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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

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

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

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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

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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

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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

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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

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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

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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

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