- Open Access
3-D flow of a compressible viscous micropolar fluid with spherical symmetry: uniqueness of a generalized solution
© Mujaković and Dražić; licensee Springer. 2014
- Received: 30 December 2013
- Accepted: 26 June 2014
- Published: 3 October 2014
We consider nonstationary 3-D flow of a compressible viscous heat-conducting micropolar fluid in the domain that is the subset of bounded with two concentric spheres that present the solid thermoinsulated walls. In the thermodynamical sense the fluid is perfect and polytropic. If we assume that the initial density and temperature are strictly positive and that the initial data are sufficiently smooth spherically symmetric functions then our problem has a generalized solution for a sufficiently small time interval. We study the problem in the Lagrangian description and prove the uniqueness of its generalized solution.
- micropolar fluid
- spherical symmetry
- generalized solution
The theory of micropolar fluids was introduced by Ahmed Cemal Eringen in 1960 . Eringen suggested many possible applications of the micropolar fluid, but from the mathematical point of view the theory is still in an early stage of development. The results for incompressible flow are very well systematized in the book of Lukaszewicz , but the theory for compressible flows, especially for flows involving temperature, is still in its beginnings.
In this paper we analyze compressible flow of isotropic, viscous, and heat-conducting micropolar fluid which is in the thermodynamical sense perfect and polytropic. The model for this type of flow was first considered by Mujaković in  where she developed the one-dimensional model. In the same work, the local existence and uniqueness of a generalized solution for homogeneous boundary conditions were proved. In the work  the existence of a solution global in time for the described problem was proved. Mujaković also analyzed the regularity and stabilization for the model, as well as the Cauchy problem . In her recent works, for example , the problem with non-homogeneous boundary condition for density, microrotation, and heat flux was analyzed.
Other authors, for example Chen et al. in  and  or Easwaran and Majumdar in , analyzed different kinds of problems concerning micropolar fluid, as well as in the three-dimensional case, but without the temperature. Till now the described model of compressible micropolar fluid (model with temperature) in the three-dimensional case has been considered just in  by Dražić and Mujaković in the spherically symmetric case.
Spherically symmetric flow for a classical fluid was considered for example in –, and . Uniqueness of the solution for the problems with a classical fluid in the spherically symmetric case was proved in . For a micropolar fluid in the one-dimensional case, the uniqueness results are given for example in  for the Cauchy problem. The uniqueness problem for the micropolar fluid was also considered in  but for a fluid which is not heat-conducting.
In this work we prove the uniqueness of the solution for the problem presented in  where we proved the local existence of generalized spherically symmetric solution for the flow of described fluid in the domain to be subset of bounded with two concentric spheres that present solid thermoinsulated walls, assuming that the initial density and temperature are strictly positive and that the initial data are smooth enough spherically symmetric functions.
( is radius of the smaller boundary sphere (see )).
In this paper we study the uniqueness of a generalized solution which is defined as follows.
Our proof of the uniqueness does not depend on the size of the time existence interval . Because of that, hereafter we will take .
The aim of this paper is to prove the following result.
As has already been mentioned, the analogous theorems for the one-dimensional case have been proved in  and . In  was considered the one-dimensional problem with the same type of boundary and initial conditions as in this article. In  there is a proof of the uniqueness theorem for the Cauchy problem of the described fluid in the one-dimensional case. In the proof of Theorem 2.2 we use the method described in , where it has been applied for the one-dimensional case of a classical fluid (problem without the microrotation variable ω) and also we use some ideas from  and .
(is the radius of smaller boundary sphere; see) for each.
Because of simplicity, hereafter we will consider the specific volume instead of density ρ.
where , , , , , , and .
for , . We also have .
The proof of Theorem 2.2 is based on getting four inequalities which we use to control the bounds of the values of the functions u, v, ω, and θ. These inequalities we prove in the following four lemmas.
and Theorem 2.2 is proved.
