Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Stationary combustion modes with allowance for heat losses


An attempt is made to investigate the number of possible stationary combustion modes in a continuous-flow semi-infinite pipe with allowance for heat losses through the walls. Cases of a zero-order reaction in the reaction mixture or similarity of the concentration and temperature fields are considered. The equations are averaged with respect to the transverse coordinate η. Within the framework of these approximations it is found that the number of stationary combustion modes is determined by the roots θn of some function. The roots θ2k correspond only to trivial unstable solutions. The roots θ2k−1 correspond to modes possible within broad regions of variation of the parameters characterizing the temperature of the mixture, the mixture feed rate, and the rate of heat removal. These regions intersect, forming zones where several stationary modes coexist. In these zones, apart from monotonic solutions there may also be solutions that initially make several oscillations. It is shown that the latter are obviously unstable and, in the last analysis, lead to one of the monotonic modes. The common case of not more than three roots is examined in detail.

If the heat release function can change sign, then a similar picture is also observed in the absence of heat losses through the walls (the roots θ2k−1 and θ2k may change roles). In this case it is no longer necessary to average the equations with respect to η, since there will not be any corresponding derivatives.

This is a preview of subscription content, log in to check access.


  1. 1.

    D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics [in Russian], Izd-vo AN SSSR, 1947.

  2. 2.

    I. M. Gel'fand, “Some problems of quasi-linear equations,” Uspekhi matem. nauk, vol. 14, no. 2, 1959.

  3. 3.

    A. G. Istratov and V. B. Librovich, “Stability of solutions in the steady-state theory of thermal explosion,” PMM, vol. 23, no. 2, 1963.

  4. 4.

    S. A. Kaganov, “Stationary theory of thermal autoignition,” PMTF, no. 1, 1963.

  5. 5.

    R. M. Zaidel and Ya. B. Zel'dovich, “Possible regimes of steady-state combustion,” PMTF, no. 4, 1962.

  6. 6.

    A. G. Merzhanov and A. K. Filbnenko, “Thermal autoignition of a homogeneous gas mixture in a flow,” DAN SSSR, vol. 152, no. 1, 1963.

  7. 7.

    E. A. Chernova, “Stationary combustion,” PMTF [Journal of Applied Mechanics and Technical Physics], no. 6, 1966.

  8. 8.

    S. A. Kaganov, “Flame propagation processes,” PMTF [Journal of Applied Mechanics and Technical Physics], no. 4, 1965.

  9. 9.

    V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Gostekhizdat, 1949.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chernova, E.A. Stationary combustion modes with allowance for heat losses. J Appl Mech Tech Phys 8, 55–58 (1967).

Download citation


  • Stationary Mode
  • Feed Rate
  • Temperature Field
  • Heat Release
  • Heat Loss