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On Green's function of the linearized viscous transonic equation

USSR Computational Mathematics and Mathematical Physics
Publication Date
DOI: 10.1016/0041-5553(72)90013-4
  • Mathematics


Abstract GREEN'S function of the linearized viscous transonic equation for the problem of flow past a symmetric plane finite body is obtained in explicit form. Quite severe changes in the flow parameters often occur in narrow regions adjacent to a shock wave. The flow parameter gradients in such regions can be so large that the influence of viscosity and thermal conductivity has to be taken into account, as well as the non-linear features of the motion. Such flows are termed short waves. With the theory of short waves is associated the theory of transonic flows. A general theory of short waves and its connection with transonic flows was outlined in [1]. In some concrete physical problems, examples have been described of short waves arising in stationary flows, where dissipative processes occur. The Mach reflection of weak shock waves from a wedge was discussed in [2]. In [3–5], the interaction of a weak shock wave with a boundary layer was examined. In [1], the asymptotic picture of sonic unidealized gas flow past a finite body was considered. In the papers cited, the conclusions were based on an equation first derived by Lipmann, Ashkenase and Cowl (see [6], Chapter V, Section 5) when describing the structure of a weak shock wave arising at a wing during the onset of a shock stall. The solutions of this equation which are supported by physical considerations are as a rule connected with the behaviour of its linear part. But no satisfactory strict mathematical proof of this fact has yet been offered. In the present paper we construct the Green's function of the linear part of the equation for the external boundary problem of flow past a finite symmetric body. It has quite a simple form, which makes it possible to investigate further the Lipmann, Ashkenase and Cowl equation in integral form. It should be mentioned that this equation is being increasingly referred to in the literature as the viscous-transonic, or simply VT, equation. The VT equation describing rotationally symmetric flows was first obtained and investigated in [7].

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