A Three-Dimensional Model of a Spherically Symmetric, Compressible Micropolar Fluid Flow with a Real Gas Equation of State

Abstract

In this work, we analyze a spherically symmetric 3D flow of a micropolar, viscous, polytropic, and heat-conducting real gas. In particular, we take as a domain the subset of $R^3$ bounded by two concentric spheres that present solid thermoinsulated walls. Also, here, we consider the generalized equation of state for the pressure in the sense that the pressure depends, as a power function, on the mass density. The model is based on the conservation laws for mass, momentum, momentum moment, and energy, as well as the equation of state for a real gas, and it is derived first in the Eulerian and then in the Lagrangian description. Through the application of the Faedo–Galerkin method, a numerical solution to a corresponding problem is obtained, and numerical simulations are performed to demonstrate the behavior of the solutions under various parameters and initial conditions in order to validate the method. The results of the simulations are discussed in detail.

1. Introduction

In this work, we derive a three-dimensional flow model for a micropolar real gas that is spherically symmetric. The model builds on the basic compressible micropolar fluid model, incorporating temperature [1] and assuming a spherically symmetric solution as in [2] by imposing a generalized equation of state for the pressure. The resulting system of four partial differential equations is first obtained in an Eulerian description and then translated into a Lagrangian mass description, which is more convenient for further analysis. The problem is then solved numerically using the Faedo–Galerkin approach. Numerical tests were performed to determine the contribution of the pressure parameters to the solution.

A generalized gas model in which temperature is considered and pressure obeys the power law with respect to mass density, i.e., $P=R\theta {\rho }^p$, where P is pressure, R is gas constant, $\theta$ is absolute temperature, and $\rho$ mass density, was introduced in [3] and considered in the contexts of micropolar fluid in [4] and a micropolar reactive fluid in [5]. In classical, non-micropolar models, this generalized gas model was considered, for example, in [6,7].

The micropolar model accounts for the micro-effects in fluids caused by the rotation of the particles that make up the fluid [8]. The interaction of particles is considered through the incorporation of microrotation velocity in the governing equations. These microrotational contributions generate internal moments, which allow for the description of particle interactions on a microscopic level. In this way, the model accounts not only for the transfer of linear momentum but also for angular momentum, providing a more accurate analysis of fluids with pronounced microstructural features. In the classical fluid model, these effects are neglected for simplicity, but sometimes, this simplification leads to suboptimal results when modeling real-world problems. The micropolar model proved to be a good upgrade to the classical model [9], but it was not too complex to make an analysis and a simulation unattainable [1].

In certain situations where the size of the domain is comparable to the size of the particles, almost any gas can exhibit micropolar characteristics. On the microscale and the nanoscale, microrotations and internal moments of particles become dominant factors in fluid dynamics, making traditional models less accurate. This approach is particularly relevant in fields such as microfluidics and nanotechnology, where particle interactions and system geometry play a critical role in fluid behavior [10]. Under such conditions, even conventional gases can demonstrate micropolar properties, as microscopic rotations influence the transfer of energy and momentum.

Micropolar behavior can also be observed in reactive mixtures, such as fuel gases, where the interaction of microscopic particles or non-homogeneous components during combustion can generate microrotations that affect heat-transfer and combustion dynamics. This is especially significant in high-temperature environments or systems involving turbulent flows, where particle interactions contribute to the overall energy exchange [11].

Additionally, micropolar gases have a wide range of applications in fields such as meteorology, industrial filtration, and space technology. Examples include atmospheric gases with suspended particles of dust or aerosols, polluted gases in industrial plants containing ash or soot particles, and rarefied gases in space applications where microparticles exhibit rotational effects [1]. These examples highlight the importance of the micropolar theory for describing systems where particle interactions within the gas play a key role in the transfer of momentum and energy.

Micropolar models are also useful for understanding the behavior of aerosols in medical inhalers, where tiny particles of liquid or medication exhibit microrotational effects, as well as in colloidal suspensions in gases, which are relevant for simulations and analyses in nanotechnology or pharmaceutical industries [12].

Although many of the applications mentioned are still at a theoretical level, the growing interest in micropolarity—particularly with the rapid advancement in nanotechnology—suggests that a transition to real-world engineering applications is imminent. As these technologies develop, the relevance and importance of micropolar models are expected to increase, bringing them closer to practical implementation [13].

