Abstract
In this work, the three-dimensional model for the compressible micropolar fluid flow is considered, whereby it is assumed that the fluid is viscous, perfect, and heat conducting. The flow between two coaxial thermoinsulated cylinders, which leads to a cylindrically symmetric model with homogeneous boundary data for velocity, microrotation, and heat flux, is analyzed.
The corresponding PDE system is formulated in the Lagrangian setting, and it is proven that this system has a generalized solution locally in time.
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1 Introduction
In this paper, we analyze the compressible flow of an isotropic, viscous, and heat-conducting micropolar fluid, whereby we consider the flow between two coaxial thermoinsulated solid cylinders. We also assume that the fluid is perfect and polytropic in the thermodynamical sense.
The micropolar fluid is a type of fluid which exhibits microrotational effects, as well as microrotational inertia, and it can be perceived as a collection of rigid particles suspended in a viscous medium, which can rotate about the centroid of the volume element. Consequently, it belongs to the class of viscous fluids with a non-symmetric stress tensor, hence the law of conservation of angular momentum must be taken into account. Therefore, in addition to the standard hydrodynamic and thermodynamic variables (mass density, velocity, and temperature), the microrotation vector is introduced to describe the micro phenomena. The micropolar fluid was introduced by Eringen (see [1]) as an extension of the Navier–Stokes model, capable of treating phenomena at the microlevel (see [2]).
As today’s science is increasingly engaged with micro and nanotechnology, the need has emerged for models that can handle the impact of the scale. That is why the micropolar continuum has begun to be intensively studied in the last few years. Let us note that the micropolar fluid model has been applied as the model for blood flow (see [3]), water-based nanofluids (see [4]), mimicking physical phenomena of bacteria (see [5]), the behavior of the epididymal material (see [6]), describing lubricants with additives, the motion of the synovial fluid in the joints (see [7]), etc. Recently, the micropolar fluid model has been used to effectively treat some heating problems as well (see [8] and [9]).
The aforementioned model was first analyzed in the one-dimensional case by N. Mujaković in [10] and for the first time in [11] in the three-dimensional case with the assumptions of spherical symmetry of the solution by I. Dražić and N. Mujaković. For the recent progress in the mathematical analysis of these two models, we refer to [12] and [13], and for general theory to [14].
In this paper, assuming that initial functions are cylindrically symmetric and smooth enough, we prove the local existence of the generalized cylindrically symmetric solution to the governing system with homogeneous boundary conditions for velocity, microrotation, and heat flux. In the proof, we follow the ideas from [10] and [11], whereby we apply the Faedo–Galerkin method together with some ideas from [15], where this method was applied for the classical fluid model in the one-dimensional case. Let us mention that the utilization of the Faedo–Galerkin method does not require additional restrictions to initial data, unlike in other approaches. Let us also note that the obtained model is the generalization of the model for the classical fluid considered by Qin in [16] and [17].
The paper is organized as follows. In the next section we state the problem, define the generalized solution to our problem, and present the main result. In the third section, in line with the Faedo–Galerkin method, we introduce the approximate problem and form a series of approximate solutions. In the forth section, we derive some a priori estimates for the obtained approximate solutions, which are the base for the final proof of our main result (the local existence theorem), which is presented in the last section of this paper.
2 Statement of the problem and the main result
The governing initial boundary value problem is derived in [18] and reads as follows:
defined on the domain \(Q_{T}=\,]0,L[\,\times \,]0,T[\,\), where
Variables in this system are mass density ρ, velocity v, microrotation ω, and temperature θ, and they are defined by
where
Here, the positive constant \(j_{I}\) is microinertia density, λ and μ are the coefficients of viscosity, \(\mu _{r}\), \(c_{0}\), \(c_{a}\), and \(c_{d}\) are the coefficients of microviscosity, k (\(k\geq 0\)) is the heat conduction coefficient, the positive constant R is the specific gas constant, and the positive constant \(c_{v}\) is the specific heat for a constant volume. Coefficients of viscosity and microviscosity have the following properties:
Let us note that equations (1)–(8) are local forms of conservation laws and, with boundary conditions (11)–(13), we describe the acting of the solid thermo-insulated walls.
