一类具有非等熵 Dusty 气体的两相流模型 Riemann 解的压力消失极限

Abstract

Abstract:

This paper studies the cavitation and concentration phenomena of the Riemann solutions for a reduced two-phase mixtures model with non-isentropic dusty gas state as the pressure vanishes. Firstly, we construct the Riemann entropy solutions by characteristic analysis method in $ (p, u, s) $ coordinate system. Secondly, we conclude that, the pressureless limit of Riemann solutions for the reduced two-phase mixtures model is just the Riemann solutions for the reduced 2-dimensional pressureless gas dynamics model. Finally, we present numerical simulations which are consistent with our theoretical analysis.

Key words: non-isentropic, two-phase mixtures model, pressureless limit, Riemann problem, Delta shock wave, vacuum state

CLC Number:

O175.23

Jin Daiguang,He Shaohong,Wu Yuyan,Jiang Weifeng. The Vanishing Pressure Limit of Riemann Solutions for a Class of Two-Phase Flow Models with Non-Isentropic Dusty Gases[J].Acta mathematica scientia,Series A, 2025, 45(2): 371-388.

Trendmd

Figures/Tables 5

References 42

[1]	Bereux F, Bonnetier E, LeFloch P G. Gas dynamics system: two special cases. SIAM Journal on Mathematical Analysis, 1997, 28(3): 499-515
[2]	Bilicki Z, Kestin J. Physical aspects of the relaxation model in two-phase flow. Proceedings of the Royal Society of London A Mathematical and Physical Sciences, 1990, 428(1875): 379-397
[3]	Minemura K, Uchiyama T. Three-dimensional calculation of air-water two-phase flow in centrifugal pump impeller based on a bubbly flow model. Journal of Fluids Engineering, 1993, 115(4): 776-771
[4]	Sainsaulieu L. Euler system modeling vaporizing sprays. Progress in Astronautics 1993, 152: 280-280
[5]	Zhang T T. The invariant region for the special gas dynamics system. Nonlinear Analysis: Real World Applications, 2017, 38: 68-77
[6]	Frost D L. Heterogeneous/particle-laden blast waves. Shock Waves, 2018, 28(3): 439-449
[7]	Elperin T, Ben-Dor G, Igra O. Head-on collision of normal shock waves in dusty gases. International Journal of Heat and Fluid Flow, 1987, 8(4): 303-312
[8]	Gretler W, Regenfelder R. Similarity solution for variable energy shock waves in a dusty gas under isothermal flow-field condition. Fluid Dynamics Research, 2003, 32(3): Article 69
[9]	Higashino F, Suzuki T. The effect of particles on blast waves in a dusty gas. Zeitschrift für Naturforschung A, 1980, 35(12): 1330-1336
[10]	Guo L H, Sheng W C, Zhang T. The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system. Communications on Pure and Applied Analysis, 2010, 9: 431-458
[11]	Weinan E, Rykov Y G, Sinai Y G. Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Communications in Mathematical Physics, 1996, 177: 349-380
[12]	Brenier Y, Grenier E. Sticky particles and scalar conservation laws. SIAM journal on numerical analysis, 1998, 35(6): 2317-2328
[13]	Shandarin S F, Zeldovich Y B. The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium. Reviews of Modern Physics, 1989, 61(2): 185-220
[14]	Hu J X. One-dimensional Riemann problem for equations of constant pressure fluid dynamics with measure solutions by the viscosity method. Acta Applicandae Mathematica, 1999, 55: 209-229
[15]	Korchinski D J. Solution of a Riemann Problem for a 2 x 2 System of Conservation Laws Possessing No Classical Weak Solution. New York: Adelphi University, 1997
[16]	Tan D C, Zhang T. Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws. Journal of Differential Equations, 1994, 111: 203-254; 255-282
[17]	Tan D C, Zhang T, Chang T, Zheng Y X. Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws. Journal of Differential Equations, 1994, 112(1): 1-32
[18]	Yang H C, Zhang Y Y. New developments of delta shock waves and its applications in systems of conservation laws. Journal of Differential Equations, 2012, 252(11): 5951-5993
[19]	李华惠, 邵志强. 压力消失时具有广义 Chaplygin 气体的 Aw-Rascle 交通模型 Riemann 解的极限. 数学物理学报, 2017, 37A(5): 917-930
	Li H H, Shao Z Q. Limit of Riemann solution for Aw-Raschle traffic model with generalized Chaplygin gas when pressure disappears. Acta Math Sci, 2017, 37A(5): 917-930
[20]	邵志强. 一维具有阻尼和摩擦项的可压缩流体欧拉方程组当压力消失时黎曼解的极限. 数学物理学报, 2022, 42A(4): 1150-1172
