Upstream Influence in Similarity Reductions of the Unsteady Navier–Stokes Equations

Similarity reductions of unsteady two-dimensional flow are almost always constructed from the boundary-layer equations, in which streamwise diffusion and the transverse momentum balance are discarded at the outset. We ask when such a reduction is also a reduction of the full Navier–Stokes equations, using the diffusion-time-scale variables \(\eta=y/\delta(x)\), \(\tau=\nu t/[\delta(x)]^2\) of Sun [32]. All chain rules collapse onto the stretching operator \(\mathcal{D}=\eta\partial_\eta+2\tau\partial_\tau\), both momentum equations transform in closed form, and the discarded terms enter through a single coefficient \(\varepsilon=\delta'^2\). For the family \(U=Cx^m\) the reduction closes in \( (\eta,\tau) \) alone if and only if \(m=1\) or \(m=-1\). The second case is a diverging channel, and there the reduced problem is elliptic in \( (\eta,\tau) \), with principal symbol \( (\xi_\eta^2+\varepsilon s^2)^2 \) degenerating only on \(\tau=0\): the similarity time is not an evolution variable, data must be posed at both ends, and marching in \(\tau\) is inconsistent. Solving the resulting boundary-value problem, we find that the influence of the terminal condition decays as \(\exp[-(T-\tau)/L]\), and that \(L\) collapses on the wedge half-angle, \(L\simeq2.2\,\theta_w\), to within 14% over a factor of 16 in Reynolds number, the prefactor depending on the sense of the flow but the scaling not. The upstream influence is therefore geometric rather than viscous: it vanishes with the divergence angle and is essentially independent of Re, so that marching is safe in slender geometries at any Reynolds number and unsafe in wide ones however viscous. Three exact solutions in elementary functions, obtained from the invariance of \(\zeta=\eta/\sqrt{\tau}=y/\sqrt{\nu t}\) under \( D \), and the Jeffery–Hamel steady limit, serve to verify the reduction throughout.