This problem is written in the Lagrangian description, which is much simpler in comparison to the Eulerian description. Moreover, using the Lagrangian coordinates, we eliminate the hyperbolic part of the system, and the density equation becomes explicitly solvable once the velocity has been determined. At the same time, other equations remain parabolic. Because of this coordinate transform, described in [18], we have the property
where
and \(a>0\) is the radius of the smaller boundary cylinder.
The main purpose of this work is to prove that problem (1)–(13) has a generalized (weak) solution in the domain \(Q_{T_{0}}=\,]0,L[\,\times ]0,T_{0}[\) for sufficiently small time \(T_{0}>0\).
Let us first introduce the vectors
and the definition of a generalized solution.
Definition 1
A generalized solution of problem (1)–(13) in the domain \(Q_{T}\) is the function
where
that satisfies equations (1)–(8) a.e. in \(Q_{T}\) and conditions (9)–(13) in the sense of traces.
It is important to note that function (26) has the properties of a strong solution. Because of the embedding and interpolation theorems (e.g., in [19] and [20]), we also have:
The aim of this paper is to prove the following theorem.
Theorem 1
Let the functions
satisfy the conditions
where \(m \in \mathbf{R}^{+}\). Then there exists \(T_{0}\), \(0< T_{0} \leq T\), such that problem (1)–(13) has a generalized solution in \(Q_{0}=Q_{T_{0}}\), having the property
For the function r, we have
Remark 1
Because of the embedding \(\mathrm{H}^{1}(]0,L[)\subset \mathrm{C}([0,L])\), we can conclude that there exists \(M\in \mathbf{R}^{+}\) such that
for \(x\in [0,L]\).
The function \(r_{0}\), introduced by (24), belongs to the space \(\mathrm{H}^{2}(]0,L[)\subset \mathrm{C}^{1}([0,L])\), and we have
where \(a_{1}=M^{-2}\), \(M_{1}=(ma)^{-1}\), a is from (24) and m from (36).
The proof of Theorem 1 is essentially based on the Faedo–Galerkin method. We first define the approximate problem (for each \(n\in \mathbf{N}\)). Then we derive uniform (in n) a priori estimates for approximate solutions, where we utilize the techniques applied in [10, 15], and [11] to similar models. Using the obtained estimates and results of weak compactness, we extract the subsequence of approximate solutions, which has a limit in some weak sense on \(\,]0,L[\,\times ]0,T_{0}[\) for sufficiently small \(T_{0}>0\).
3 Approximate solutions
In [18], we have already introduced the Faedo–Galerkin approximations to problem (1)–(13), where we used them to find the numerical solution to problem (1)–(13). For the reader’s convenience, we will describe it here briefly.
As we have already pointed out, we shall find a local generalized solution to problem (1)–(13) as a limit of approximate solutions
where \(\rho ^{n}\), \(v^{r n}\), \(v^{\varphi n}\), \(v^{z n}\), \(\omega ^{r n}\), \(\omega ^{\varphi n}\), \(\omega ^{z n}\), and \(\theta ^{n}\) are the approximations of the functions ρ, \(v^{r}\), \(v^{\varphi }\), \(v^{z}\), \(\omega ^{r}\), \(\omega ^{\varphi }\), \(\omega ^{z}\), and θ, respectively. We define them by
We also define
where \(r_{0}(x)\) is defined by (24). \(v_{i}^{r n}\), \(v _{i}^{\varphi n}\), \(v_{i}^{z n}\), \(i=1,2,\ldots ,n\), \(\omega _{j} ^{r n}\), \(\omega _{j}^{\varphi n}\), \(\omega _{j}^{z n}\), \(j=1,\ldots ,n\) and \(\theta _{k}^{n}\), \(k=0,\ldots ,n\), are unknown smooth functions defined on the interval \([0,T_{n}]\), \(T_{n}\leq T\).
