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.

[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.

show all references

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.

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.
[1]

Tai-Ping Liu, Zhouping Xin, Tong Yang. Vacuum states for compressible flow. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 1-32. doi: 10.3934/dcds.1998.4.1

[2]

Thomas H. Otway. Compressible flow on manifolds. Conference Publications, 2001, 2001 (Special) : 289-294. doi: 10.3934/proc.2001.2001.289

[3]

Šárka Nečasová. Stokes and Oseen flow with Coriolis force in the exterior domain. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 339-351. doi: 10.3934/dcdss.2008.1.339

[4]

Jian Zhai, Jianping Fang, Lanjun Li. Wave map with potential and hypersurface flow. Conference Publications, 2005, 2005 (Special) : 940-946. doi: 10.3934/proc.2005.2005.940

[5]

Michael Renardy. Backward uniqueness for linearized compressible flow. Evolution Equations & Control Theory, 2015, 4 (1) : 107-113. doi: 10.3934/eect.2015.4.107

[6]

Gui-Qiang Chen, Bo Su. A viscous approximation for a multidimensional unsteady Euler flow: Existence theorem for potential flow. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1587-1606. doi: 10.3934/dcds.2003.9.1587

[7]

Joachim Naumann, Jörg Wolf. On Prandtl's turbulence model: Existence of weak solutions to the equations of stationary turbulent pipe-flow. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1371-1390. doi: 10.3934/dcdss.2013.6.1371

[8]

Manuel Falconi, E. A. Lacomba, C. Vidal. The flow of classical mechanical cubic potential systems. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 827-842. doi: 10.3934/dcds.2004.11.827

[9]

Shuxing Chen, Gui-Qiang Chen, Zejun Wang, Dehua Wang. A multidimensional piston problem for the Euler equations for compressible flow. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 361-383. doi: 10.3934/dcds.2005.13.361

[10]

Tong Tang, Yongfu Wang. Strong solutions to compressible barotropic viscoelastic flow with vacuum. Kinetic & Related Models, 2015, 8 (4) : 765-775. doi: 10.3934/krm.2015.8.765

[11]

Young-Pil Choi. Compressible Euler equations interacting with incompressible flow. Kinetic & Related Models, 2015, 8 (2) : 335-358. doi: 10.3934/krm.2015.8.335

[12]

Reimund Rautmann. Lower and upper bounds to the change of vorticity by transition from slip- to no-slip fluid flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1101-1109. doi: 10.3934/dcdss.2014.7.1101

[13]

Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651

[14]

Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757

[15]

Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007

[16]

Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357

[17]

Shijin Ding, Junyu Lin, Changyou Wang, Huanyao Wen. Compressible hydrodynamic flow of liquid crystals in 1-D. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 539-563. doi: 10.3934/dcds.2012.32.539

[18]

Anna Kaźmierczak, Jan Sokolowski, Antoni Zochowski. Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes. Mathematical Control & Related Fields, 2018, 8 (1) : 89-115. doi: 10.3934/mcrf.2018004

[19]

Shuguang Shao, Shu Wang, Wen-Qing Xu, Bin Han. Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle. Kinetic & Related Models, 2016, 9 (4) : 767-776. doi: 10.3934/krm.2016015

[20]

Takayuki Kubo, Yoshihiro Shibata, Kohei Soga. On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3741-3774. doi: 10.3934/dcds.2016.36.3741

2016 Impact Factor: 0.801

Metrics

  • PDF downloads (11)
  • HTML views (56)
  • Cited by (0)

Other articles
by authors

[Back to Top]