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.
Keywords: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
ω - microrotation velocity
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
The problem we consider here is based on local forms of the conservation laws for mass, momentum, momentum moment and energy, which are stated respectively as follows:
The scalar product of tensors A and B is defined by
The linear constitutive equations for stress tensor, couple stress tensor and heat flux density vector are respectively in the forms:
λ, μ - coefficients of viscosity,
k - heat conduction coefficient
are constants with the properties
Assuming that the fluid is perfect and polytropic, for pressure and internal energy we have the equations
Let , , denote the domain bounded by two concentric spheres with radii a and b. The boundary of the described domain is . We shall consider the problem (1)-(11) in the region (where is arbitrary) with the following initial conditions:
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.
2 Spherically symmetric form and the main result
We first derive the spherically symmetric form of (1)-(7) and (10)-(11). A spherically symmetric solution of (1)-(7) has the form:
with the following initial and boundary conditions
To investigate the local existence, it is convenient to transform the system (16)-(19) to that in Lagrangian coordinates. The Eulerian coordinates are connected to the Lagrangian coordinates by the relation
We introduce the new function η by
From (16) we can easily get the equation
It is useful to introduce the next coordinate
and the following functions
Similarly as in , for a new coordinate we get
We assume the inequalities
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.
Theorem 2.1Let the functions
For the functionr, it holds
From (40) and (41) we get
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.
3 Approximate solutions
We shall find a local generalized solution to the problem (30)-(38) as a limit of approximate solutions
Since and are sufficiently smooth functions, we can conclude that the function is continuous on the rectangle with the property . Because of aforementioned, we can conclude that there exists such , that
Evidently, the boundary conditions
According to the Faedo-Galerkin method, we take the following approximate conditions:
then we have
Lemma 3.1For eachthere exists such, that the Cauchy problem (79)-(86) has a unique solution defined on. The functions, anddefined by the formulas (57), (62) and (63) belong to the class, and satisfy conditions (71)-(73).
From (77) and (78) we can also easily conclude that
Proof The statements follow from (90)-(91), (41), (53) and (55). □
4 A priori estimates
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.
We also use the notation
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 .
Proof From (58) follows
and using Remark 2.2 we get (95) immediately. □
Using (92), we get (96). □
In what follows, we use the inequalities
Taking into account (96), (71), (73), and (97) we obtain (98). □
Lemma 4.4 (, Lemma 5.3)
Proof As in  Lemma 5.5,  pp.63-66 and in  Lemma 5.6, multiplying (65), (66) and (67) respectively by , and and taking into account (57), (62) and (63), after summation over and addition of the obtained equations, we get
Taking into account (92)-(94) and (95)-(100), we estimate the terms on the right-hand side of (102). For instance,
Applying the Young inequality, we get
Using these inequalities with sufficiently small ε and estimates (92)-(94), from (102) we get (101). □
and we can conclude that
We compare the solution of the problem (109)-(110) with the solution of the Cauchy problem
and from (101) it follows
Now, using the inequalities (97) for the function v, we easily get
Using (116), we derive the following estimates:
where C and B are from (103). Assuming that
and using (117) and (118) from (58) and (60), we get (104)-(105).
From (119) and (120), we can easily conclude that
and from (95), (100) and (99) it follows
Using (121), (122), (103), (116) and applying the Young inequality, we get
From Lemmas 4.7 and 4.8 we easily derive the following statements.
5 The proof of Theorem 2.1
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.
Lemma 5.1There exists a function
The functionrsatisfies the conditions
Using (104), (58), (116), (103) and (107), we obtain
and because of that we have (135). □
Lemma 5.2There exists a function
The functionρsatisfies the conditions
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). □
Lemma 5.3There exist functions
The functionsv, ωandθsatisfy the conditions
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). □
Proof Let be subsequence defined by Lemmas 5.1, 5.2 and 5.3. Taking into account (144), (149) and strong convergencies (132), (133), (145) and (151) we get that (30) follows immediately from (59). We can write Equation (65) in the form
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). □
Remark 5.1 Taking into account (150) and (58), we can easily prove that the function r defined by Lemma 5.1 has the form
where v is from Lemma 5.3.
Proof Because of the inclusion (see Remark 2.1), in the same way as in , we conclude that for each there exists , , such that for holds
The conclusions of Theorem 2.1 are an immediate consequence of the above lemmas. □
The authors declare that they have no competing interests.
The paper is the result of joint work of both authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
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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