The approximation \(\rho ^{n}\) of the function ρ becomes the solution to the initial problem
and it can be written in the form
We also have
Since \(r^{n}\) and \(v^{r n}\) are sufficiently smooth functions, we can conclude that the function \(\rho ^{n}\) is continuous on the rectangle \([0,L]\times [0,T_{n}]\) with the property \(\rho ^{n}(x,0)=\rho _{0}(x) \geq m>0\). Because of the aforementioned, we can conclude that there exists such \(T_{n}\), \(0< T_{n}\leq T\), that
Evidently, the boundary conditions
are satisfied, which is in accordance with boundary conditions (11)–(13) of the starting problem.
We take the initial conditions for \(v^{r n}\), \(v^{\varphi n}\), \(v^{z n}\), \(\omega ^{r n}\), \(\omega ^{\varphi n}\), \(\omega ^{z n}\), and \(\theta ^{n}\) in the form:
where \(v_{0}^{r n}\), \(v_{0}^{\varphi n}\), \(v_{0}^{z n}\), \(\omega _{0}^{r n}\), \(\omega _{0}^{\varphi n}\), \(\omega _{0}^{z n}\), and \(\theta _{0}^{n}\) are defined by
and \(v_{0i}^{r}\), \(v_{0i}^{\varphi }\), \(v_{0i}^{z}\), \(\omega _{0j}^{r}\), \(\omega _{0j}^{\varphi }\), \(\omega _{0j}^{z}\), and \(\theta _{0k}\) are the Fourier coefficients of the functions \(v_{0}^{r}\), \(v_{0}^{\varphi }\), \(v_{0}^{z}\), \(\omega _{0}^{r}\), \(\omega _{0}^{\varphi }\), \(\omega _{0} ^{z}\), and \(\theta _{0}\), respectively.
According to the Faedo–Galerkin method, we take the following approximate conditions:
for \(i_{1},i_{2},i_{3},j_{1},j_{2},j_{3}=1,\ldots ,n\), \(k=0,1,\ldots ,n\).
To simplify the problem, we introduce the functions \(z^{n}_{m}(t)\) and \(\lambda ^{n}_{pq}(t)\) by
Now, (47) and (49) could be written in the form
Taking into account (58)–(72), from (62)–(68) we obtain, for \(n^{2}+8n+1\)-tuple \(\mathbf{u}_{n}\) (with coordinates \(v_{i_{1}}^{r n}\), \(v_{i_{2}}^{ \varphi n}\), \(v_{i_{3}}^{z n}\), \(\omega _{j_{1}}^{r n}\), \(\omega _{j_{2}}^{\varphi n}\), \(\omega _{j_{3}}^{z n}\), \(\theta _{k} ^{n}\), \(z_{m}^{n}\), \(\lambda ^{n}_{pq}\), \(i_{1}\), \(i_{2}\), \(i_{3}\), \(j_{1}\), \(j_{2}\), \(j_{3}\), m, p, \(q=1,\ldots ,n\), \(k=0,1,\ldots ,n\)), the following differential equation:
with the initial conditions \(\mathbf{u}_{n}(0)\) defined by
Let us notice that the function F on the right-hand side of (73) satisfies the conditions of the Cauchy–Picard theorem, and we can conclude that problem (73)–(76) has a unique smooth solution on a sufficiently small domain \([0,T_{n}[\), \(T_{n}\leq T\). Because of (36), (40), and (41), we easily obtain the following statements.
Lemma 1
For each \(n\in N\), there exists such \(T_{n}, 0< T_{n}\leq T\), and \(Q_{n}=\,]0,L[\,\times ]0,T_{n}[\), that the functions \(v^{r n}\), \(v^{\varphi n}\), \(v^{z n}\), \(\omega ^{r n}\), \(\omega ^{\varphi n}\), \(\omega ^{z n}\), and \(\theta ^{n}\) belong to the class \(\mathrm{C} ^{\infty }(\overline{Q}_{n})\) and satisfy conditions (55)–(57).
