Was Squire wrong - New 3D Oblique Spatially Evolving Instability Modes

In 1933 Squire showed that for time-evolving perturbations, 2D instabilities occur at the smallest Reynolds number – the critical Reynolds number. We prove that this is not necessarily so for spatially evolving 3D modes, for which we introduce both a complex streamwise and spanwise wavenumber giving rise to oblique 3D modes. Such modes admit in the $x-z$-plane neutral stability lines (NSL), which are oblique to the main flow direction, und perpendicular to it shows maximum growth. As such extending Squires idea by invoking symmetry methods in parameter space the key result is that oblique 3D instabilities at a Reynolds number below the critical 2D Reynolds number exist. Other than for temporally evolving modes, however, for spatially evolving modes the additional condition of group velocity (GV) v_g must be taken into account, which states that the GV has to propagate in the direction of the spatially increasing mode. For 3D instabilities the vector v_g of the GV must thus point into the unstable region, i.e. cross the above mentioned NSL in the $x-z$-plane. Details on the extension of Squire's theory based on Lie symmetries will be present and exemplified using plane Couette flow, which is known to not admit a time-evolving instability.