American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020193

The nonstationary flows of micropolar fluids with thermal convection: An iterative approach

Charles Amorim 1, , Miguel Loayza 1,, and  Marko A. Rojas-Medar 2,

1. 

Departamento de Matemática, Universidade Federal de Pernambuco, Recife, PE, Brazil
2. 

Departamento de Matemática, Universidad de Tarapacá, Arica, Chile

* Corresponding author: miguel@dmat.ufpe.br

Received  October 2019 Revised  January 2020 Published  June 2020

Fund Project: This work was partially supported by CAPES-PRINT, 88887.311962/2018-00.
Charles Amorim was supported by CNPQ/Brazil

We consider a problem that describes the motion of a viscous incompressible and heat-conducting micropolar fluids in a bounded domain $ \Omega \subset \mathbb{R}^3 $. We use an iterative method to analyze the existence, uniqueness, and regularity of the solutions. We also determine the convergence rates in several norms.

Keywords: Thermal convection, existence and uniqueness, regularity, convergence rates.
Mathematics Subject Classification: Primary: 35Q35, 65M12, 65M15, 76D03; Secondary: 76R05, 76M99.
Citation: Charles Amorim, Miguel Loayza, Marko A. Rojas-Medar. The nonstationary flows of micropolar fluids with thermal convection: An iterative approach. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020193
References:
[1]

C. Amrouche and V. Girault, On the existence and regularity of the solution of Stokes problem in arbitrary dimension, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 171-175.  doi: 10.3792/pjaa.67.171.  Google Scholar

[2]

J. Boussinesq, Théorie Analytique de la Chaleur II, Gauthier-Villars, 1903. Google Scholar

[3]

A. C. Eringen, Simple microfluids, Internat. J. Engrg. Sci., 2 (1964), 205-217.  doi: 10.1016/0020-7225(64)90005-9.  Google Scholar

[4]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.  Google Scholar

[5]

D. D. Joseph, Stability of Fluid Motions. I, Springer Tracts in Natural Philosophy, 27, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-80991-0.  Google Scholar

[6]

Y. Kagei and M. Skowron, Nonstationary flows of nonsymmetric fluids with thermal convection, Hiroshima Math. J., 23 (1993), 343-363.  doi: 10.32917/hmj/1206128257.  Google Scholar

[7]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Mathematics and its Applications, 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.  Google Scholar

[8]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uralćeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[9]

J.-L. Lions, Quelques méthodes de résolution des problémes aux limites non linéares, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[10]

G. Łukaszewicz, On the existence, uniqueness and asymptotic properties for solutions of flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 13 (1989), 105-120.   Google Scholar

[11]

G. Łukaszewicz and W. Waluś, On stationary flows of asymmetric fluids with heat convection, Math. Methods Appl. Sci., 11 (1989), 343-351.  doi: 10.1002/mma.1670110304.  Google Scholar

[12]

G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[13]

L. G. Petrosyan, Some problems of mechanics of fluids with antisymmetric stress tensor, Izd. Erevan Univ., (1984). Google Scholar

[14]

M. A. Rojas-Medar and E. E. Ortega-Torres, The equations of a viscous asymmetric fluid: An interactive approach, ZAMM Z. Angew. Math. Mech., 85 (2005), 471-489.  doi: 10.1002/zamm.199910189.  Google Scholar

[15]

M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.  doi: 10.1002/mana.19971880116.  Google Scholar

[16]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[17]

A. G. Zarubin, An iterative method for the approximate solution of an initial-boundary value problem for heat convection equations, Comput. Math. Math. Phys., 33 (1993), 1077-1085.   Google Scholar

show all references

References:
[1]

C. Amrouche and V. Girault, On the existence and regularity of the solution of Stokes problem in arbitrary dimension, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 171-175.  doi: 10.3792/pjaa.67.171.  Google Scholar

[2]

J. Boussinesq, Théorie Analytique de la Chaleur II, Gauthier-Villars, 1903. Google Scholar

[3]

A. C. Eringen, Simple microfluids, Internat. J. Engrg. Sci., 2 (1964), 205-217.  doi: 10.1016/0020-7225(64)90005-9.  Google Scholar

[4]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.  Google Scholar

[5]

D. D. Joseph, Stability of Fluid Motions. I, Springer Tracts in Natural Philosophy, 27, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-80991-0.  Google Scholar

[6]

Y. Kagei and M. Skowron, Nonstationary flows of nonsymmetric fluids with thermal convection, Hiroshima Math. J., 23 (1993), 343-363.  doi: 10.32917/hmj/1206128257.  Google Scholar

[7]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Mathematics and its Applications, 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.  Google Scholar

[8]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uralćeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[9]

J.-L. Lions, Quelques méthodes de résolution des problémes aux limites non linéares, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[10]

G. Łukaszewicz, On the existence, uniqueness and asymptotic properties for solutions of flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 13 (1989), 105-120.   Google Scholar

[11]

G. Łukaszewicz and W. Waluś, On stationary flows of asymmetric fluids with heat convection, Math. Methods Appl. Sci., 11 (1989), 343-351.  doi: 10.1002/mma.1670110304.  Google Scholar

[12]

G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[13]

L. G. Petrosyan, Some problems of mechanics of fluids with antisymmetric stress tensor, Izd. Erevan Univ., (1984). Google Scholar

[14]

M. A. Rojas-Medar and E. E. Ortega-Torres, The equations of a viscous asymmetric fluid: An interactive approach, ZAMM Z. Angew. Math. Mech., 85 (2005), 471-489.  doi: 10.1002/zamm.199910189.  Google Scholar

[15]

M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.  doi: 10.1002/mana.19971880116.  Google Scholar

[16]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar

[17]

A. G. Zarubin, An iterative method for the approximate solution of an initial-boundary value problem for heat convection equations, Comput. Math. Math. Phys., 33 (1993), 1077-1085.   Google Scholar

[1]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310
[2]

Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020367
[3]

Xiuli Xu, Xueke Pu. Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 987-1010. doi: 10.3934/dcdsb.2020150
[4]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320
[5]

Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020289
[6]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[7]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326
[8]

Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039
[9]

Thomas Y. Hou, Dong Liang. Multiscale analysis for convection dominated transport equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 281-298. doi: 10.3934/dcds.2009.23.281
[10]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
[11]

Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 1-27. doi: 10.3934/dcds.2009.23.1
[12]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384
[13]

Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030
[14]

Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034
[15]

Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161
[16]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[17]

Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083
[18]

Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341
[19]

Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230
[20]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004

2019 Impact Factor: 1.27

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences