November  2003, 3(4): 541-563. doi: 10.3934/dcdsb.2003.3.541

Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow

1. 

Department of Mathematics, University of Houston, Houston, Texas 77204-3008, United States, United States

2. 

Department of Mathematics, The Hebrew University, Jerusalem, Israel

Received  September 2002 Revised  January 2003 Published  August 2003

The two-fluid equations for two-phase flow, a model derived by averaging, analogy and experimental observation, have the property (in at least some commonly-occurring derivations) of losing hyperbolicity in their principal parts, those representing the chief entries in modeling conservation of mass and transfer of momentum and energy.
Much attention has centered on reformulating details of the model to avoid this awkwardness. This paper takes a different approach: a study of the nonhyperbolic operator itself. The objective is to understand the nature of ill-posedness in nonlinear, as distinct from linearized, models.
We present our initial study of the nonlinear operator that occurs in the two-fluid equations for incompressible two-phase flow. Our research indicates that one can solve Riemann problems for these nonlinear, nonhyperbolic equations. The solutions involve singular shocks, very low regularity solutions of conservation laws (solutions with singular shocks, however, are not restricted to nonhyperbolic equations). We present evidence, based on asymptotic treatment and numerical solution of regularized equations, that these singular solutions occur in the two-fluid model for incompressible two-phase flow. The Riemann solutions found using singular shocks have a reasonable physical interpretation.
Citation: Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541
[1]

Adán J. Corcho. Ill-Posedness for the Benney system. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 965-972. doi: 10.3934/dcds.2006.15.965

[2]

Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863

[3]

Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 253-259. doi: 10.3934/dcds.2011.30.253

[4]

Xavier Carvajal, Mahendra Panthee. On ill-posedness for the generalized BBM equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4565-4576. doi: 10.3934/dcds.2014.34.4565

[5]

Tsukasa Iwabuchi, Kota Uriya. Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $|u|^2$. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1395-1405. doi: 10.3934/cpaa.2015.14.1395

[6]

Piero D'Ancona, Mamoru Okamoto. Blowup and ill-posedness results for a Dirac equation without gauge invariance. Evolution Equations & Control Theory, 2016, 5 (2) : 225-234. doi: 10.3934/eect.2016002

[7]

In-Jee Jeong, Benoit Pausader. Discrete Schrödinger equation and ill-posedness for the Euler equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 281-293. doi: 10.3934/dcds.2017012

[8]

Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985

[9]

Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415

[10]

Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437

[11]

G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327

[12]

Yannis Angelopoulos. Well-posedness and ill-posedness results for the Novikov-Veselov equation. Communications on Pure & Applied Analysis, 2016, 15 (3) : 727-760. doi: 10.3934/cpaa.2016.15.727

[13]

Antonio Fasano, Marco Gabrielli, Alberto Gandolfi. Investigating the steady state of multicellular spheroids by revisiting the two-fluid model. Mathematical Biosciences & Engineering, 2011, 8 (2) : 239-252. doi: 10.3934/mbe.2011.8.239

[14]

Jean-François Crouzet. 3D coded aperture imaging, ill-posedness and link with incomplete data radon transform. Inverse Problems & Imaging, 2011, 5 (2) : 341-353. doi: 10.3934/ipi.2011.5.341

[15]

Bernadette N. Hahn. Dynamic linear inverse problems with moderate movements of the object: Ill-posedness and regularization. Inverse Problems & Imaging, 2015, 9 (2) : 395-413. doi: 10.3934/ipi.2015.9.395

[16]

Shunlian Liu, David M. Ambrose. Sufficiently strong dispersion removes ill-posedness in truncated series models of water waves. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3123-3147. doi: 10.3934/dcds.2019129

[17]

Marcel Braukhoff. Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness. Kinetic & Related Models, 2019, 12 (2) : 445-482. doi: 10.3934/krm.2019019

[18]

Qiangchang Ju, Hailiang Li, Yong Li, Song Jiang. Quasi-neutral limit of the two-fluid Euler-Poisson system. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1577-1590. doi: 10.3934/cpaa.2010.9.1577

[19]

B. Wiwatanapataphee, Theeradech Mookum, Yong Hong Wu. Numerical simulation of two-fluid flow and meniscus interface movement in the electromagnetic continuous steel casting process. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1171-1183. doi: 10.3934/dcdsb.2011.16.1171

[20]

Jin Lai, Huanyao Wen, Lei Yao. Vanishing capillarity limit of the non-conservative compressible two-fluid model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1361-1392. doi: 10.3934/dcdsb.2017066

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (22)

[Back to Top]