Moreover, we have \(\rho ^{n}\in \mathrm{C}(\overline{Q}_{n})\), \(r^{n}\in \mathrm{C}^{1}(\overline{Q}_{n})\), and
on \(\overline{Q}_{n}\). The constants m, a, \(a_{1}\), M, and \(M_{1}\) are introduced by (36), (40), and (41).
4 Some properties of approximate solutions
In the previous section, we showed that, for each \(n\in \mathbf{N}\), there exists \(T_{n}, 0< T_{n}\leq T\), such that the set \(Q_{n}\) is a domain of the nth approximate solution introduced in Lemma 1. Our first goal is to find such \(T_{0}\), \(0< T_{0}\leq T\), that for each \(n\in \mathbf{N}\) a solution \(\mathbf{u}_{n}\) to problem (73)–(76) is defined on \([0,T_{0}]\). Therefore, the approximate functions (43)–(47) and (49) also exist on \(Q_{0}\). For that we need some interrelationships between the functions \(\rho ^{n}\), \(v^{r n}\), \(v ^{\varphi n}\), \(v^{z n}\), \(\omega ^{r n}\), \(\omega ^{\varphi n}\), \(\omega ^{z n}\), and \(\theta ^{n}\) which we, using the ideas adapted from [11], state in the following lemmas.
Hereafter, we denote by \(C>0\) or \(C_{i}>0\) (\(i=1,2,\ldots\)) a generic constant, independent of \(n\in \mathbf{N}\), which may have different values in different places.
For simplicity reasons, we use the notations
as well as
for \(\beta =0,1,\dots \). We use the same notations for the vector \(\mathbf{W}=(\omega ^{r n}, \omega ^{\varphi n}, \omega ^{z n})\).
In what follows, we use the inequalities
which are valid for the function f vanishing at \(x=0\) and \(x=L\), and with the first derivative vanishing at some point \(x\in [0,L]\). These inequalities satisfy the functions \(v^{r n}\), \(v^{\varphi n}\), \(v^{z n}\), \(\omega ^{r n}\), \(\omega ^{\varphi n}\), and \(\omega ^{z n}\). The function \(\theta ^{n}\) satisfies only (85). Let us note that inequalities (84) and (85) follow from the Gagliardo–Ladyzhenskaya, the Friedrichs, and the Poincaré inequalities, adapted in accordance with the spaces of functions used in this work.
Hereafter, we use \(T_{n}\), \(0< T_{n}\leq T\), from Lemma 1.
Lemma 2
For \(t\in [0,T_{n}]\), we have
Proof
From (47) we have
and by using Remark 1 we immediately obtain (86). □
Lemma 3
For \(t\in [0,T_{n}]\), we have
Proof
After multiplying (65), (66), and (67), respectively, by \(j_{I}\omega ^{r n}_{j_{1}}\), \(j_{I}\omega ^{\varphi n}_{j_{2}}\), and \(j_{I}\omega ^{z n}_{j_{3}}\), summing over \(j_{1},j_{2},j_{3}=1,\ldots,n\), and by using formula (50), integration by parts, and boundary conditions, we obtain
Now, we will use (77) and (78) as well as the Young inequality with the parameter \(\varepsilon >0\) applied to the integrals on the right-hand side of (89). We obtain
Integrating (90) over \([0,t],0< t\leq T_{n}\), and taking into account that
from (90), for sufficiently small \(\varepsilon >0\), we obtain (88). □
Lemma 4
For \(t \in [0,T_{n}]\), we have
Proof
