# American Institute of Mathematical Sciences

March  2018, 17(2): 539-555. doi: 10.3934/cpaa.2018029

## Subsonic irrotational inviscid flow around certain bodies with two protruding corners

 Institute of Mathematics, Academia Sinica, 6F Astronomy-Mathematics Building, No. 1 Sec. 4 Roosevelt Rd., Taipei 10617, Taiwan

* Corresponding author

Received  February 2017 Revised  February 2017 Published  March 2018

We prove non-existence of nontrivial uniformly subsonic inviscid irrotational flows around several classes of solid bodies with two protruding corners, in particular vertical and angled flat plates; horizontal plates are the only case where solutions exists. This fills the gap between classical results on bodies with a single protruding corner on one hand and recent work on bodies with three or more protruding corners.

Thus even with zero viscosity and slip boundary conditions solids can generate vorticity, in the sense of having at least one rotational but no irrotational solutions. Our observation complements the commonly accepted explanation of vorticity generation based on Prandtl's theory of viscous boundary layers.

Citation: Volker Elling. Subsonic irrotational inviscid flow around certain bodies with two protruding corners. Communications on Pure and Applied Analysis, 2018, 17 (2) : 539-555. doi: 10.3934/cpaa.2018029
##### References:
 [1] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Mathematical Library, 1967. [2] S. Bernstein, Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre, Math. Ann., 59 (1904), 20-76. [3] L. Bers, Existence and uniqueness of a subsonic flow past a given profile, Comm. Pure. Appl. Math., 7 (1954), 441-504. [4] P. Berg, The existence of subsonic helmholtz flows of a compressible fluid, Comm. Pure Appl. Math., 15 (1962), 289-347. [5] V. Elling, Non-existence of subsonic and incompressible flows in non-straight infinite angles, Submitted. [6] V. Elling, Nonexistence of irrotational flow around solids with protruding corners, Submitted to Proceedings of HYP2016. [7] V. Elling, Nonexistence of low-mach irrotational inviscid flows around polygons, J. Diff. Eqns., 262 (2017), 2705-2721. [8] R. Finn and D. Gilbarg, Asymptotic behaviour and uniqueness of plane subsonic flows, Comm. Pure Appl. Math., 10 (1957), 23-63. [9] F. I. Frankl and M. Keldysh, Die äussere Neumann'sche Aufgabe für nichtlineare elliptische Differentialgleichungen mit Anwendung auf die Theorie der Flügel im kompressiblen Gas, Bull. Acad. Sci. URSS, 12 (1934), 561-607. [10] item[Gri85]{grisvard} (MR775683) P. Grisvard, Elliptic Problems in Nonsmooth Domains Pitman, 1985. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., A Series of Comprehensive Studies in Mathematics, vol. 224, Springer, 1983. [12] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524. [13] G. Lieberman, Hölder continuity of the gradient at a corner for the capillary problem and related results, Pac. J. Math., 133 (1988), 115-135. [14] G. S. S. Ludford, The behaviour at infinity of the potential function of a two-dimensional subsonic compressible flow, J. Math. Phys., 30 (1951-1952), 117-130. [15] C. B. Morrey, On the solution of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., (1938), 126-166. [16] C. B. Morrey, On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations., Amer. J. of Math., 80 (1958), 198-218. [17] V. G. Maz'ya and B. A. Plamenevskii, Estimates in $L_p$ and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points, Amer. Math. Soc. Transl., 123 (1984), 1-88. [18] L. Prandtl, Über die Entstehung von Wirbeln in der idealen Flüssigkeit, mit Anwendung auf die Tragflügeltheorie und andere Aufgaben, Vorträge aus dem Gebiet der Hydro-und Aerodynamik, Springer, 1924. [19] M. Shiffman, On the existence of subsonic flows of a compressible fluid, J. Rat. Mech. Anal., 1 (1952), 605-652.

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##### References:
 [1] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Mathematical Library, 1967. [2] S. Bernstein, Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre, Math. Ann., 59 (1904), 20-76. [3] L. Bers, Existence and uniqueness of a subsonic flow past a given profile, Comm. Pure. Appl. Math., 7 (1954), 441-504. [4] P. Berg, The existence of subsonic helmholtz flows of a compressible fluid, Comm. Pure Appl. Math., 15 (1962), 289-347. [5] V. Elling, Non-existence of subsonic and incompressible flows in non-straight infinite angles, Submitted. [6] V. Elling, Nonexistence of irrotational flow around solids with protruding corners, Submitted to Proceedings of HYP2016. [7] V. Elling, Nonexistence of low-mach irrotational inviscid flows around polygons, J. Diff. Eqns., 262 (2017), 2705-2721. [8] R. Finn and D. Gilbarg, Asymptotic behaviour and uniqueness of plane subsonic flows, Comm. Pure Appl. Math., 10 (1957), 23-63. [9] F. I. Frankl and M. Keldysh, Die äussere Neumann'sche Aufgabe für nichtlineare elliptische Differentialgleichungen mit Anwendung auf die Theorie der Flügel im kompressiblen Gas, Bull. Acad. Sci. URSS, 12 (1934), 561-607. [10] item[Gri85]{grisvard} (MR775683) P. Grisvard, Elliptic Problems in Nonsmooth Domains Pitman, 1985. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., A Series of Comprehensive Studies in Mathematics, vol. 224, Springer, 1983. [12] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524. [13] G. Lieberman, Hölder continuity of the gradient at a corner for the capillary problem and related results, Pac. J. Math., 133 (1988), 115-135. [14] G. S. S. Ludford, The behaviour at infinity of the potential function of a two-dimensional subsonic compressible flow, J. Math. Phys., 30 (1951-1952), 117-130. [15] C. B. Morrey, On the solution of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., (1938), 126-166. [16] C. B. Morrey, On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations., Amer. J. of Math., 80 (1958), 198-218. [17] V. G. Maz'ya and B. A. Plamenevskii, Estimates in $L_p$ and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points, Amer. Math. Soc. Transl., 123 (1984), 1-88. [18] L. Prandtl, Über die Entstehung von Wirbeln in der idealen Flüssigkeit, mit Anwendung auf die Tragflügeltheorie und andere Aufgaben, Vorträge aus dem Gebiet der Hydro-und Aerodynamik, Springer, 1924. [19] M. Shiffman, On the existence of subsonic flows of a compressible fluid, J. Rat. Mech. Anal., 1 (1952), 605-652.
Left: a protruding corner in a solid (shaded). Center: horizontal plate; ${\boldsymbol{\rm{v}}}={\boldsymbol{\rm{v}}}_\infty$ is the trivial solution. Right: angled plate
Left: flow onto a vertical plate. Right: flow onto a body symmetric across the flow axis, with y extrema not attained in corners.
Streamlines around Kármán-Trefftz lens with interior corner angle $270^\circ$, deflection $\beta =20^\circ$. Top left: $\Gamma =0$. Top right: $\Gamma$ chosen to yield bounded velocity at trailing (right) corner; rotate by $180^\circ$ to get the corresponding diagram for the leading corner.
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