Full 3D flows are, in most cases, still extremely hard to analyze. For the latest results on three-dimensional incompressible fluid flow, see [14]. In order to make models suitable for mathematical and numerical analysis, certain assumptions, such as different symmetries of a domain and solutions, are considered. Some of the classical symmetries are spherical, cylindrical, and helical. In this work, we studied a domain bounded by two concentric spheres and assumed a spherical symmetry of the solution. Spherically symmetric gas flows are used, for example, to model gaseous stars [15,16,17]. A micropolar, compressible flow model with spherical symmetry was derived in [2], a numerical solution using a finite difference scheme was developed in [18], and a free boundary problem was considered in [19]. More generally, spherically symmetric compressible flow was first studied in [20], and subsequently, there were a variety of models that considered spherically symmetric flows; e.g., a viscous radiative and reactive gas was studied in [21], a free boundary problem for the equations of motion of a compressible gas with density-dependent viscosity was studied in [22], and a viscous polytropic gas was studied in [23].

To delineate the novelty of this work in relation to prior studies, we emphasize the following key contributions. This work significantly extends our previous research (see, for example, [4]) by advancing from a one-dimensional to a fully three-dimensional, spherically symmetric model of a viscous, micropolar, heat-conducting real gas. While earlier studies focused on the ideal gas equation of state or were limited to simpler models, we now generalize fluid dynamics by incorporating a more realistic equation of state, where the pressure is modeled as a power-law function of mass density. This synthesis provides a more accurate and comprehensive representation of fluid behavior under spherical symmetry, making it the first such detailed exploration of this type of fluid model.

Additionally, we establish the applicability of the Faedo–Galerkin method for solving this complex system, demonstrating its robustness for both simple and complex initial conditions. Previous studies have not applied this numerical method to such a sophisticated configuration of micropolar fluid dynamics, particularly for real gases, making this a notable contribution in terms of numerical approaches.

Moreover, unlike earlier work that primarily relied on Lagrangian descriptions, we perform simulations in Eulerian coordinates for the spherically symmetric problem. To the best of our knowledge, this approach has not been previously employed for micropolar fluids, offering novel insights into fluid behavior and enabling a new form of solution visualization.

Finally, by comparing simulations using the generalized equation of state with those based on the ideal gas law, we investigate the influence of the pressure exponent on the solutions. This comparison underscores the importance of the generalized equation of state in providing a more accurate depiction of physical behaviors across different flow conditions.

The paper is organized as follows. The general mathematical model for the micropolar real gas with spherical symmetry is described in Section 2.In Section 3 and Section 4, the spherically symmetric model is derived in the Eulerian description and in the Lagrangian description. In Section 5, we construct a series of approximate semi-discretized problems using the Faedo–Galerkin technique, and in Section 6, we develop a numerical method for solving the system and present the results of numerical experiments, with a particular emphasis on studying the contribution of a generalized equation of state.

2. The General Mathematical Model for a Micropolar Real Gas with Spherical Symmetry

To derive our model, we started with the balance laws for mass, momentum, momentum moment, and energy, which can be found, for example, in [1], and through which the behavior of the gas is described:

$$\begin{array}{cc}\hfill & \dot{\rho }=-\rho \nabla ·\mathbf{v},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & \rho \dot{\mathbf{v}}=\nabla ·\mathbf{T}+\rho \mathbf{f},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & \rho j_I\dot{\omega }=\nabla ·\mathbf{C}+{\mathbf{T}}_x+\rho \mathbf{g},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \rho \dot{E}=-\nabla ·\mathbf{q}+\mathbf{T}:\nabla \mathbf{v}+\mathbf{C}:\nabla \omega -{\mathbf{T}}_x·\omega .\hfill \end{array}$$

(4)

Here, $\rho$, $\mathbf{v}$, $\omega$, and E represent the mass density, velocity, microrotation velocity, and internal energy density, respectively, and they are considered in the domain $Q_T=\mathsf{\Omega }\times$]$0,T[,$, where $T>0$ is arbitrary, and $\mathsf{\Omega }\subset {\mathbf{R}}^3$.

In micropolar fluids, the inclusion of the microrotation vector or particle spin vector is essential for capturing the rotational behavior of fluid particles, which is not accounted for in classical fluid models. This vector represents the angular velocity of fluid particles about their own axes, allowing the model to describe both translational and rotational degrees of freedom. By incorporating this vector, the model can account for the internal angular momentum and the associated effects on the overall dynamics of the fluid.

In the system of (1)–(4), $\mathbf{f}$ and $\mathbf{g}$ are the body force and body couple density, respectively, and the positive constant $j_I$ is the microinertia density. For simplicity reasons, we assumed that

$$\mathbf{f}=\mathbf{g}=0.$$

(5)

This simplification allowed us to focus on the intrinsic dynamics of the micropolar fluid due to its microstructure and microrotation effects without the added complexity of external forces and moments. This approach is consistent with previous analyses that focus on the existence and uniqueness of solutions under simplified conditions (see [2,23]). While this assumption is common in foundational studies of micropolar fluids, and it facilitates mathematical tractability, we acknowledge that it limits the direct applicability of the model to real-world situations where external forces and moments cannot be neglected. In practical applications, factors such as gravity, electromagnetic fields, and pressure gradients can significantly influence fluid behavior. Future work could extend the current model by incorporating non-zero body force and couple-density functions to better represent specific real-world scenarios.

$\mathbf{T}=\left({\mathbf{T}}_{ij}\right)$ and $\mathbf{C}=\left({\mathbf{C}}_{ij}\right)$ are the stress tensor and couple stress tensor, respectively, which are given through the constitutive equations for the micropolar fluid by

$${\mathbf{T}}_{ij}=(-P+\lambda {\mathbf{v}}_{k,k}){\delta }_{ij}+\mu \left({\mathbf{v}}_{i,j}+{\mathbf{v}}_{j,i}\right)+{\mu }_r\left({\mathbf{v}}_{j,i}-{\mathbf{v}}_{i,j}\right)-2{\mu }_r{\epsilon }_{mij}{\omega }_m,$$

(6)

$${\mathbf{C}}_{ij}=c_0{\omega }_{k,k}{\delta }_{ij}+c_d\left({\omega }_{i,j}+{\omega }_{j,i}\right)+c_a\left({\omega }_{j,i}-{\omega }_{i,j}\right).$$

(7)

Here, P denotes pressure, the symbol ${\delta }_{ij}$ is the Kronecker delta, $\lambda$ and $\mu$ are coefficients of viscosity, and ${\mu }_r$, $c_0$, $c_d$, and $c_a$ are coefficients of microviscosity. The viscosity and microviscosity coefficients are connected via the Clausius–Duhamel inequalities, as follows:

$$\mu \ge 0,3\lambda +2\mu \ge 0,{\mu }_r\ge 0,$$

(8)

$$c_d\ge 0,3c_0+2c_d\ge 0,|c_d-c_a|\le c_d+c_a.$$

(9)

In Equations (3) and (4), the vector ${\mathbf{T}}_x$ is an axial vector with Cartesian components given by ${\left({\mathbf{T}}_x\right)}_i={\epsilon }_{ijk}{\mathbf{T}}_{jk}$, where ${\epsilon }_{ijk}$ represents the Levi–Civita symbol. We adopt the Einstein notation for summation. The colon operator in Equation (4) denotes the scalar product of tensors, such as $\mathbf{T}:\nabla \mathbf{v}={\mathbf{T}}_{ij}{\mathbf{v}}_{i,j}$.

The differential (dot) operator in Equations (1)–(4) denotes material derivative defined by

$$\dot{\mathbf{b}}={\partial }_t\mathbf{b}+\left({\nabla }_{\mathbf{x}}\mathbf{b}\right)·\mathbf{v},$$

(10)

where $\mathbf{b}=\mathbf{b}(\mathbf{x},t)$ is any vector field, $\mathbf{x}=\mathbf{x}\left(t\right)$ is the position at time t, and $\mathbf{v}={\displaystyle \frac{d\mathbf{x}}{dt}}$ is the flow velocity. For a scalar field, b, the material derivative is defined by

$$\dot{b}={\partial }_{t}b+\left({\nabla }_{\mathbf{x}}b\right)·\mathbf{v}.$$

(11)

In order to thoroughly examine all the effects of micropolarity, in this paper, we will additionally require that ${\mu }_r\ne 0$; i.e., we will assume that

$${\mu }_r>0.$$

(12)

In addition to conservation laws (1)–(4) and constitutive Equations (6) and (7), the model for micropolar real gas also includes the following equation:

$$\mathbf{q}=-k_{\theta }\nabla \theta ,$$

(13)

which presents the Fourier law, in which $\mathbf{q}$ is the heat flux and $\theta$ is the absolute temperature, as well as the state equation for the real gas

$$P=R{\rho }^p\theta ,$$

(14)

which implies

$$E=c_v\theta .$$

(15)

Equation (15) presents the assumption that the fluid is heat-conducting.