First, we multiply (62) and (63), respectively, by \(v^{r n}_{i_{1}}\) and \(v^{\varphi n}_{i_{2}}\), sum over \(i_{1},i_{2}=1,\ldots,n\), and add (68) for \(k=0\) (multiplied by \(c_{v}\)). After we integrate the resulting equality over \([0,L ]\) and employ integration by parts, we get
Integrating (93) over \([0,t]\), \(0< t\leq T_{n}\), using
the Young inequality, and properties (77)–(79), from (93) we have
Since, because of (84), we have
taking into account (88), from (95) we easily obtain (92). □
Lemma 5
For \((x,t)\in \overline{Q}_{n}\), we have
Proof
Let \(t\in [0,T_{n} ]\) be fixed, but arbitrary. As the function \(\theta ^{n}\) is continuous with respect to the variable \(x\in [0,L ]\) (see Lemma 1), there exist such \(x_{1}(t),x_{2}(t)\in [0,L ]\) that we have
Now, using the Hölder inequality, we obtain
which implies
Analogously, using the function \(M_{n}(t)\), we obtain
which together with (101) implies
Now, using (92), from (102) we immediately obtain (97). □
Lemma 6
For \(t\in [0,T_{n}]\), we have
Proof
Taking the derivative of the function \(\rho ^{n}\) represented by (49) with respect to x and by using (36) and (77)–(79), we obtain
With the help of (84) and (85) applied to the function \(v^{r n}\), the Hölder, and the Young inequalities as well as (86), we obtain (104). □
Lemma 7
For \(t\in [0,T_{n}]\), we have
Proof
We apply a similar procedure as in [11], Lemma 4.6, and in [10], Lemma 5.6. Multiplying (62)–(68), respectively, by \(-\frac{(\pi i _{1})^{2}}{L^{2}}v_{i_{1}}^{r n}\), \(-\frac{(\pi i_{2})^{2}}{L^{2}}v _{i_{2}}^{\varphi n}\), \(-\frac{(\pi i_{3})^{2}}{L^{2}}v_{i_{3}} ^{z n}\), \(-\frac{(\pi j_{1})^{2}}{L^{2}}\omega _{j_{1}}^{r n}\), \(-\frac{(\pi j_{2})^{2}}{L^{2}}\omega _{j_{2}}^{\varphi n}\), \(-\frac{(\pi j_{3})^{2}}{L^{2}}\omega _{j_{3}}^{z n}\), and \(-\frac{(\pi k)^{2}}{L^{2}}\theta _{k}^{n}\), after summation over \(i_{1},i_{2},i_{3},j_{1},j_{2},j_{3},k=1,2,\ldots,n\), using (50) and addition of the obtained equations, we obtain
where
Now, we will estimate integrals \(I_{1}\)–\(I_{61}\). Taking into account (77)–(79) as well as (84) and the Young inequality, we obtain
In the same way, we estimate integrals \(I_{7}\), \(I_{12}\), \(I_{15}\), \(I_{17}\), \(I_{21}\), \(I_{22}\), \(I_{26}\), \(I_{28}\), \(I_{31}\), and \(I_{33}\).
We estimate integrals \(I_{2}\), \(I_{5}\), \(I_{8}\), \(I_{10}\), \(I_{13}\), \(I_{16}\), \(I_{18}\), \(I_{23}\), \(I_{27}\), \(I_{29}\), \(I_{32}\), and \(I_{34}\) by using the Young inequality, and in some cases by using (77) and (78). For instance, we have
Now, we estimate integral \(I_{3}\). To do it, we need (84), the Hölder, and the Young inequalities, as well as (78) and (104). We have
We use the same approach in the estimates of integrals \(I_{9}\), \(I_{14}\), \(I_{19}\), \(I_{24}\), \(I_{30}\), and \(I_{35}\).
To estimate integral \(I_{4}\), we need the Hölder and the Young inequalities, as well as (77), (78), (84), (97), and (104). We have
In the same way we estimate integrals \(I_{36}\) and \(I_{37}\).
