March  2020, 28(1): 47-66. doi: 10.3934/era.2020004

The existence of solutions for a shear thinning compressible non-Newtonian models

College of Science, Liaoning University of Technology, Jinzhou 121001, China

* Corresponding author: Yukun Song

Received  September 2019 Revised  November 2019 Published  March 2020

Fund Project: The first author is supported by the National Nature Science Foundation of China No. 11572146, the Education Department Foundation of Liaoning Province No. JQL201715411 and the Natural Science Foundation of Liaoning Province No. 20180550585.

This paper is concerned with the initial boundary value problem for a shear thinning fluid-particle interaction non-Newtonian model with vacuum. The viscosity term of the fluid and the non-Newtonian gravitational force are fully nonlinear. Under Dirichlet boundary for velocity and the no-flux condition for density of particles, the existence and uniqueness of strong solutions is investigated in one dimensional bounded intervals.

Citation: Yukun Song, Yang Chen, Jun Yan, Shuai Chen. The existence of solutions for a shear thinning compressible non-Newtonian models. Electronic Research Archive, 2020, 28 (1) : 47-66. doi: 10.3934/era.2020004
References:
[1]

J. Ballew and K. Trivisa, Suitable weak solutions and low stratification singular limit for a fluid particle interaction model, Q. Appl. Math., 70 (2012), 469-494.  doi: 10.1090/S0033-569X-2012-01310-2.  Google Scholar

[2]

C. BarangerL. BoudinP. E. Jabin and S. Mancini, A modeling of biospray for the upper airways. CEMRACS 2004-mathematics and applications to biology and medicine, ESAIM: Proc., 14 (2005), 41-47.   Google Scholar

[3]

G. Böhme, Non-Newtonian Fluid Mechanics, Appl. Math. Mech., North-Holland, Amsterdam, 1987.  Google Scholar

[4]

J. A. CarrilloT. Karper and K. Trivisa, On the dynamics of a fluid-particle model: The bubbling regime, Nonlinear Analysis: Real World Applications, 74 (2011), 2778-2801.  doi: 10.1016/j.na.2010.12.031.  Google Scholar

[5]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid particle interaction model, Commun. Partial. Differ. Equations, 31 (2006), 1349-1379.  doi: 10.1080/03605300500394389.  Google Scholar

[6]

J. A. CarrilloT. Goudon and P. Lafitte., Simulation of fluid and particles flows: Asymptotic preserving schemes for bubbling and flowing regimes, J. Comput. Phys, 227 (2008), 7929-7951.  doi: 10.1016/j.jcp.2008.05.002.  Google Scholar

[7]

T. ChevalierS. RodtsX. ChateauC. Chevalier and P. Coussot, Breaking of non-Newtonian character in flows through a porous medium, Physical Review E, 89 (2014), 023002.   Google Scholar

[8]

R. P. Chhabra, Bubbles, Drops, and Particles in Non-Newtonian Fluids, Second Edition. Talor & Francis, New York, 2007. Google Scholar

[9]

R. P. Chhabra and J. F. Richardson, Non-Newtonian Flow and Applied Rheology, (Second edition), Oxford, 2008. Google Scholar

[10]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 228 (2006), 377-411.  doi: 10.1016/j.jde.2006.05.001.  Google Scholar

[11]

E. Feireisl and H. Petzeltová, Large time behavior of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal., 150 (1999), 77-96.  doi: 10.1007/s002050050181.  Google Scholar

[12]

E. FeireislA. Novotný and H. Petzeltová., On the existence of globally defined weak solution to the Navier-Stokes equations, J.Math.Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar

[13]

J. GachelinG. MiñoH. BerthetA. LindnerA. Rousselet and É. Clément, Non-Newtonian viscosity of Escherichia coli suspensions, Physical Review Letters, 110 (2013), 268103.   Google Scholar

[14]

B. Guo and P. Zhu, Partial regularity of suitable weak solutions to the system of the incompressible non-Newtonian fluids, J.Differential Equations, 178 (2002), 281-297.  doi: 10.1006/jdeq.2000.3958.  Google Scholar

[15]

R. Ji and Y. Wang, Mass concentration phonomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations, Discrete and Continuous Dynamical Systems, 39 (2019), 1117-1133.  doi: 10.3934/dcds.2019047.  Google Scholar

[16]

O. A. Ladyzhenskaya, New equations for the description of viscous incompressible fluids and solvability in the large of the boundary value problems for them, In Boundary Value Problems of Mathematical Physics, vol. V, Amer. Math. Soc., Providence, RI, 1970. Google Scholar

[17]

H. Lan and R. Lian, Regularity to the spherically symmetric compressible Navier-Stokes equations with density-dependent viscosity, Boundary Value Problems, 85 (2018), 1-13.  doi: 10.1186/s13661-018-1007-x.  Google Scholar

[18] P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol.2, Compressible models, Oxford University Press, Oxford, 1998.   Google Scholar
[19]