The paper was made with financial support of scientific project ‘Mathematical and numerical modelling of compressible micropolar fluid flow’ (220.127.116.11.03), University of Rijeka, Croatia.
- Eringen AC: Simple microfluids. Int. J. Eng. Sci. 1964, 2(2):205-217. 10.1016/0020-7225(64)90005-9MathSciNetView ArticleGoogle Scholar
- Lukaszewicz G: Micropolar Fluids: Theory and Applications. Birkhäuser, Boston; 1999.View ArticleGoogle Scholar
- Mujaković N: One-dimensional flow of a compressible viscous micropolar fluid: a local existence theorem. Glas. Mat. 1998, 33: 71-91.Google Scholar
- Mujaković N: One-dimensional flow of a compressible viscous micropolar fluid: a global existence theorem. Glas. Mat. 1998, 33: 199-208.Google Scholar
- Mujaković N: One-dimensional compressible viscous micropolar fluid model: stabilization of the solution for the Cauchy problem. Bound. Value Probl. 2010., 2010: 10.1155/2010/796065Google Scholar
- Mujaković N: Nonhomogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: regularity of the solution. Bound. Value Probl. 2008., 2008: 10.1155/2008/189748Google Scholar
- Chen M: Blowup criterion for viscous, compressible micropolar fluids with vacuum. Nonlinear Anal., Real World Appl. 2012, 13(2):850-859. 10.1016/j.nonrwa.2011.08.021MathSciNetView ArticleGoogle Scholar
- Chen M, Huang B, Zhang J: Blowup criterion for the three-dimensional equations of compressible viscous micropolar fluids with vacuum. Nonlinear Anal. TMA 2013, 79: 1-11. 10.1016/j.na.2012.10.013MathSciNetView ArticleGoogle Scholar
- Easwaran C, Majumdar S: A uniqueness theorem for compressible micropolar flows. Acta Mech. 1987, 68: 185-191. 10.1007/BF01190882MathSciNetView ArticleGoogle Scholar
- Dražić I, Mujaković N: 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: a local existence theorem. Bound. Value Probl. 2012., 2012: 10.1186/1687-2770-2012-69Google Scholar
- Hoff D: Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data. Indiana Univ. Math. J. 1992, 41: 1225-1302. 10.1512/iumj.1992.41.41060MathSciNetView ArticleGoogle Scholar
- Jiang S, Zhang P: On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations. Commun. Math. Phys. 2001, 215: 559-581. 10.1007/PL00005543View ArticleGoogle Scholar
- Fujita-Yashima H, Benabidallah R: Equation à symétrie sphérique d’un gaz visqueux et calorifère avec la surface libre. Ann. Mat. Pura Appl. 1995, 168: 75-117. 10.1007/BF01759255MathSciNetView ArticleGoogle Scholar
- Yanagi S: Asymptotic stability of the spherically symmetric solutions for a viscous polytropic gas in a field of external forces. Transp. Theory Stat. Phys. 2000, 29(3-5):333-353. 10.1080/00411450008205878MathSciNetView ArticleGoogle Scholar
- Fujita-Yashima H, Benabidallah R: Unicité de la solution de equation monodimensionnelle ou a symetrie spherique d’un gaz visqueux et calorifere. Rend. Circ. Mat. Palermo 1993, XLII: 195-218. 10.1007/BF02843945MathSciNetView ArticleGoogle Scholar
- Mujaković N: Uniqueness of a solution of the Cauchy problem for one-dimensional compressible viscous micropolar fluid model. Appl. Math. E-Notes 2006, 6: 113-118.MathSciNetGoogle Scholar
- Lions JL, Dautray R: Functional and Variational Methods. Springer, Berlin; 2000.Google Scholar
- Lions JL, Dautray R: Evolution Problems I. Springer, Berlin; 2000.Google Scholar
- Antontsev SN, Kazhikhov AV, Monakhov VN: Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. North-Holland, Amsterdam; 1990.Google Scholar
This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.