As mentioned, we observe the model (1)–(15) in a three-dimensional case but with the assumption that the model is spherically symmetric. Therefore, for the spatial domain of our system, we take

$$\mathsf{\Omega }=\{\mathbf{x}\in {\mathbf{R}}^3,a<|\mathbf{x}|<b\},a>0,$$

(16)

where a and b are the radii of two concentric spheres, and we assume that the solution ($\rho$,$\mathbf{v}$,$\omega$,$\theta$) depends only on the time variable t and the space variable $r=\left|\mathbf{x}\right|$, $\mathbf{x}=(x_1,x_2,x_3)\in {\mathbf{R}}^3$. Therefore, as obtained in [2], we are looking at the solution ($\rho$,$\mathbf{v}$,$\omega$,$\theta$) to the problem (1)–(15) in the following form:

$$\rho (\mathbf{x},t)=\rho (r,t),$$

(17)

$$\mathbf{v}(\mathbf{x},t)=v(r,t){\displaystyle \frac{\mathbf{x}}{r}},$$

(18)

$$\omega (\mathbf{x},t)=\omega (r,t){\displaystyle \frac{\mathbf{x}}{r}},$$

(19)

$$\theta (\mathbf{x},t)=\theta (r,t).$$

(20)

3. Derivation of the Model in the Eulerian Description

Taking into account (17)–(20), the system (1)–(15) will take a simpler form. To begin with, for the divergence and gradient of the vector field $\mathbf{v}$, the following equations hold:

$$\nabla ·\mathbf{v}=v_r+{\displaystyle \frac{2}{r}}v,$$

(21)

$$\nabla \mathbf{v}={\displaystyle \frac{v}{r}}\mathbf{I}+\left({\displaystyle \frac{v_r}{r^2}}-{\displaystyle \frac{v}{r^3}}\right)\mathbf{x}\otimes \mathbf{x},$$

(22)

where $\mathbf{I}$ is the identity matrix, and ⊗ is a tensor product of vectors.

Using (21) and (22), for the stress tensor $\mathbf{T}$, defined by (6), we obtain

$$\mathbf{T}=\left(-R{\rho }^p\theta +\lambda \left(v_r+2{\displaystyle \frac{v}{r}}\right)+2\mu {\displaystyle \frac{v}{r}}\right)\mathbf{I}+2\mu \left({\displaystyle \frac{v_r}{r^2}}-{\displaystyle \frac{v}{r^3}}\right)\mathbf{x}\otimes \mathbf{x}-2{\mu }_r{\displaystyle \frac{\omega }{r}}{\mathbf{x}}_{skw},$$

(23)

where ${\left({\mathbf{x}}_{skw}\right)}_{ij}={\epsilon }_{mij}{x}_m$, and for its divergence

$$\nabla ·\mathbf{T}=\left(-R{\left({\rho }^p\theta \right)}_r+\lambda \left(v_{rr}+2{\displaystyle \frac{v_r}{r}}-2{\displaystyle \frac{v}{r^2}}\right)+2\mu \left(v_{rr}+2{\displaystyle \frac{v_r}{r}}-2{\displaystyle \frac{v}{r^2}}\right)\right){\displaystyle \frac{\mathbf{x}}{r}}.$$

(24)

Similarly, for the couple stress tensor $\mathbf{C}$, defined by (7), the following equalities hold:

$$\mathbf{C}=\left(c_0{\omega }_r+2c_0{\displaystyle \frac{\omega }{r}}+2c_d{\displaystyle \frac{\omega }{r}}\right)\mathbf{I}+2c_d\left({\displaystyle \frac{{\omega }_r}{r^2}}-{\displaystyle \frac{\omega }{r^3}}\right)\mathbf{x}\otimes \mathbf{x},$$

