Abstract
We consider inviscid flow with isentropic coefficient greater than one. For flow along smooth infinite protruding corners, we attempt to impose a nonzero limit for velocity at infinity at the upstream wall. We prove that the problem does not have any irrotational uniformly subsonic solutions, whereas rotational flows do exist. This can be considered a case of a slip-condition solid “generating” vorticity in inviscid flow.
Similar content being viewed by others
References
Ahlfors, L.V.: Lectures on Quasiconformal Mappings. University lecture Series, vol. 38, 2nd edn. American Mathematical Society, Providence (2006)
Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge Mathematical Library, Cambridge (1967)
Bers, L.: Existence and uniqueness of a subsonic flow past a given profile. Commun. Pure Appl. Math. 7, 441–504 (1954)
Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Interscience Publishers, Geneva (1948)
Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics, 3rd edn. Springer, Berlin (1992)
d’Alembert, J.L.R.: Essai d’une nouvelle théorie de la résistance des fluides (1752). Reprint by Hachette Livre, France. ISBN-13: 978-2012542839
Elling, V.: Compressible vortex sheets separating from solid boundaries. Discr. Contin. Dyn. Syst. (Ser. A) 36(12), 6781–6797 (2016)
Finn, R., Gilbarg, D.: Asymptotic behaviour and uniqueness of plane subsonic flows. Commun. Pure Appl. Math. 10, 23–63 (1957)
Frankl, F.I., Keldysh, M.: Die äussere Neumann’sche Aufgabe für nichtlineare elliptische Differentialgleichungen mit Anwendung auf die Theorie der Flügel im kompressiblen Gas. Bull. Acad. Sci. URSS 12, 561–607 (1934)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. A Series of Comprehensive Studies in Mathematics, vol. 224, 2nd edn. Springer, Berlin (1983)
Helmholtz, H.: Über discontinuirliche Flüssigkeits-Bewegungen, Monatsberichte der Königlich Preussischen, pp. 215–228. Akademie der Wissenschaften, Berlin (1868)
Hui, Y., Yin, H.: A Liouville-type theorem for subsonic flows around an infinite long ramp. Q. Appl. Math. 72(2), 253–265 (2014)
Kirchhoff, G.: Zur Theorie freier Flüssigkeitsstrahlen. J. Reine Angew. Math. 70, 289–298 (1869)
Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34, 481–524 (1981)
Levi-Civita, T.: Scie e leggi di resistenzia. Rend. Circ. Mat. Palermo 23, 1–37 (1907)
Lieberman, G.: Hölder continuity of the gradient at a corner for the capillary problem and related results. Pac. J. Math. 133(1), 115–135 (1988)
Morrey, C.B.: On the solution of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938)
Prandtl, L.: Über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandlungen des III. Internationalen Mathematiker-Kongresses, Heidelberg (1904)
Rayleigh, L.: On the resistance of fluids. Philos. Mag. 11, 430–441 (1876)
Shiffman, M.: On the existence of subsonic flows of a compressible fluid. J. Ration. Mech. Anal. 1, 605–652 (1952)
Shi, D.: Capillary surfaces at a reentrant corner. Pac. J. Math. 224(2), 321–353 (2006)
van Dyke, M.: An Album of Fluid Motion. The Parabolic Press, Stanford (1982)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Elling, V. Nonexistence of compressible irrotational inviscid flows along infinite protruding corners. Z. Angew. Math. Phys. 69, 59 (2018). https://doi.org/10.1007/s00033-018-0956-3
Received:
Published:
DOI: https://doi.org/10.1007/s00033-018-0956-3