Mathematical model and method of solving the generalized Dirichle problem of heat exchange of a cut count


  • M. Berdnyk



Fourier series, Dirichlet boundary value problem, Laplace integral transformation, relaxation time


The choice of thermal protection of the rocket fairing is approached with special care, because the fairing must protect against aerodynamic heating, radiation, temperature changes. Currents with large Mach numbers are accompanied by gas-dynamic and physicochemical effects. When flowing around the blunt body, a shock wave is formed, which departs from the body, remaining in the vicinity of the frontal point almost equidistant to its surface. Physico-chemical effects are due to rising temperatures caused by the inhibition of gas by the shock wave. At the same time there is a transition of kinetic energy of a stream rushing in thermal, fluctuating degrees of freedoms of gas molecules are excited, its dissociation and even ionization begins. Therefore, among the problems of great theoretical and practical interest is the problem of studying the temperature fields arising in the fairings for missiles in the form of a truncated cone, which rotate around its axis, given the finiteness of the rate of heat propagation. In the article the mathematical model of calculation of temperature fields for a truncated cone is constructed for the first time which approximately models distribution of temperature fields which arise in fairings for rockets, with taking into account the angular velocity and the final speed heat distribution in the form of a boundary value problem of mathematical physics for hyperbolic equation of thermal conduc-tivity with boundary conditions Dirichlet. A new integral transformation for a two-dimensional finite space is constructed, in the application of which the temperature field in the form of a convergent series is found. The solution found can be used to predict the possible value of thermomechanical stresses, to promote the correct choice of technological parameters, objective control, allows to identify ways to improve the thermal protection of fairings for missiles.


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