Examples of steady subsonic flows in a convergent-divergent approximate nozzle. (English) Zbl 1139.35069
Summary: We construct special solutions of the full Euler system for steady compressible flows in a convergent-divergent approximate nozzle and study the stability of the purely subsonic flows. For a given pressure \(p_{0}\) prescribed at the entry of the nozzle, as the pressure \(p_{1}\) at the exit decreases, the flow patterns in the nozzle change continuously: there appear subsonic flow, subsonic-sonic flow, transonic flow and transonic shocks. Our results indicate that, to determine a subsonic flow in a two-dimensional nozzle, if the Bernoulli constant is uniform in the flow field, then this constant should not be prescribed if the pressure, density at the entry and the pressure at the exit of the nozzle are given; if the Bernoulli constant and both the pressures at the entrance and the exit are given, the average of the density at the entrance is then totally determined.
MSC:
| 35L60 | First-order nonlinear hyperbolic equations |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76H05 | Transonic flows |
References:
| [1] | Bers, L., Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Surv. Appl. Math., vol. 3 (1958), Wiley: Wiley New York · Zbl 0083.20501 |
| [2] | Chen, G.-Q.; Feldman, M., Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc., 16, 461-494 (2003) · Zbl 1015.35075 |
| [3] | Courant, R.; Friedrichs, K. O., Supersonic Flow and Shock Waves (1948), Interscience Publ.: Interscience Publ. New York · Zbl 0041.11302 |
| [4] | Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss., vol. 224 (1983), Springer: Springer Berlin, New York · Zbl 0691.35001 |
| [5] | Li, T.-T.; Zheng, S.-M.; Tan, Y.-J.; Shen, W.-X., Boundary Value Problems with Equivalued Surface and Resistivity Well-Logging, Pitman Res. Notes Math., vol. 382 (1998), Longman: Longman Harlow · Zbl 0919.35001 |
| [6] | L. Liu, On subsonic compressible flows in a two-dimensional duct, Nonlinear Anal. (2007), in press; L. Liu, On subsonic compressible flows in a two-dimensional duct, Nonlinear Anal. (2007), in press |
| [7] | L. Liu, H. Yuan, Stability of cylindrical transonic shocks for two-dimensional steady compressible Euler flows, J. Hyperbolic Differ. Equ. (2007), in press; L. Liu, H. Yuan, Stability of cylindrical transonic shocks for two-dimensional steady compressible Euler flows, J. Hyperbolic Differ. Equ. (2007), in press |
| [8] | Sibner, L. M.; Sibner, R. J., Transonic flows on axially symmetric torus, J. Math. Anal. Appl., 72, 362-382 (1979) · Zbl 0445.76045 |
| [9] | Taylor, M., Partial Differential Equations, vol. 3 (1996), Springer: Springer New York · Zbl 0869.35002 |
| [10] | Whitham, B., Linear and Nonlinear Waves (1974), John Wiley: John Wiley New York · Zbl 0373.76001 |
| [11] | Woods, L. C., The Theory of Subsonic Plane Flow (1961), Cambridge Univ. Press: Cambridge Univ. Press London · Zbl 1219.76003 |
| [12] | Yuan, H., A remark on determination of transonic shocks in divergent nozzles for steady compressible Euler flows, Nonlinear Anal. Real World Appl., 9, 316-325 (2008) · Zbl 1137.76029 |
| [13] | Yuan, H., On transonic shocks in two-dimensional variable-area ducts for steady Euler system, SIAM J. Math. Anal., 38, 1343-1370 (2006) · Zbl 1121.35081 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