J. Málek, J. Nečas, M. Rokyta and M. R$\dot{\rm u}$žička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman and Hall, New York. 1996.  Google Scholar

[20]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Commun. Math. Phys., 281 (2008), 573-596.  doi: 10.1007/s00220-008-0523-4.  Google Scholar

[21]

M. Pokorný, Cauchy problem for the non-newtonian viscous incompressible fluid, Applications of Mathematics, 41 (1996), 169-201.   Google Scholar

[22]

O. Rozanova, Nonexistence results for a compressible non-Newtonian fluid with magnetic effects in the whole space, J. Math. Anal. Appl., 371 (2010), 190-194.  doi: 10.1016/j.jmaa.2010.05.013.  Google Scholar

[23]

W. K. Sartory, Three-component analysis of blood sedimentation by the method of characteristics, Math. Biosci., 33 (1977), 145-165.   Google Scholar

[24]

X. Shi, Some results of boundary problem of non-Newtonian fluids, Systems Sci. Math. Sci., 9 (1996), 107-119.   Google Scholar

[25]

A. Spannenberg and K. P. Galvin, Continuous differential sedimentation of a binary suspension, Chem. Eng. Aust., 21 (1996), 7-11.   Google Scholar

[26]

Y. SongS. Chen and F. Liu, The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential, Open Mathematics, 16 (2018), 1466-1477.  doi: 10.1515/math-2018-0122.  Google Scholar

[27]

E. M. ToryK. H. KarlsenR. Bürger and S. Berres, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80.  doi: 10.1137/S0036139902408163.  Google Scholar

[28]

Y. Wang and J. Xiao, Well/ill posedness for the dissipative Navier-Stokes system in generalized carleson measure spaces, Advances in Nonlinear Analysis, 8 (2019), 203-224.  doi: 10.1515/anona-2016-0042.  Google Scholar

[29]

H. Yuan and X. Xu, Existence and uniqueness of solutions for a class of non-Newtonian fluids with singularity and vacuum, J. Differential Equations, 245 (2008), 2871-2916.  doi: 10.1016/j.jde.2008.04.013.  Google Scholar

[30]

B. M. YunL. P. DasiC. K. Aidun and A. P. Yoganathan, Computational modelling of flow through prosthetic heart valves using the entropic lattice-Boltzmann method, Journal of Fluid Mechanics, 743 (2014), 170-201.   Google Scholar

[31]

J. Zhang, C. Song and H. Li, Global solutions for the one-dimensional compressible Navier-Stokes-Smoluchowski system, Journal of Mathematical Physics, 58 (2017), 051502, 19pp. doi: 10.1063/1.4982360.  Google Scholar

[32]

C. ZhaoS. Zhou and Y. Li, Trajectory attractor and global attractor for a two-dimensional incompressible non-Newtonian fluid, J.Math.Anal.Appl., 325 (2007), 1350-1362.  doi: 10.1016/j.jmaa.2006.02.069.  Google Scholar

show all references

References:
[1]

J. Ballew and K. Trivisa, Suitable weak solutions and low stratification singular limit for a fluid particle interaction model, Q. Appl. Math., 70 (2012), 469-494.  doi: 10.1090/S0033-569X-2012-01310-2.  Google Scholar

[2]

C. BarangerL. BoudinP. E. Jabin and S. Mancini, A modeling of biospray for the upper airways. CEMRACS 2004-mathematics and applications to biology and medicine, ESAIM: Proc., 14 (2005), 41-47.   Google Scholar

[3]

G. Böhme, Non-Newtonian Fluid Mechanics, Appl. Math. Mech., North-Holland, Amsterdam, 1987.  Google Scholar

[4]

J. A. CarrilloT. Karper and K. Trivisa, On the dynamics of a fluid-particle model: The bubbling regime, Nonlinear Analysis: Real World Applications, 74 (2011), 2778-2801.  doi: 10.1016/j.na.2010.12.031.  Google Scholar

[5]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid particle interaction model, Commun. Partial. Differ. Equations, 31 (2006), 1349-1379.  doi: 10.1080/03605300500394389.  Google Scholar

[6]

J. A. CarrilloT. Goudon and P. Lafitte., Simulation of fluid and particles flows: Asymptotic preserving schemes for bubbling and flowing regimes, J. Comput. Phys, 227 (2008), 7929-7951.  doi: 10.1016/j.jcp.2008.05.002.  Google Scholar

[7]

T. ChevalierS. RodtsX. ChateauC. Chevalier and P. Coussot, Breaking of non-Newtonian character in flows through a porous medium, Physical Review E, 89 (2014), 023002.   Google Scholar

[8]

R. P. Chhabra, Bubbles, Drops, and Particles in Non-Newtonian Fluids, Second Edition. Talor & Francis, New York, 2007. Google Scholar

[9]

