doi: 10.3934/era.2020122

Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle

1. 

School of Mathematics, Jilin University, Changchun 130012, China

2. 

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

* Corresponding author: Mingjun Zhou

Received  August 2020 Revised  October 2020 Published  November 2020

Fund Project: Mingjun Zhou is supported by the National Natural Science Foundation of China (Grant Nos. 11925105 and 12001227), Jingxue Yin is supported by the National Natural Science Foundation of China Grant No. 11771156, Guangdong Basic and Applied Basic Research Foundation Grant No. 2020B1515310013, Science and Technology Program of Guangzhou No. 2019050001, and the Natural Science Foundation of Guangzhou Grant No. 201804010391

This paper focuses on two-dimensional continuous subsonic-sonic potential flows in a semi-infinitely long nozzle with a straight lower wall and an upper wall which is convergent at the outlet while straight at the far fields. It is proved that if the variation rate of the cross section of the nozzle is suitably small, there exists a unique continuous subsonic-sonic flows in the nozzle such that the sonic curve intersects the upper wall at a fixed point and the velocity of the flow is along the normal direction at the sonic curve. Furthermore, the sonic curve is free, where the flow is singular in the sense that the flow speed is only Hölder continuous and the flow acceleration blows up. Additionally, the asymptotic behaviors of the flow speed at the far fields is shown.

Citation: Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, doi: 10.3934/era.2020122
References:
[1]

L. Bers, Existence and uniqueness of a subsonic flow past a given profile, Comm. Pure Appl. Math., 7 (1954), 441-504.  doi: 10.1002/cpa.3160070303.  Google Scholar

[2]

L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958.  Google Scholar

[3]

G.-Q. ChenC. M. DafermosM. Slemrod and D. Wang, On two-dimensional sonic-subsonic flow, Comm. Math. Phys., 271 (2007), 635-647.  doi: 10.1007/s00220-007-0211-9.  Google Scholar

[4]

C. ChenL. DuC. Xie and Z. Xin, Two dimensional subsonic Euler flows past a wall or a symmetric body, Arch. Ration. Mech. Anal., 221 (2016), 559-602.  doi: 10.1007/s00205-016-0968-0.  Google Scholar

[5]

G.-Q. ChenF.-M. Huang and T.-Y. Wang, Subsonic-sonic limit of approximate solutions to multidimensional steady Euler equations, Arch. Ration. Mech. Anal., 219 (2016), 719-740.  doi: 10.1007/s00205-015-0905-7.  Google Scholar

[6]

G.-Q. ChenF.-M. HuangT.-Y. Wang and W. Xiang, Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles, Adv. Math., 346 (2019), 946-1008.  doi: 10.1016/j.aim.2019.02.002.  Google Scholar

[7]

C. Chen and C. Xie, Existence of steady subsonic Euler flows through infinitely long periodic nozzles, J. Differential Equations, 252 (2012), 4315-4331.  doi: 10.1016/j.jde.2011.12.015.  Google Scholar

[8]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y., 1948.  Google Scholar

[9]

G. Dong, Nonlinear Partial Differential Equations of Second Order, Translations of Mathematical Monographs. 95, American Mathematical Society, Providence, RI, 1991. doi: 10.1090/mmono/095.  Google Scholar

[10]

L. Du and B. Duan, Subsonic Euler flows with large vorticity through an infinitely long axisymmetric nozzle, J. Math. Fluid Mech., 18 (2016), 511-530.  doi: 10.1007/s00021-016-0255-8.  Google Scholar

[11]

L. DuC. Xie and Z. Xin, Steady subsonic ideal flows through an infinitely long nozzle with large vorticity, Comm. Math. Phys., 328 (2014), 327-354.  doi: 10.1007/s00220-014-1951-y.  Google Scholar

[12]

L. DuZ. Xin and W. Yan, Subsonic flows in a multi-dimensional nozzle, Arch. Ration. Mech. Anal., 201 (2011), 965-1012.  doi: 10.1007/s00205-011-0406-2.  Google Scholar

[13]

R. Finn and D. Gilbarg, Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations, Acta Math., 98 (1957), 265-296.  doi: 10.1007/BF02404476.  Google Scholar

[14]

F. HuangT. Wang and Y. Wang, On multi-dimensional sonic-subsonic flow, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 2131-2140.  doi: 10.1016/S0252-9602(11)60389-5.  Google Scholar

[15]

A. Kuz'min, Boundary Value Problems for Transonic Flow, John Wiley and Sons, Ltd, 2002. Google Scholar

[16]

Y. Nie and C. Wang, Continuous subsonic-sonic flows in convergent nozzles with straight solid walls, Nonlinearity, 29 (2016), 86-130.  doi: 10.1088/0951-7715/29/1/86.  Google Scholar

[17]

