Exact Navier–Stokes Solutions for Strained Shear Layers on a Stretching Sheet

A recent similarity solution of the two-dimensional unsteady laminar boundary-layer equations [B. H. Sun, Phys. Fluids 36, 083616 (2024)], expressed in terms of Kummer functions, was proposed as the solution for a semi-infinite flat plate impulsively set into motion, but its boundary conditions cannot be satisfied at any finite time. We show that the solution is, in fact, a Burgers–Townsend strained shear layer superposed on the decaying, zero-pressure-gradient straining flow u = x/t, v = −y/t. For this class the nonlinear terms cancel identically, and the shear component obeys a linear equation that a Lundgren-type change of variables reduces to the heat equation; the Kummer functions are the fractional repeated-integral error functions i^2/3 erfc. The two forms of the published solution are copies of a single flat-wall solution related by Prandtl’s transposition theorem, and the flat-wall representative is an exact solution of the full Navier–Stokes equations, not merely of the boundary-layer equations. The correct physical setting is a sheet stretching at the decaying rate 1/(t + t₀), the S = −1 point of the Wang–Andersson family of unsteady stretching-sheet flows. Within this setting we solve three well-posed initial–boundary-value problems in closed form: (1) impulsive translation of the stretching sheet, whose wall shear stress crosses over from the Rayleigh law to −1.370 ρU_s√ν/t; (2) the decay of a pre-existing shear layer, governed by the exact invariant \( $s^3\!\int_0^\infty Y u'\ \)dY and (3) arbitrary wall histories, including power-law ramps with wall-shear factor √3 Γ(p + 1)/Γ(p + 1/2) and an oscillating sheet that relaxes to the classical Stokes layer. All three problems are two-dimensional – both velocity components are nonzero and the field depends on x – in contrast to Stokes’ first and second problems, which are one-dimensional and are recovered only as limits. The published Kummer solution is identified as the self-similar late-time attractor of problem (1). The unsteady Blasius problem for a finite plate remains open.