## Abstract

A laminar boundary layer in supersonic flow can evolve spontaneously from an undisturbed form to a separated state, where an adverse pressure gradient thickens the boundary layer, thus displacing the external streamlines, which leads to the original pressure gradient. A linearized study by Lighthill was later generalized by Stewartson in a triple-deck analysis, in which the equations for the main deck are $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0, u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = -\frac{\mathrm{d}p}{\mathrm{d}x} + \frac{\partial^2u}{\partial y^2},$ subject to $u = v = 0 \operatorname{at} y = 0, u \rightarrow y \operatorname{as} x \rightarrow - \infty,$ and $u \rightarrow y - \int^x_{-\infty} p(t)dt \operatorname{as} y \rightarrow \infty.$ The problem is here studied by means of the series expansions $u = y - \sum^\infty_{n=1} a_n \mathrm{e}^{nkx}f'_n(y), v = \sum^\infty_{n=1} nka_n \mathrm{e}^{nkx}f_n(y), p = \sum^\infty_{n=1} \mathrm{a}_n \mathrm{e}^{nkx}.$ This gives a sequence of equations for the $f_n(y)$, of which the first 15 have been solved. Appropriate series for the pressure $p$ and the skin-friction $\tau$ have been derived and analysed, and previous numerical solutions of the partial differential equations by Williams have been well confirmed, in some instances to greater accuracy. Among the conclusions reached are the following. (i) The value of p at separation is calculated to be 1.025 947 44. (ii) As $x \rightarrow \infty, p$ tends to a constant value $p_0 = 1.7903$, compared with the value 1.800 given by Williams. (iii) In the separated region, the most negative value taken by $\tau$ is - 0.149 408 1.

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