Y. Nie and C. Wang, Continuous subsonic-sonic flows in a convergent nozzle, Acta Math. Sin. (Engl. Ser.), 34 (2018), 749-772.  doi: 10.1007/s10114-017-7341-6.  Google Scholar

[18]

C. Wang, Continuous subsonic-sonic flows in a general nozzle, J. Differential Equations, 259 (2015), 2546-2575.  doi: 10.1016/j.jde.2015.03.036.  Google Scholar

[19]

C. Wang, A free boundary problem of a degenerate elliptic equation and subsonic-sonic flows with general sonic curves, SIAM J. Math. Anal., 51 (2019), 4977-5010.  doi: 10.1137/19M1255860.  Google Scholar

[20]

C. Wang and Z. Xin, On a degenerate free boundary problem and continuous subsonic-sonic flows in a convergent nozzle, Arch. Ration. Mech. Anal., 208 (2013), 911-975.  doi: 10.1007/s00205-012-0607-3.  Google Scholar

[21]

C. Wang and M. Zhou, A degenerate elliptic problem from subsonic-sonic flows in general nozzles, J. Differential Equations, 267 (2019), 3778-3796.  doi: 10.1016/j.jde.2019.04.026.  Google Scholar

[22]

C. Xie and Z. Xin, Global subsonic and subsonic-sonic flows through infinitely long nozzles, Indiana Univ. Math. J., 56 (2007), 2991-3023.  doi: 10.1512/iumj.2007.56.3108.  Google Scholar

[23]

C. Xie and Z. Xin, Existence of global steady subsonic Euler flows through infinitely long nozzles, SIAM J. Math. Anal., 42 (2010), 751-784.  doi: 10.1137/09076667X.  Google Scholar

[24]

C. Xie and Z. Xin, Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles, J. Differential Equations, 248 (2010), 2657-2683.  doi: 10.1016/j.jde.2010.02.007.  Google Scholar

show all references

References:
[1]

L. Bers, Existence and uniqueness of a subsonic flow past a given profile, Comm. Pure Appl. Math., 7 (1954), 441-504.  doi: 10.1002/cpa.3160070303.  Google Scholar

[2]

L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958.  Google Scholar

[3]

G.-Q. ChenC. M. DafermosM. Slemrod and D. Wang, On two-dimensional sonic-subsonic flow, Comm. Math. Phys., 271 (2007), 635-647.  doi: 10.1007/s00220-007-0211-9.  Google Scholar

[4]

C. ChenL. DuC. Xie and Z. Xin, Two dimensional subsonic Euler flows past a wall or a symmetric body, Arch. Ration. Mech. Anal., 221 (2016), 559-602.  doi: 10.1007/s00205-016-0968-0.  Google Scholar

[5]

G.-Q. ChenF.-M. Huang and T.-Y. Wang, Subsonic-sonic limit of approximate solutions to multidimensional steady Euler equations, Arch. Ration. Mech. Anal., 219 (2016), 719-740.  doi: 10.1007/s00205-015-0905-7.  Google Scholar

[6]

G.-Q. ChenF.-M. HuangT.-Y. Wang and W. Xiang, Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles, Adv. Math., 346 (2019), 946-1008.  doi: 10.1016/j.aim.2019.02.002.  Google Scholar

[7]

C. Chen and C. Xie, Existence of steady subsonic Euler flows through infinitely long periodic nozzles, J. Differential Equations, 252 (2012), 4315-4331.  doi: 10.1016/j.jde.2011.12.015.  Google Scholar

[8]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y., 1948.  Google Scholar

[9]

G. Dong, Nonlinear Partial Differential Equations of Second Order, Translations of Mathematical Monographs. 95, American Mathematical Society, Providence, RI, 1991. doi: 10.1090/mmono/095.  Google Scholar

[10]

L. Du and B. Duan, Subsonic Euler flows with large vorticity through an infinitely long axisymmetric nozzle, J. Math. Fluid Mech., 18 (2016), 511-530.  doi: 10.1007/s00021-016-0255-8.  Google Scholar

[11]

L. DuC. Xie and Z. Xin, Steady subsonic ideal flows through an infinitely long nozzle with large vorticity, Comm. Math. Phys., 328 (2014), 327-354.  doi: 10.1007/s00220-014-1951-y.  Google Scholar

[12]

L. DuZ. Xin and W. Yan, Subsonic flows in a multi-dimensional nozzle, Arch. Ration. Mech. Anal., 201 (2011), 965-1012.  doi: 10.1007/s00205-011-0406-2.  Google Scholar

[13]

R. Finn and D. Gilbarg, Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations, Acta Math., 98 (1957), 265-296.  doi: 10.1007/BF02404476.  Google Scholar

[14]