We base the estimate of integral \(I_{6}\) on (77), (78), (84), and (85) as well as on the Young inequality. We obtain
We perform the estimates of integrals \(I_{39}\), \(I_{40}\), \(I_{42}\), \(I_{43}\), \(I_{45}\), \(I_{46}\), \(I_{48}\), \(I_{49}\), \(I_{50}\), \(I_{51}\), and \(I_{56}\)–\(I_{58}\) analogously.
We still have to estimate integrals \(I_{11}\), \(I_{20}\), \(I_{25}\), \(I_{38}\), \(I_{41}\), \(I_{44}\), \(I_{47}\), \(I_{52}\)–\(I_{55}\), and \(I_{59}\)–\(I_{61}\). To do this, we use (77), (85), and the Young and the Hölder inequalities. For instance, we have
Using the obtained estimates with a sufficiently small ε together with (77)–(79), from (106) we obtain (105). □
Lemma 8
There exists such \(T_{0}\) (\(0< T_{0}\leq T)\) that, for each \(n\in \mathbf{N}\), the Cauchy problem (73)–(76) has a unique solution defined on \([0,T_{0}]\). Moreover, for the functions \(\mathbf{V}^{n}\), \(\mathbf{W}^{n}\), \(\theta ^{n}\), \(\rho ^{n}\), and \(r^{n}\), we have
where a, \(a_{1}\), m, and M are defined by (36) and (40)–(41).
Proof
To obtain estimate (113), we use a similar approach as in [11], Lemma 4.7, [10], Lemma 5.6, and [15] pp. 64–67. First, we introduce the function
where \(C_{1}\) is the constant introduced in (105). Using Lemma 7, we find that function \(y_{n}\) satisfies the differential inequality
Let C̅ be a constant defined by
where \(\mathbf{V}_{0}=(v_{0}^{r},v_{0}^{\varphi },v_{0}^{z})\) and \(\mathbf{W}_{0}=(\omega _{0}^{r},\omega _{0}^{\varphi },\omega _{0}^{z})\). It is easy to see that we have
Now, we compare the solution of problem (121)–(123) with the solution of the Cauchy problem
Let \([0,T'[\), \(0< T'\leq T\) be an existence interval of the solution to problem (124)–(125). Because of the property of the maximal solution for problem (121)–(123), we conclude that
Let \(T_{0}\) be such that \(0< T_{0}\leq T'\). From (126) we have
which together with (120) and (105) implies
Integrating (128) over \([0,t]\), \(0< t\leq T_{0}\) and using (123), we immediately obtain (113).
Now, using inequalities (84)–(85) for the function \(v^{r n}\) and (113), we derive the following estimate:
In the same way we get
where \(C_{1}\) and \(C_{2}\) are taken from (113). Estimates (129) and (130) are also valid for the functions \(v^{\varphi n}\), \(v^{z n}\), \(\omega ^{r n}\), \(\omega ^{\varphi n}\), and \(\omega ^{z n}\).
With the help of (41), (129), and (130), we can easily conclude that for
from (47) and (49) we obtain (114) and (115).
Because of (96) and the same inequality for the function W, from (113) for \(t\in [0,T_{0} ]\), we obtain
Also, taking into account (113) from (93), for \(t\in [0,T_{0}]\), we have
Therefore, we obtain
Estimates (132) and (135) give (119). Finally, from (104), (97), and (86), we immediately get (116), (117), and (118), respectively. □
Lemma 9
Let \(T_{0}\) be defined by Lemma 8. Then, for each \(n\in \mathbf{N}\), we have
Proof
First, we multiply (62)–(68) respectively by \(\frac{dv_{i}^{r n}}{\,dt}\), \(\frac{dv_{i}^{\varphi n}}{\,dt}\), \(\frac{dv_{i}^{z n}}{\,dt}\), \(\frac{\omega _{j}^{r n}}{\,dt}\), \(\frac{\omega _{j}^{\varphi n}}{\,dt}\), \(\frac{\omega _{j}^{z n}}{\,dt}\), \(\frac{\theta _{k}^{n}}{\,dt}\), summarize over \(i_{1},i_{2},i_{3},j_{1},j_{2},j_{3}=1,2,\ldots,n\), \(k=0,1,2,\ldots,n\). After we integrate the obtained equality over \([0,L ]\), we get the equality which is very similar to (106). We estimate it in the same way as in the proof of Lemma 7 and obtain (136).
