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[J]. Electronic Research Archive, 2020, 28(1): 47-66. doi: 10.3934/era.2020004
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.
[1] | Suitable weak solutions and low stratification singular limit for a fluid particle interaction model. Q. Appl. Math. (2012) 70: 469-494. |
[2] | A modeling of biospray for the upper airways. CEMRACS 2004-mathematics and applications to biology and medicine. ESAIM: Proc. (2005) 14: 41-47. |
[3] | G. Böhme, Non-Newtonian Fluid Mechanics, Appl. Math. Mech., North-Holland, Amsterdam, 1987. |
[4] | On the dynamics of a fluid-particle model: The bubbling regime. Nonlinear Analysis: Real World Applications (2011) 74: 2778-2801. |
[5] | Stability and asymptotic analysis of a fluid particle interaction model. Commun. Partial. Differ. Equations (2006) 31: 1349-1379. |
[6] | Simulation of fluid and particles flows: Asymptotic preserving schemes for bubbling and flowing regimes. J. Comput. Phys (2008) 227: 7929-7951. |
[7] | Breaking of non-Newtonian character in flows through a porous medium. Physical Review E (2014) 89: 023002. |
[8] | R. P. Chhabra, Bubbles, Drops, and Particles in Non-Newtonian Fluids, Second Edition. Talor & Francis, New York, 2007. |
[9] | R. P. Chhabra and J. F. Richardson, Non-Newtonian Flow and Applied Rheology, (Second edition), Oxford, 2008. |
[10] | Existence results for viscous polytropic fluids with vacuum. J. Differential Equations (2006) 228: 377-411. |
[11] | Large time behavior of solutions to the Navier-Stokes equations of compressible flow. Arch. Ration. Mech. Anal. (1999) 150: 77-96. |
[12] | On the existence of globally defined weak solution to the Navier-Stokes equations. J.Math.Fluid Mech. (2001) 3: 358-392. |
[13] | Non-Newtonian viscosity of Escherichia coli suspensions. Physical Review Letters (2013) 110: 268103. |
[14] | Partial regularity of suitable weak solutions to the system of the incompressible non-Newtonian fluids. J.Differential Equations (2002) 178: 281-297. |
[15] | Mass concentration phonomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems (2019) 39: 1117-1133. |
[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. |
[17] | Regularity to the spherically symmetric compressible Navier-Stokes equations with density-dependent viscosity. Boundary Value Problems (2018) 85: 1-13. |
[18] | (1998) Mathematical Topics in Fluid Dynamics, Vol.2.Compressible models, Oxford University Press. |
[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. |
[20] | Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations. Commun. Math. Phys. (2008) 281: 573-596. |
[21] | Cauchy problem for the non-newtonian viscous incompressible fluid. Applications of Mathematics (1996) 41: 169-201. |
[22] | Nonexistence results for a compressible non-Newtonian fluid with magnetic effects in the whole space. J. Math. Anal. Appl. (2010) 371: 190-194. |
[23] | Three-component analysis of blood sedimentation by the method of characteristics. Math. Biosci. (1977) 33: 145-165. |
[24] | Some results of boundary problem of non-Newtonian fluids. Systems Sci. Math. Sci. (1996) 9: 107-119. |
[25] | Continuous differential sedimentation of a binary suspension. Chem. Eng. Aust. (1996) 21: 7-11. |
[26] | The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential. Open Mathematics (2018) 16: 1466-1477. |
[27] | Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math. (2003) 64: 41-80. |
[28] | Well/ill posedness for the dissipative Navier-Stokes system in generalized carleson measure spaces. Advances in Nonlinear Analysis (2019) 8: 203-224. |
[29] | Existence and uniqueness of solutions for a class of non-Newtonian fluids with singularity and vacuum. J. Differential Equations (2008) 245: 2871-2916. |
[30] | Computational modelling of flow through prosthetic heart valves using the entropic lattice-Boltzmann method. Journal of Fluid Mechanics (2014) 743: 170-201. |
[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 |
[32] | Trajectory attractor and global attractor for a two-dimensional incompressible non-Newtonian fluid. J.Math.Anal.Appl. (2007) 325: 1350-1362. |