F. HuangT. Wang and Y. Wang, On multi-dimensional sonic-subsonic flow, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 2131-2140.  doi: 10.1016/S0252-9602(11)60389-5.  Google Scholar

[15]

A. Kuz'min, Boundary Value Problems for Transonic Flow, John Wiley and Sons, Ltd, 2002. Google Scholar

[16]

Y. Nie and C. Wang, Continuous subsonic-sonic flows in convergent nozzles with straight solid walls, Nonlinearity, 29 (2016), 86-130.  doi: 10.1088/0951-7715/29/1/86.  Google Scholar

[17]

Y. Nie and C. Wang, Continuous subsonic-sonic flows in a convergent nozzle, Acta Math. Sin. (Engl. Ser.), 34 (2018), 749-772.  doi: 10.1007/s10114-017-7341-6.  Google Scholar

[18]

C. Wang, Continuous subsonic-sonic flows in a general nozzle, J. Differential Equations, 259 (2015), 2546-2575.  doi: 10.1016/j.jde.2015.03.036.  Google Scholar

[19]

C. Wang, A free boundary problem of a degenerate elliptic equation and subsonic-sonic flows with general sonic curves, SIAM J. Math. Anal., 51 (2019), 4977-5010.  doi: 10.1137/19M1255860.  Google Scholar

[20]

C. Wang and Z. Xin, On a degenerate free boundary problem and continuous subsonic-sonic flows in a convergent nozzle, Arch. Ration. Mech. Anal., 208 (2013), 911-975.  doi: 10.1007/s00205-012-0607-3.  Google Scholar

[21]

C. Wang and M. Zhou, A degenerate elliptic problem from subsonic-sonic flows in general nozzles, J. Differential Equations, 267 (2019), 3778-3796.  doi: 10.1016/j.jde.2019.04.026.  Google Scholar

[22]

C. Xie and Z. Xin, Global subsonic and subsonic-sonic flows through infinitely long nozzles, Indiana Univ. Math. J., 56 (2007), 2991-3023.  doi: 10.1512/iumj.2007.56.3108.  Google Scholar

[23]

C. Xie and Z. Xin, Existence of global steady subsonic Euler flows through infinitely long nozzles, SIAM J. Math. Anal., 42 (2010), 751-784.  doi: 10.1137/09076667X.  Google Scholar

[24]

C. Xie and Z. Xin, Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles, J. Differential Equations, 248 (2010), 2657-2683.  doi: 10.1016/j.jde.2010.02.007.  Google Scholar

[1]

Guanming Gai, Yuanyuan Nie, Chunpeng Wang. A degenerate elliptic problem from subsonic-sonic flows in convergent nozzles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021070

[2]

Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837

[3]

Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013

[4]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[5]

Donatella Danielli, Marianne Korten. On the pointwise jump condition at the free boundary in the 1-phase Stefan problem. Communications on Pure & Applied Analysis, 2005, 4 (2) : 357-366. doi: 10.3934/cpaa.2005.4.357

[6]

Qi An, Chuncheng Wang, Hao Wang. Analysis of a spatial memory model with nonlocal maturation delay and hostile boundary condition. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5845-5868. doi: 10.3934/dcds.2020249

[7]

Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095

[8]

Hiroko Morimoto. Survey on time periodic problem for fluid flow under inhomogeneous boundary condition. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 631-639. doi: 10.3934/dcdss.2012.5.631

[9]

Mahdi Boukrouche, Grzegorz Łukaszewicz. On global in time dynamics of a planar Bingham flow subject to a subdifferential boundary condition. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 3969-3983. doi: 10.3934/dcds.2014.34.3969

[10]

Yoshikazu Giga, Przemysław Górka, Piotr Rybka. Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 493-519. doi: 10.3934/dcds.2010.26.493

[11]

Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128

[12]

Jiayue Zheng, Shangbin Cui. Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4397-4410. doi: 10.3934/dcdsb.2020103

[13]

Chunpeng Wang. Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1041-1060. doi: 10.3934/dcds.2016.36.1041

[14]

Eun Heui Kim. Boundary gradient estimates for subsonic solutions of compressible transonic potential flows. Conference Publications, 2007, 2007 (Special) : 573-579. doi: 10.3934/proc.2007.2007.573

[15]

Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353

[16]

Aram L. Karakhanyan. Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 261-277. doi: 10.3934/dcds.2016.36.261

[17]

Pierangelo Ciurlia. On a general class of free boundary problems for European-style installment options with continuous payment plan. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1205-1224. doi: 10.3934/cpaa.2011.10.1205

[18]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101

[19]

Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generalized Wentzell boundary conditions for second order operators with interior degeneracy. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 697-715. doi: 10.3934/dcdss.2016023

[20]

Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021129

 Impact Factor: 0.263

Metrics

  • PDF downloads (47)
  • HTML views (178)
  • Cited by (0)

Other articles
by authors

[Back to Top]