We formulate and solve the conjugate problem for Lagrange multipliers connected with designing a Laval nozzle optimal contour, including its subsonic part. In approximation of an ideal (inviscid and nonheat-conducting) gas, the sought contour provides a thrust maximum under a number of constraints; in particular, given total nozzle length and gas mass flow. For a contour of a contracting (subsonic) part of the nozzle (which is suspected to be optimal) we take an abrupt contraction. Because of the constraint on the nozzle length, the abrupt contraction can be a region of boundary extremum with positive permissible variations of the longitudinal (“axial”) coordinate of the contour. To clarify whether this is true, we use the method of Lagrange multipliers and formulate the conjugate problem for finding the Lagrange multipliers. As in the case of quasilinear Euler equations governing a flow, the linear equations in the conjugate problem are elliptic (hyperbolic) in the subsonic (supersonic) flow domain. The requirement that the conjugate problem be solvable for any contour highlights some features that can be of interest not only for this special problem, but, possibly for the general theory of mixed type partial differential equations.
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A. L. Gonor and A. N. Kraiko, “Some results of the study of optimal shape under supersonic and hypersonic velocities” [in Russian], In: Theory of Optimal Aerodynamic Shapes, pp. 455–492, Mir, Moscow (1969).
A. N. Kraiko, Variational Problems of Gas Dynamics [in Russian], Nauka, Moscow (1979).
A. N. Kraiko, N. I. Tillyaeva, and S. A. Shcherbakov, “Comparison of integrated characteristics and shapes of profiled contours of Laval nozzles with “smooth” and “abrupt” contractions” [in Russian], Izv. Akad. Nauk SSSR, Mekh. Zhid. Gaz. No. 4, 129-137 (1986); English transl.: Fluid Dyn. 21, No. 4, 615–623 (1986).
A. V. Bitsadze, Some Classes of Partial Differential Equations [in Russian], Nauka, Moscow (1981); English transl.: Gordon & Breach Sci., New York etc. (1988).
M. M. Smirnov, Equations of Mixed Type [in Russian], Vysshaya Shkola, Moscow (1985); English transl.: Am. Math. Soc., Providence, RI (1977).
G. G. Chernyi, Gas Dynamics [in Russian], Nauka, Moscow (1988); English transl.: CRC Press, (1994).
A. N. Kraiko, Theoretical Gasdynamics. Classical and Topical [in Russian], Torus Press, Moscow (2010).
A. A. Kraiko, A. N. Kraiko, K. S. P’yankov, and N. I. Tillyaeva, “Contouring the nozzles producing a uniform supersonic flow or a thrust maximum in the presence of a curvilinear sonic line” [in Russian], Izv. Ross. Akad. Nauk, Mekh. Zhid. Gaz. 47, No. 2, 97-113 (2012); English transl.: Fluid Dynam. 47, No. 2, 223-238 (2012).
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Translated from Problemy Matematicheskogo Analiza 80, April 2015, pp. 31-45.
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Kraiko, A.N., Tillyaeva, N.I. Conjugate Problem for Lagrange Multipliers and Consequences for Partial Differential Equations of Mixed Type. J Math Sci 208, 181–198 (2015). https://doi.org/10.1007/s10958-015-2436-z
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DOI: https://doi.org/10.1007/s10958-015-2436-z