Optimization of the Shape of a Three-Dimensional Body for Radiation Heat Flux

A solution is derived for the variational problem on the shape of a three-dimensional slender body subjected to a minimal radiation heat flux during its motion in the atmosphere with a constant hypersonic velocity. The distribution of radiation flux over the surface is approximated in the form of dependence on the local surface slope. The limitations on the body shape are provided by the isoperimetric conditions for the bulk of the body, for the area of its wettable surface, for the area of its bottom cross section, and for the perimeter of the bottom cross section, as well as the limitation on the limiting value of the wave drag. It is found that the transverse contour of the optimal body may have the form of a circle, of a star-shaped transverse contour, and of a contour consisting of arcs of a circle and line segments. It is demonstrated that the use of optimal three-dimensional bodies enables one to significantly (by more than $50\%$) reduce the radiation heat flux to the body surface compared with bodies of revolution which have the same preassigned geometric characteristics.