R. P. Chhabra and J. F. Richardson, Non-Newtonian Flow and Applied Rheology, (Second edition), Oxford, 2008. Google Scholar

[10]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differential Equations, 228 (2006), 377-411.  doi: 10.1016/j.jde.2006.05.001.  Google Scholar

[11]

E. Feireisl and H. Petzeltová, Large time behavior of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal., 150 (1999), 77-96.  doi: 10.1007/s002050050181.  Google Scholar

[12]

E. FeireislA. Novotný and H. Petzeltová., On the existence of globally defined weak solution to the Navier-Stokes equations, J.Math.Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar

[13]

J. GachelinG. MiñoH. BerthetA. LindnerA. Rousselet and É. Clément, Non-Newtonian viscosity of Escherichia coli suspensions, Physical Review Letters, 110 (2013), 268103.   Google Scholar

[14]

B. Guo and P. Zhu, Partial regularity of suitable weak solutions to the system of the incompressible non-Newtonian fluids, J.Differential Equations, 178 (2002), 281-297.  doi: 10.1006/jdeq.2000.3958.  Google Scholar

[15]

R. Ji and Y. Wang, Mass concentration phonomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations, Discrete and Continuous Dynamical Systems, 39 (2019), 1117-1133.  doi: 10.3934/dcds.2019047.  Google Scholar

[16]

O. A. Ladyzhenskaya, New equations for the description of viscous incompressible fluids and solvability in the large of the boundary value problems for them, In Boundary Value Problems of Mathematical Physics, vol. V, Amer. Math. Soc., Providence, RI, 1970. Google Scholar

[17]

H. Lan and R. Lian, Regularity to the spherically symmetric compressible Navier-Stokes equations with density-dependent viscosity, Boundary Value Problems, 85 (2018), 1-13.  doi: 10.1186/s13661-018-1007-x.  Google Scholar

[18] P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol.2, Compressible models, Oxford University Press, Oxford, 1998.   Google Scholar
[19]

J. Málek, J. Nečas, M. Rokyta and M. R$\dot{\rm u}$žička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman and Hall, New York. 1996.  Google Scholar

[20]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Commun. Math. Phys., 281 (2008), 573-596.  doi: 10.1007/s00220-008-0523-4.  Google Scholar

[21]

M. Pokorný, Cauchy problem for the non-newtonian viscous incompressible fluid, Applications of Mathematics, 41 (1996), 169-201.   Google Scholar

[22]

O. Rozanova, Nonexistence results for a compressible non-Newtonian fluid with magnetic effects in the whole space, J. Math. Anal. Appl., 371 (2010), 190-194.  doi: 10.1016/j.jmaa.2010.05.013.  Google Scholar

[23]

W. K. Sartory, Three-component analysis of blood sedimentation by the method of characteristics, Math. Biosci., 33 (1977), 145-165.   Google Scholar

[24]

X. Shi, Some results of boundary problem of non-Newtonian fluids, Systems Sci. Math. Sci., 9 (1996), 107-119.   Google Scholar

[25]

A. Spannenberg and K. P. Galvin, Continuous differential sedimentation of a binary suspension, Chem. Eng. Aust., 21 (1996), 7-11.   Google Scholar

[26]

Y. SongS. Chen and F. Liu, The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential, Open Mathematics, 16 (2018), 1466-1477.  doi: 10.1515/math-2018-0122.  Google Scholar

[27]

E. M. ToryK. H. KarlsenR. Bürger and S. Berres, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80.  doi: 10.1137/S0036139902408163.  Google Scholar

[28]

Y. Wang and J. Xiao, Well/ill posedness for the dissipative Navier-Stokes system in generalized carleson measure spaces, Advances in Nonlinear Analysis, 8 (2019), 203-224.  doi: 10.1515/anona-2016-0042.  Google Scholar

[29]

H. Yuan and X. Xu, Existence and uniqueness of solutions for a class of non-Newtonian fluids with singularity and vacuum, J. Differential Equations, 245 (2008), 2871-2916.  doi: 10.1016/j.jde.2008.04.013.  Google Scholar

[30]

B. M. YunL. P. DasiC. K. Aidun and A. P. Yoganathan, Computational modelling of flow through prosthetic heart valves using the entropic lattice-Boltzmann method, Journal of Fluid Mechanics, 743 (2014), 170-201.   Google Scholar

[31]

J. Zhang, C. Song and H. Li, Global solutions for the one-dimensional compressible Navier-Stokes-Smoluchowski system, Journal of Mathematical Physics, 58 (2017), 051502, 19pp. doi: 10.1063/1.4982360.  Google Scholar

[32]

C. ZhaoS. Zhou and Y. Li, Trajectory attractor and global attractor for a two-dimensional incompressible non-Newtonian fluid, J.Math.Anal.Appl., 325 (2007), 1350-1362.  doi: 10.1016/j.jmaa.2006.02.069.  Google Scholar

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