From (48), after we use the Hölder inequality as well as (77) and (78), we get
Using (113) and (119) from (139), by integrating over \([0,T_{0} ]\), we easily obtain (137). Estimates (138) follow directly from (47) and (49). □
Using the results from Lemmas 8 and 9, we easily derive the following statements.
Proposition 1
Let \(T_{0}\) be defined by Lemma 8. Then, for the sequence
we have:
-
(i)
\(\{r^{n}\}\) is bounded in \(\mathrm{\mathrm{L}}^{\infty }(Q _{0})\), \(\mathrm{L}^{\infty }(0, T_{0}; \mathrm{H}^{2}(]0,L[))\) and \(\mathrm{H}^{2}(Q_{0})\);
-
(ii)
\(\{ \frac{\partial r^{n}}{\partial x} \}\) is bounded in \(\mathrm{L} ^{\infty }(Q_{0})\);
-
(iii)
\(\{\rho ^{n}\}\) is bounded in \(\mathrm{L}^{\infty }(Q_{0})\), \(\mathrm{L}^{\infty }(0, T_{0}; \mathrm{H}^{1}(]0,L[))\), and \(\mathrm{H}^{1}(Q_{0})\);
-
(iv)
\(\{\mathbf{V}^{n}\}\) and \(\{\mathbf{W}^{n}\}\) are bounded in \((\mathrm{L}^{\infty }(0, T_{0}; \mathrm{H}^{1}(]0,L[)) ) ^{3}\), \((\mathrm{H}^{1}(Q_{0}) )^{3}\), and \((\mathrm{L} ^{2}(0, T_{0}; \mathrm{\mathrm{H}}^{2}(]0,L[)) )^{3}\);
-
(v)
\(\{\theta ^{n}\}\) is bounded in \(\mathrm{L}^{\infty }(0, T _{0}; \mathrm{H}^{1}(]0,L[))\), \(\mathrm{H}^{1}(Q_{0})\), and \(\mathrm{L}^{2}(0, T_{0}; \mathrm{\mathrm{H}}^{2}(]0,L[))\).
5 The proof of Theorem 1
To prove Theorem 1, we extract the convergent subsequence of sequence (140) and show that the limit of this subsequence is a solution to our problem. The proof is very similar to the proof of Theorem 2.1 in [11]; therefore, we omit the details of some proofs hereafter and refer to the corresponding results from [11].
Let \(T_{0}\in \mathbf{R}^{+}\) be defined by Lemma 8. Theorem 1 follows directly from the following lemmas.
Lemma 10
(in [11], Lemma 5.1)
There exist a function
and a subsequence (for simplicity reasons denoted again as \(\{r^{n}\}\)) of \(\{r^{n}\}\) with the properties
The function r satisfies the conditions
where \(r_{0}\) is defined by (24).
Lemma 11
(in [11], Lemma 5.2)
There exists a function
and a subsequence (for simplicity reasons denoted again as \(\{\rho ^{n}\}\)) of \(\{\rho ^{n}\}\) with the properties
The function ρ satisfies the conditions
Lemma 12
There exist functions \(\mathbf{V}=(v^{r},v^{\varphi },v^{z},\omega ^{r})\), \(\mathbf{W}=(\omega ^{r},\omega ^{\varphi },\omega ^{z})\), and θ such that
and a subsequence of \(\{(\mathbf{V}^{n}, \mathbf{W}^{n}, \theta ^{n}) \}\) (for simplicity reasons denoted again as \(\{(\mathbf{V}^{n}, \mathbf{W}^{n}, \theta ^{n})\}\)) of \(\{(\mathbf{V}^{n}, \mathbf{W}^{n}, \theta ^{n})\}\) with the properties:
The functions V, W, and θ satisfy the conditions
where \(\mathbf{V}_{0}= (v^{r}_{0},v^{\varphi }_{0},v^{z}_{0} )\), \(\mathbf{W}_{0}= (\omega ^{r}_{0},\omega ^{\varphi }_{0},\omega ^{z}_{0} )\), and \(\theta _{0}\) are defined by (35).
