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 & 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. Google Scholar

[2]

S. Bernstein, Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre, Math. Ann., 59 (1904), 20-76.   Google Scholar

[3]

L. Bers, Existence and uniqueness of a subsonic flow past a given profile, Comm. Pure. Appl. Math., 7 (1954), 441-504.   Google Scholar

[4]

P. Berg, The existence of subsonic helmholtz flows of a compressible fluid, Comm. Pure Appl. Math., 15 (1962), 289-347.   Google Scholar

[5]

V. Elling, Non-existence of subsonic and incompressible flows in non-straight infinite angles, Submitted. Google Scholar

[6]

V. Elling, Nonexistence of irrotational flow around solids with protruding corners, Submitted to Proceedings of HYP2016. Google Scholar

[7]

V. Elling, Nonexistence of low-mach irrotational inviscid flows around polygons, J. Diff. Eqns., 262 (2017), 2705-2721.   Google Scholar

[8]

R. Finn and D. Gilbarg, Asymptotic behaviour and uniqueness of plane subsonic flows, Comm. Pure Appl. Math., 10 (1957), 23-63.   Google Scholar

[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.   Google Scholar

[10]

item[Gri85]{grisvard} (MR775683) P. Grisvard, Elliptic Problems in Nonsmooth Domains Pitman, 1985. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[15]

C. B. Morrey, On the solution of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., (1938), 126-166.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

[19]

M. Shiffman, On the existence of subsonic flows of a compressible fluid, J. Rat. Mech. Anal., 1 (1952), 605-652.   Google Scholar

show all references

References:
[1]

G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Mathematical Library, 1967. Google Scholar

[2]

S. Bernstein, Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre, Math. Ann., 59 (1904), 20-76.   Google Scholar

[3]

L. Bers, Existence and uniqueness of a subsonic flow past a given profile, Comm. Pure. Appl. Math., 7 (1954), 441-504.   Google Scholar

[4]

P. Berg, The existence of subsonic helmholtz flows of a compressible fluid, Comm. Pure Appl. Math., 15 (1962), 289-347.   Google Scholar

[5]

V. Elling, Non-existence of subsonic and incompressible flows in non-straight infinite angles, Submitted. Google Scholar

[6]

V. Elling, Nonexistence of irrotational flow around solids with protruding corners, Submitted to Proceedings of HYP2016. Google Scholar

[7]

V. Elling, Nonexistence of low-mach irrotational inviscid flows around polygons, J. Diff. Eqns., 262 (2017), 2705-2721.   Google Scholar

[8]

R. Finn and D. Gilbarg, Asymptotic behaviour and uniqueness of plane subsonic flows, Comm. Pure Appl. Math., 10 (1957), 23-63.   Google Scholar

[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.   Google Scholar

[10]

item[Gri85]{grisvard} (MR775683) P. Grisvard, Elliptic Problems in Nonsmooth Domains Pitman, 1985. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[15]

C. B. Morrey, On the solution of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., (1938), 126-166.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

[19]

M. Shiffman, On the existence of subsonic flows of a compressible fluid, J. Rat. Mech. Anal., 1 (1952), 605-652.   Google Scholar

Figure 1.  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
Figure 2.  Left: flow onto a vertical plate. Right: flow onto a body symmetric across the flow axis, with y extrema not attained in corners.
Figure 3.  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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