(25)

$$\nabla ·\mathbf{C}=\left(c_0{\omega }_{rr}+2c_0\left({\displaystyle \frac{{\omega }_r}{r}}-{\displaystyle \frac{\omega }{r^2}}\right)+2c_d\left({\omega }_{rr}+2{\displaystyle \frac{{\omega }_r}{r}}-2{\displaystyle \frac{\omega }{r^2}}\right)\right){\displaystyle \frac{\mathbf{x}}{r}}.$$

(26)

Also, since ${\mathbf{T}}_x=(T_{23}-T_{32},T_{31}-T_{13},T_{12}-T_{21})$, from (23) follows

$${\mathbf{T}}_x=-4{\mu }_r{\displaystyle \frac{\omega }{r}}\mathbf{x}.$$

(27)

Next, from (22) and (23), for the scalar product of tensors $\mathbf{T}$ and $\nabla \mathbf{v}$, we obtain

$$\mathbf{T}:\nabla \mathbf{v}=\left(-R{\rho }^p\theta +\lambda \left(v_r+2{\displaystyle \frac{v}{r}}\right)+2\mu {\displaystyle \frac{v}{r}}\right)\left(v_r+2{\displaystyle \frac{v}{r}}\right)+2\mu v_r\left(v_r-{\displaystyle \frac{v}{r}}\right),$$

(28)

and, similarly,

$$\mathbf{C}:\nabla \omega =\left(c_0{\omega }_r+2c_0{\displaystyle \frac{\omega }{r}}+2c_d{\displaystyle \frac{\omega }{r}}\right)\left({\omega }_r+2{\displaystyle \frac{\omega }{r}}\right)+2c_d{\omega }_r\left({\omega }_r-{\displaystyle \frac{\omega }{r}}\right).$$

(29)

The material derivatives of the functions $\rho$, $\mathbf{v}$, $\omega$, and $\theta$ have the following spherically symmetric forms (obtained in [2]):

$$\dot{\rho }={\rho }_t+v{\rho }_r,$$

(30)

$$\dot{\mathbf{v}}=(v_t+v{v}_r){\displaystyle \frac{\mathbf{x}}{r}},$$

(31)

$$\dot{\omega }=({\omega }_t+v{\omega }_r){\displaystyle \frac{\mathbf{x}}{r}},$$

(32)

$$\dot{\theta }={\theta }_t+v{\theta }_r.$$

(33)

Now, by substituting (21), (24), (26), and (27)–(33) into the system (1)–(4) and using (5), we obtain the model in the Eulerian description:

$$\begin{array}{cc}\hfill & {\rho }_t+{\left(v\rho \right)}_r+{\displaystyle \frac{2}{r}}\rho v=0,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & \rho \left(v_t+v{v}_r\right)=-R{\left({\rho }^p\theta \right)}_r+(\lambda +2\mu ){\left(v_r+2{\displaystyle \frac{v}{r}}\right)}_r,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & \rho j_I\left({\omega }_t+v{\omega }_r\right)=-4{\mu }_r\omega +(c_0+2c_d){\left({\omega }_r+2{\displaystyle \frac{\omega }{r}}\right)}_r,\hfill \end{array}$$

(36)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \rho c_v\left({\theta }_t+v{\theta }_r\right)=k\left({\theta }_{rr}+{\displaystyle \frac{2}{r}}{\theta }_r\right)-R{\rho }^p\theta \left(v_r+2{\displaystyle \frac{v}{r}}\right)+(\lambda +2\mu ){\left(v_r+2{\displaystyle \frac{v}{r}}\right)}^2\hfill \\ & -4\mu {\displaystyle \frac{v}{r}}\left(2v_r+{\displaystyle \frac{v}{r}}\right)+(c_0+2c_d){\left({\omega }_r+2{\displaystyle \frac{\omega }{r}}\right)}^2-4c_d{\displaystyle \frac{\omega }{r}}\left(2{\omega }_r+{\displaystyle \frac{\omega }{r}}\right)+4{\mu }_r{\omega }^2.\hfill \end{array}\end{array}$$

(37)

We propose the following initial conditions:

$$\rho (r,0)={\rho }_0\left(r\right),v(r,0)=v_0\left(r\right),\omega (r,0)={\omega }_0\left(r\right),\theta (r,0)={\theta }_0\left(r\right),$$

