# Unique Solvability in the Lavrent’ev–Bitsadze Model for Two Problems of Weakly Supersonic Symmetric Flow with Detached Shock Wave Past a Wedge

### Abstract

We study two problems on planar steady-state weakly supersonic potential flows of an ideal perfect gas with detached shock wave. In the first problem, we consider the flow past a finite wedge in an unbounded stream; the second problem deals with the flow past an infinite wedge in a steady jet. The free-stream velocity is close to the speed of sound, and therefore, the entropy increments $$\Delta S(\psi )=O(\varepsilon ^2)$$ on the shock wave and the derivative of the entropy with respect to the stream function are disregarded. In the velocity hodograph plane, the cross-coupled sub- and supersonic flow behind a shock wave is described by a solution of a Tricomi type problem. On part of the boundary depicting the shock wave, the directional derivative condition is set for the stream function. It is proved that its direction is not tangent to the domain boundary. The uniqueness of the solution for the problems under consideration follows from the “strong” Hopf maximum principle for uniformly elliptic equations. Replacing the Chaplygin equation with the Lavrent’ev–Bitsadze equation leads to two Hilbert problems for analytic functions with piecewise constant boundary conditions. The solutions of the Hilbert problems are expressed using the Schwarz operator.

This is a preview of subscription content, access via your institution. Fig. 1. Fig. 2. Fig. 3.
1. 1.

If the free-stream velocity exceeds some constant independent of the heat capacity ratio $$k$$, then on the subsonic segment of the shock polar there exists a point at which the vector $$\mathbf {\nu }$$ touches the shock polar, so that the vectors $$\mathbf {\nu }$$ on the neighboring sections lie on different sides of the shock polar [5, p. 46; 6, p. 45].

2. 2.

Under the condition

$$\frac {{\psi _{\eta }}}{{\psi _\beta }} =\beta \frac {7\eta +{{\eta }_\infty }}{5\eta +3{{\eta }_\infty }},$$

Guderley [9, p. 145 of the Russian translation] produced an asymptotic formula on the shock polar $$\beta =$$ $$({{\eta }_\infty }-\eta )\sqrt {({{\eta }_\infty }+\eta )/2}$$ as $${M_\infty }\to 1$$ which implies that the touching occurs only for $$\eta ={{\eta }_\infty }$$.

3. 3.

Instead of the last argument, one can refer to the “extremum principle” : “A solution of the Tricomi problem for Eq. (5) cannot attain a positive maximum or a negative minimum on an interval of the straight line $$\lambda =1$$.”

## REFERENCES

1. 1

Chaplygin, S.A., O gazovykh struyakh. Sobr. soch. T. II (On Gas Jets. Collected Works. Vol. II), Moscow–Leningrad: Gos. Izd. Tekh.-Teor. Lit., 1933.

2. 2

Frankl, F.I., On S.A. Chaplygin’s problems for mixed sub- and supersonic flows, Izv. Akad. Nauk SSSR. Ser. Mat., 1945, vol. 9, no. 2, pp. 121–143.

3. 3

Shifrin, E.G., On the problem of the flow of a supersonic jet past an infinite wedge, Izv. Akad. Nauk SSSR. Mekh. Zhidk. Gaza, 1969, no. 2, pp. 88–93.

4. 4

Bers, L., Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Chichester: John Wiley and Sons, 1958. Translated under the title: Matematicheskie voprosy dozvukovoi i okolozvukovoi gazovoi dinamiki, Moscow: Izd. Inostr. Lit., 1961.

5. 5

Shifrin, E.G. and Belotserkovskii, O.M., Transonic Vortical Gas Flows, Chichester–New York–Brisbane–Toronto–Singapore: Wiley, 1994.

6. 6

Shifrin, E.G., Potentsial’nye i vikhrevye transzvukovye techeniya ideal’nogo gaza (Potential and Vortical Transonic Ideal Gas Flows), Moscow: Fizmatlit, 2001.

7. 7

Smirnov, M.M., Uravneniya smeshannogo tipa (Mixed Type Equations), Moscow: Nauka, 1970.

8. 8

Serrin, J., Mathematical Principles of Classical Fluid Mechanics, Heidelberg: Springer-Verlag, 1959. Translated under the title: Matematicheskie osnovy klassicheskoi mekhaniki zhidkosti, Moscow: Fizmatlit, 1963.

9. 9

Guderley, K.G., Theorie Schallnaher Strömungen, Berlin: Springer-Verlag, 1957. Translated under the title: Teoriya okolozvukovykh techenii, Moscow: Izd. Inostr. Lit., 1960.

10. 10

Hopf, E., Elementary Bemerkungen über die Lösungen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus, in Sitz. Ber. Preuss. Akad. Wiss., Berlin, Math.-Phys. Kl. 19, 1927, pp. 147–152.

11. 11

Courant, R., Partial Differential Equations, Chichester: Wiley-VCH, 1962. Translated under the title: Uravneniya s chastnymi proizvodnymi, Moscow: Mir, 1964.

12. 12

Hopf, E., A remark on linear elliptic differential equations of second order, Proc. Am. Math. Soc., 1952, vol. 3, pp. 791–793.

13. 13

Bitsadze, A.V., On some mixed type problems, Dokl. Akad. Nauk SSSR, 1950, vol. 70, no. 4, pp. 561–565.

14. 14

Gakhov, F.D., Kraevye zadachi (Boundary Value Problems), Moscow: Gos. Izd. Fiz.-Mat. Lit., 1963.

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## ACKNOWLEDGMENTS

This paper was created with support from the Moscow Center for Fundamental and Applied Mathematics.

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Correspondence to E. I. Moiseev or E. G. Shifrin.

## Additional information

Translated by V. Potapchouck

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