	Shao Z Q. Limit of Riemann solution for one-dimensional compressible fluid Euler equations with damping and friction terms when pressure disappears. Acta Math Sci, 2022, 42A(4): 1150-1172
[21]	Yang H C. Riemann problems for a class of coupled hyperbolic systems of conservation laws. Journal of Differential Equations, 1999, 159(2): 447-484
[22]	Sheng W C, Zhang T. The Riemann Problem for the Transportation Equations in Gas Dynamics. American Mathematical Society, 1999
[23]	Ding X, Wang Z. Existence and uniqueness of discontinuous solutions defined by Lebesgue-Stieltjes integral. Sci China Ser A, 1996, 39(8): 807-819
[24]	Li J Q. Note on the compressible Euler equations with zero temperature. Applied Mathematics Letters, 2001, 14(4): 519-523
[25]	Chen G Q, Liu H L. Formation of $ \delta $-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM Journal on Mathematical Analysis, 2003, 34(4): 925-938
[26]	Sheng W C, Wang G J, Yin G. Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes. Nonlinear Analysis: Real World Applications, 2015, 22: 115-128
[27]	Shen C, Sun M N. Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model. Journal of Differential Equations, 2010, 249(12): 3024-3051
[28]	Yang H C, Wang J H. Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas. Journal of Mathematical Analysis and Applications, 2014, 413(2): 800-820
[29]	Zhang Q L. The vanishing pressure limit of solutions to the simplified Euler equations for isentropic fluids. Annals of Applied Mathematics, 2012, 28(1): 115-126
[30]	Zhang Y, Fan S, Zhang Y Y. Concentration and cavitation in the vanishing pressure limit of solutions to a 3$ \times $3 generalized Chaplygin gas equations. Mathematical Modelling of Natural Phenomena, 2022, 17: Article 10
[31]	Shen C. The singular limits of solutions to the Riemann problem for the liquid-gas two-phase isentropic flow model. Journal of Mathematical Physics, 2020, 61(8): Article 081502
[32]	Song Y T, Guo L H. General limiting behavior of Riemann solutions to the non-isentropic Euler equations for modified Chaplygin gas. Journal of Mathematical Physics, 2020, 61(4): Article 041506
[33]	Song Y T, Guo L H. Behavior of Riemann solutions of extended chaplygin gas under the limiting condition. Acta Applicandae Mathematicae, 2021, 174: 1-17
[34]	Zhang Y, Pang Y C, Wang J H. Concentration and cavitation in the vanishing pressure limit of solutions to the generalized Chaplygin Euler equations of compressible fluid flow. European Journal of Mechanics-B/Fluids, 2019, 78: 252-262
[35]	Kipgen L, Singh R. $ \delta $-shocks and vacuum states in the Riemann problem for isothermal van der Waals dusty gas under the flux approximation. Physics of Fluids, 2023, 35(1): Article 016116
[36]	Yang H C, Wang J H. Concentration in vanishing pressure limit of solutions to the modified Chaplygin gas equations. Journal of Mathematical Physics, 2016, 57(11): Article 111504
[37]	Chaturvedi R K, Singh L. The phenomena of concentration and cavitation in the Riemann solution for the isentropic zero-pressure dusty gasdynamics. Journal of Mathematical Physics, 2021, 62(3): Article 033101
[38]	Lei Z T, Shao Z Q. Concentration and cavitation in the vanishing pressure limit of solutions to the relativistic Euler equations with the logarithmic equation of state. Journal of Mathematical Physics, 2023, 64(7): Article 071507
[39]	Jiang W F, Zhang Y, Li T, Chen T T. The cavitation and concentration of Riemann solutions for the isentropic Euler equations with isothermal dusty gas. Nonlinear Analysis: Real World Applications, 2023, 71: Article 103761
[40]	Pang Y C, Ge J J, Liu Z Z, Hu M. The Riemann problem for one-dimensional isentropic flow of a mixture of a non-ideal gas with small solid particles. Results in Physics, 2019, 15: Article 102587
[41]	Chen G Q, Liu H L. Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids. Physica D: Nonlinear Phenomena, 2004, 189(1/2): 141-165
[42]	Jiang W F, Jin D G, Li T, Chen T T. The singular limits of the Riemann solutions as pressure vanishes for a reduced two-phase mixtures model with non-isentropic gas state. Journal of Mathematical Physics, 2024, 65(7): Article 071503

