- Open Access
3-D flow of a compressible viscous micropolar fluid with spherical symmetry: a local existence theorem
© Dražić and Mujaković; licensee Springer 2012
- Received: 26 December 2011
- Accepted: 12 June 2012
- Published: 2 July 2012
We consider nonstationary 3-D flow of a compressible viscous heat-conducting micropolar fluid in the domain to be the subset of bounded with two concentric spheres that present solid thermoinsulated walls. In thermodynamical sense fluid is perfect and polytropic. Assuming that the initial density and temperature are strictly positive we will prove that for smooth enough spherically symmetric initial data there exists a spherically symmetric generalized solution locally in time.
- micropolar fluid
- generalized solution
- spherical symmetry
- weak and strong convergence
The theory of micropolar fluids is introduced in  by Eringen. Various problems with different initial and boundary conditions for incompressible micropolar fluid are presented in , but the theory of compressible micropolar fluid is still in the beginning. N. Mujakovic in  developed the model for one-dimensional isotropic, viscous, compressible micropolar fluid which is in thermodynamical sense perfect and polytropic. In the same work, the local existence of the solution for homogeneous boundary conditions is proved. N. Mujakovic in  and in the references cited therein proved the local and global existence of inhomogeneous boundary conditions for velocity and microrotation as well as stabilization and regularity. In  the Cauchy problem for the described problem was also considered. In the last years we find some interesting works with different kind of problems concerning micropolar fluid, e.g., [6, 7], but till now the described model of compressible micropolar fluid in three-dimensional case has not been considered.
In this work we consider the three-dimensional model with spherical symmetry. The first article in which the problem of spherical symmetry was described is , but for classical fluid. The spherical symmetry for classical fluid is also considered in articles [9–12].
In the setting of the field equations we use the Eulerian description.
In what follows we use the notation:
ρ - mass density
v - velocity
p - pressure
T - stress tensor
- an axial vector with the Cartesian components , where is Levi-Civita alternating tensora
ω - microrotation velocity
- skew tensor with Cartesian components
- microinertia density (a positive scalar field)
C - couple stress tensor
θ - absolute temperature
E - internal energy density
q - heat flux density vector
f - body force density
g - body couple density
δ - body heat density
λ, μ - coefficients of viscosity,
, , , - coefficients of microviscosity,
k - heat conduction coefficient
where R and are positive constants.
for ; the vector ν is an exterior unit normal vector.
For simplicity we also assume that and .
The initial boundary problems for the system (1)-(13) so far have not been considered in three-dimensional case. The same and similar models in one-dimensional case were considered in [3, 5, 13] and . In  the three-dimensional model was considered but for an incompressible micropolar fluid.
In this paper we prove the local existence of generalized spherically symmetric solution to the problem (1)-(13) in the domain Ω, assuming that the initial functions are also spherically symmetric. In the proof we use the Faedo-Galerkin method. We follow some ideas of  where this method was applied to a classical fluid (where microrotation is equal to zero) in one-dimensional case as well as the ideas from  and  where the same result as here was provided for one-dimensional case.
The paper is organized as follows. In the second section, we derive a spherically symmetric form of (1)-(4), introduce Lagrangian description, and present the main result. In the third section, we consider an approximate problem and get an approximate solution for each . In the forth section, we prove uniform a priori estimates for the approximate solutions. The proof of the main result is given in the fifth section.
where is a radius of the smaller boundary sphere.
Before stating the main result, we introduce the following definition.
It is easy to check that the solution (42) with properties (43)-(44) satisfies the condition for a strong solution of the described problem.
The aim of this paper is to prove the following statements.
where and .
The proof of Theorem 2.1 is essentially based on a careful examination of a priori estimates and a limit procedure. We first study, for each , an approximate problem and derive the a priori estimates for approximate solutions independent of n by utilizing a technique of Kazhikov [14, 18] and Mujakovic [3, 13] for one-dimensional case. Using the obtained a priori estimates and results of weak compactness, we extract the subsequence of approximate solutions, which, when n tends to infinity, has limit in the same weak sense on for sufficiently small , . Finally, we show that this limit is the solution to our problem.
where is defined by (40) and , are unknown smooth functions defined on an interval , .
where and are again unknown smooth functions defined on an interval , .
for are satisfied.
for , .
Lemma 3.1 For each there exists such , that the Cauchy problem (79)-(86) has a unique solution defined on . The functions , and defined by the formulas (57), (62) and (63) belong to the class , and satisfy conditions (71)-(73).
on . The constants m, a, , M and are introduced by (40), (41), (53) and (55).
Proof The statements follow from (90)-(91), (41), (53) and (55). □
Our purpose is to find , such that for each there exists a solution to the problem (79)-(86), defined on . It will be sufficient to find uniform (in ) a priori estimates for the solution defined through Lemmas 3.1 and 3.2.
In what follows we denote by or () a generic constant, not depending on , which may have different values at different places.
Some of our considerations are very similar or identical to that of  or . In these cases we omit proofs or details of proofs making references to corresponding pages of the articles  or .
and using Remark 2.2 we get (95) immediately. □
Using (92), we get (96). □
(for a function f vanishing at and and with the first derivative vanishing at some point ) that satisfy the functions , and .
Taking into account (96), (71), (73), and (97) we obtain (98). □
Lemma 4.4 (, Lemma 5.3)
With the help of (97) applied to the function , the Hoelder and Young inequalities as well as (95), we get (100). □
Using these inequalities with sufficiently small ε and estimates (92)-(94), from (102) we get (101). □
(a, , m and M are defined by (41) and (53)-(55)).
Integrating (101) over , and using estimates (110) and (115), we immediately get (103).
and using (117) and (118) from (58) and (60), we get (104)-(105).
and we can conclude that the solution of the problem (79)-(86) is defined on . □
In the same way, from (66) and (67) we obtain the estimates for and respectively. The estimates (124) and (125) follow directly from (59) and (58). □
From Lemmas 4.7 and 4.8 we easily derive the following statements.
is bounded in , and ;
is bounded in ;
is bounded in , and ;
, , are bounded in , and .
In the proofs we use some well-known facts of functional analysis (e.g., ). Let be defined by Lemma 4.7. Theorem 2.1 is a consequence of the following lemmas.
where is defined by (40).
and because of that we have (135). □
Proof Taking into account Proposition 4.1, estimates (103)-(106) and the Arzelà-Ascoli theorem, we prove in the same way as in the previous lemma the properties (143)-(147). □
Proof The conclusions (148)-(151) follow from Proposition 4.1 and embedding properties (see Remark 2.1). For verification of the boundary and initial conditions (152), (153) and (154), we use the Green formula as follows.
In the similar way, we get all the remaining properties in (152)-(154). □
Lemma 5.4 The functions r, ρ, v, ω, θ, defined by Lemmas 5.1, 5.2 and 5.3 satisfy the Equations (30)-(33) a.e. in .
and using already mentioned convergences, we can easily conclude that (160) is satisfied. In the same way, we can derive the convergences of other integrals in (159). Analogously, we get that (32) and (33) follow from (66) and (67). □
where v is from Lemma 5.3.
The conclusions of Theorem 2.1 are an immediate consequence of the above lemmas. □
We assume Einstein notation for summation.
The paper was made with financial support of the scientific project Mathematical analysis of composite and thin structures (037-0693014-2765), Ministry of Science, Education and Sports, Republic of Croatia.
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