(38)

for $r\in ]a,b[$, and the homogeneous boundary conditions

$$v(a,t)=v(b,t)=0,\omega (a,t)=\omega (b,t)=0,{\theta }_r(a,t)={\theta }_r(b,t)=0,$$

(39)

for $t\in ]0,T[$. It is important to note that the boundary conditions for v and ${\theta }_r$ are physically justified, as they reflect the assumption that the flow occurs between two concentric spheres with solid, thermally insulated walls. Additionally, $\omega (a,t)=\omega (b,t)=0$ represents a standard boundary condition for microrotation. These homogeneous boundary conditions for microrotation are widely used in the literature, as demonstrated in works such as [1,2], and others.

4. The Spherically Symmetric Model in the Lagrangian Description

As mentioned, the system (34)–(39) presents the model in the Eulerian description; i.e., the unknown functions are described as functions of the time coordinate t and the spatial coordinate $r\in [a,b]$, where r represents the position of the material point at time t. For the mathematical analysis of the problem (34)–(39), it is more convenient to transform it into a Lagrangian description, in which the unknowns are functions of the time coordinate t and the spatial coordinate $\xi \in [a,b]$, where $\xi$ represents the starting position of the material point considered. To do this, we followed the procedure described in [2].

Let us emphasize that the Lagrangian description is also significant from a numerical standpoint. This approach offers substantial advantages for solution stability. By eliminating convective terms from the equations of motion, we reduce numerical errors associated with convection. Moreover, in conjunction with the Faedo–Galerkin method, this approach allows for the natural tracking of individual fluid particles and their microrotations without the need for computational meshes, thereby minimizing potential sources of instability related to mesh generation and handling. Furthermore, the selection of appropriate basis functions addresses the complexity associated with implementing boundary conditions in the Lagrangian framework.

Conversely, the Eulerian description includes convective terms in the governing equations, which can potentially affect numerical stability—particularly in micropolar fluids with strong coupling between translational and rotational motions. Nevertheless, to provide a more comprehensive analysis, we will present the final solution in both descriptions.

The Eulerian coordinates $(r,t)$ are connected to the Lagrangian coordinates $(\xi ,t)$ via the relation

$$r(\xi ,t)=r_0\left(\xi \right)+{\int }_0^t\tilde{v}(\xi ,t)d\tau ,r_0\left(\xi \right)=r(\xi ,0),$$

(40)

where $\tilde{v}$ is defined by

$$\tilde{v}(\xi ,t)=v(r(\xi ,t),t).$$

(41)

Let the function $\eta$ be defined by

$$\eta \left(\xi \right)={\int }_a^{\xi }{\rho }_0\left(s\right)s^{2}ds.$$

(42)

From Equation (34) and relation (40) follows

$${\displaystyle \frac{\partial }{\partial t}}{\int }_a^{r(\xi ,t)}\rho \left(s\right)s^{2}ds=0$$

(43)

and, through the integration of (43) over $[0,t]$,

$${\int }_a^{r(\xi ,t)}\rho \left(s\right)s^{2}ds={\int }_a^{\xi }{\rho }_0\left(s\right)s^{2}ds=\eta \left(\xi \right).$$

(44)

It is important to note that, since the domain is entirely filled with fluid (with no vacuum states present), it is physically reasonable to assume that the initial mass density has a lower bound:

$${\rho }_0\left(r\right)\ge m$$

(45)

on $]a,b[$, for some $m\in {\mathbf{R}}^{+}$. Because of (45), there exists an inverse function, ${\eta }^{-1}$, so we can introduce the new coordinate, x, via the relation

$$x=L^{-1}\eta \left(\xi \right)$$

(46)

where

$$L=\eta \left(b\right)={\int }_a^b{\rho }_0\left(s\right)s^{2}ds.$$

(47)

Note that, for the coordinate x, holds $0=L^{-1}\eta \left(a\right)\le x\le L^{-1}\eta \left(b\right)=1$.