The Vanishing Pressure Limit of Riemann Solutions for a Class of Two-Phase Flow Models with Non-Isentropic Dusty Gases

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

Figures/Tables 5

References 42

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Pan Lijun, Lv Shun, Weng Shasha. Riemann Solution and Stability of Coupled Aw-Rascle-Zhang Model [J]. Acta mathematica scientia,Series A, 2024, 44(4): 885-895.
[2]	He Fen, Wang Zhen. Riemann Problem for a Class of Mixed-type Chemotaxis Models [J]. Acta mathematica scientia,Series A, 2024, 44(2): 354-360.
[3]	Zhiqiang Shao. Concentration and Cavitation in the Pressureless Limit of Euler Equations of Compressible Fluid Flow with Damping and Friction [J]. Acta mathematica scientia,Series A, 2022, 42(4): 1150-1172.
[4]	Yuan-an Zhao,Gaowei Cao,Xiaozhou Yang. Global Singular Structures of Non-Selfsimilar Riemann Solutions for Two Dimensional Non-homogeneous Burgers Equation [J]. Acta mathematica scientia,Series A, 2022, 42(4): 1122-1149.
[5]	Jia Jia. The Two-Dimensional Steady Chaplygin Gas Flows Passing a Straight Wedge [J]. Acta mathematica scientia,Series A, 2021, 41(5): 1270-1282.
[6]	Xin Liu,Xiaolei Dong. Low Mach Number Limit to One-Dimensional Non-Isentropic Compressible Viscous Micropolar Fluid Model [J]. Acta mathematica scientia,Series A, 2021, 41(5): 1445-1464.
[7]	Yufeng Chen,Tingting Chen,Zhen Wang. The Existence of the Measure Solution for the Non-Isentropic Chaplygin Gas [J]. Acta mathematica scientia,Series A, 2020, 40(4): 833-841.
[8]	Qingling Zhang,Ying Ba. The Perturbed Riemann Problem for the Pressureless Euler Equations with a Flocking Dissipation [J]. Acta mathematica scientia,Series A, 2020, 40(1): 49-62.
[9]	Lijun Pan,Xinli Han,Tong Li. The Generalized Riemann Problem for Chromatography Equations with Delta Shock Wave [J]. Acta mathematica scientia,Series A, 2019, 39(6): 1300-1313.
[10]	Chen Tingting, Qu Aifang, Wang Zhen. The Two-Dimensional Riemann Problem for Isentropic Chaplygin Gas [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1053-1061.
[11]	Li Huahui, Shao Zhiqiang. Vanishing Pressure Limit of Riemann Solutions to the Aw-Rascle Model for Generalized Chaplygin Gas [J]. Acta mathematica scientia,Series A, 2017, 37(5): 917-930.
[12]	Li Huahui, Shao Zhiqiang. Interactions of Delta Shock Waves for the Nonsymmetric Keyfitz-Kranzer System with Split Delta Functions [J]. Acta mathematica scientia,Series A, 2017, 37(4): 714-729.
[13]	Li Xin, Wang Shu, Feng Yuehong. Stability of Non-Constant Equilibrium Solutions for Bipolar Non-Isentropic Euler-Poisson Equations [J]. Acta mathematica scientia,Series A, 2016, 36(5): 978-996.
[14]	SHAO Zhi-Qiang. The Riemann Problem with Delta Initial Data for the Aw-Rascle Traffic Model with Chaplygin Pressure [J]. Acta mathematica scientia,Series A, 2014, 34(6): 1353-1371.
[15]	LI Jian-Ye, SHAO Zhi-Qiang. The Riemann Problem with Delta Initial Data for the Zero-Pressure Relativistic Euler Equations Hyperbolic Systems of Conservation Laws [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1083-1092.