Proof
As in [11], Lemma 5.3, conclusions (156)–(159) follow from Proposition 1.
To verify the boundary and initial conditions (160)–(162), we use the Green formula in the same way as in [11]. Here, we demonstrate the proof for the boundary condition (161) at \(x=0\).
Let φ be a function from \(\mathrm{C}^{\infty }([0,L])\), which is equal to zero in a neighborhood of L, with \(\varphi (0)\neq 0\) and \(u\in \mathrm{L}^{\infty }(]0,T_{0}[)\). Using the integration by parts for \(\frac{\partial \theta }{\partial x}\) and \(\frac{\partial \theta ^{n}}{\partial x}\), we obtain
Passing to the limit when \(n\to \infty \) in (164), comparing (164) and (163), and by using the convergence (158), we obtain
In a similar way, we obtain all the remaining equalities in (160)–(162). □
Lemma 13
The functions r, ρ, v, ω, θ, defined by Lemmas 10, 11, and 12 satisfy equations (1)–(8) a.e. in \(Q_{0}\).
Proof
The proof of this lemma is based on strong and weak convergences from Proposition 1. As the procedure is the same as in [11], Lemma 5.4, we will demonstrate here the idea of the proof just for equation (8), which is the most complex.
Let \(\{(r^{n},\rho ^{n},\mathbf{V}^{n},\mathbf{W}^{n},\theta ^{n}):n \in \mathbf{N}\}\) be the subsequence defined by Lemmas 10, 11, and 12, and let \(\varphi \in \mathcal{D} (]0,T_{0}[ )\), where \(\mathcal{D}\) denotes the space of test functions. We first rewrite equation (68) in the following form:
Now, we should show the convergence for each integrand on the left-hand side of (166). Here we will demonstrate the following convergence:
when \(n\to \infty \). Using integration by parts, as well as the Hölder inequality, we have
Taking into account the strong convergences \(\rho ^{n}\to \rho \) and \(r^{n}\omega ^{r n}\to r\omega ^{r}\), from (168) we easily obtain (167).
In the same way, we can derive the convergences of other integrals in (166). □
Let us note that from (47) we have
for all \(\varphi \in \mathrm{L}^{2} (Q_{0} )\), which together with (144) and (159) implies
For the function θ, we have the following property.
Lemma 14
(in [11], Lemma 5.5)
There exists \(T_{0}\), \(0< T_{0}\leq T\), such that the function θ defined by Lemma 12 satisfies the condition
The conclusions of Theorem 1 are an immediate consequence of the above lemmas.
6 Conclusion
The initial boundary problem for the 3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry and homogeneous boundary conditions for velocity, microrotation, and heat flux has a generalized solution locally in time.
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ID is currently an assistant professor at the University of Rijeka, Faculty of Engineering, Croatia. NM is a retired associate professor from the University of Rijeka, Department of Mathematics.
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This work is financed through the support of scientific research of the University of Rijeka (project No. 13.14.1.3.03 and 17.10.2.2.01).
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Dražić, I., Mujaković, N. Local existence of the generalized solution for three-dimensional compressible viscous flow of micropolar fluid with cylindrical symmetry. Bound Value Probl 2019, 16 (2019). https://doi.org/10.1186/s13661-019-1131-2
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DOI: https://doi.org/10.1186/s13661-019-1131-2