the model for treating disturbances to a quite general class of similarity-type boundary-layer flows. The flow disturbances are inherently non-parallel, but this effect is properly incorporated into the analysis.

A further motivation for this paper is that Duck et al. (1999, 2000) have shown that by permitting a spanwise component of flow within a boundary layer of the appropriate form (in particular, growing linearly with the spanwise coordinate), it is found that new families of solutions exist – even the Blasius boundary layer has a three-dimensional ‘cousin’. Therefore a further aim of this paper is to assess the stability of the different solution branches, using the ideas introduced in this paper, to give some clues as to which of the solutions may be encountered experimentally.

Several numerical methods are presented for tackling various aspects of the problem. It is shown that when algebraically growing, steady eigensolutions exist, their effect remains important in the unsteady context. We show how even infinitesimal, unsteady flow perturbations can provoke extremely large-amplitude flow responses, including in some cases truly unstable flow disturbances which grow algebraically downstream without bound in the linear context. There are some interesting parallels suggested therefore regarding mechanisms perhaps linked to bypass transition in an important class of boundary-layer flows.