Development and separation of a compressible laminar boundary layer under the action of a very sharp adverse pressure gradiant

We consider a compressible laminar boundary layer with uniform pressure when x < x0 and a prescribed large adverse pressure gradient when x > x0. The Illingworth-Stewartson transformation is applied, and the transformed external velocity u1(x) then chosen such that \[ \lambda = -\frac{x_0}{u^2_0}u_1\frac{du_1}{dx}\frac{T_w}{T_s} \] is constant, where Ts is the stagnation temperature.For large λ, when a thin sublayer exists as the layer reacts to the sharp pressure gradient, inner and outer asymptotic expansions are derived and matched for functions F and S which determine the stream function and the temperature. The equations for F and S are largely uncoupled, in that the first approximation to F is independent of S, the first approximation to S depends only on the first approximation to F, and so on.