As obtained in [2], the unknown functions expressed in the new coordinate, x, are connected to the ones in the Lagrangian coordinates via the following relations:

$$\rho (x,t)=\tilde{\rho }\left({\eta }^{-1}\left(xL\right),t\right),$$

(48)

$$v(x,t)=\tilde{v}\left({\eta }^{-1}\left(xL\right),t\right),$$

(49)

$$\omega (x,t)=\tilde{\omega }\left({\eta }^{-1}\left(xL\right),t\right),$$

(50)

$$\theta (x,t)=\tilde{\theta }\left({\eta }^{-1}\left(xL\right),t\right),$$

(51)

$$r(x,t)=r\left({\eta }^{-1}\left(xL\right),t\right),$$

(52)

together with the initial conditions

$${\rho }_0\left(x\right)={\rho }_0\left({\eta }^{-1}\left(xL\right)\right),$$

(53)

$$v_0\left(x\right)=v_0\left({\eta }^{-1}\left(xL\right)\right),$$

(54)

$${\omega }_0\left(x\right)={\omega }_0\left({\eta }^{-1}\left(xL\right)\right),$$

(55)

$${\theta }_0\left(x\right)={\theta }_0\left({\eta }^{-1}\left(xL\right)\right),$$

(56)

$$r_0\left(x\right)=r_0\left({\eta }^{-1}\left(xL\right)\right)={\eta }^{-1}\left(xL\right).$$

(57)

To be able to express the system (34)–(39) in the new coordinates, we use the following equality:

$$r_x(x,t)={\displaystyle \frac{L}{\rho (x,t)r^2(x,t)}},$$

(58)

and the following equalities, valid for the function $f\in \{\rho ,\mathbf{v},\omega ,\theta \}$:

$$f_t(x,t)=f_t(r,t)+v(r,t)f_r(r,t),$$

(59)

$$f_x(x,t)=f_r(r,t)r_x(x,t)={\displaystyle \frac{L}{\rho (x,t)r^2(x,t)}}{f}_r(r,t).$$

(60)

Finally, we obtain our model in the Lagrangian description:

$$\begin{array}{cc}\hfill & {\rho }_t=-{\displaystyle \frac{1}{L}}{\rho }^2{\left(r^{2}v\right)}_x,\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill & v_t=-{\displaystyle \frac{R}{L}}{r}^2{\left({\rho }^p\theta \right)}_x+{\displaystyle \frac{\lambda +2\mu }{L^2}}{r}^2{\left(\rho {\left(r^{2}v\right)}_x\right)}_x,\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill & \rho {\omega }_t=-{\displaystyle \frac{4{\mu }_r}{j_I}}\omega +{\displaystyle \frac{c_0+2c_d}{j_I{L}^2}}{r}^2\rho {\left(\rho {\left(r^2\omega \right)}_x\right)}_x,\hfill \end{array}$$

(63)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \rho {\theta }_t={\displaystyle \frac{k}{c_v{L}^2}}\rho {\left(r^4\rho {\theta }_x\right)}_x-{\displaystyle \frac{R}{c_{v}L}}{\rho }^{p+1}\theta {\left(r^{2}v\right)}_x+{\displaystyle \frac{\lambda +2\mu }{c_v{L}^2}}{\left(\rho {\left(r^{2}v\right)}_x\right)}^2\hfill \\ \hfill & -{\displaystyle \frac{4\mu }{c_{v}L}}\rho {\left(r{v}^2\right)}_x+{\displaystyle \frac{c_0+2c_d}{c_v{L}^2}}{\left(\rho {\left(r^2\omega \right)}_x\right)}^2-{\displaystyle \frac{4c_d}{c_{v}L}}\rho {\left(r{\omega }^2\right)}_x+{\displaystyle \frac{4{\mu }_r}{c_v}}{\omega }^2,\hfill \end{array}\end{array}$$

(64)

$$\begin{array}{cc}\hfill & \rho (x,0)={\rho }_0\left(x\right),v(x,0)=v_0\left(x\right),\omega (x,0)={\omega }_0\left(x\right),\theta (x,0)={\theta }_0\left(x\right),\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill & v(0,t)=v(1,t)=0,\omega (0,t)=\omega (1,t)=0,{\theta }_x(0,t)={\theta }_x(1,t)=0,\hfill \end{array}$$

(66)

for $x\in ]0,1[$, $t\in ]0,T[$, $T>0$.

Also, from (40), by taking $t=0$, we get

$$r_0\left(x\right)=\sqrt[3]{a^3+3L{\int }_0^x{\displaystyle \frac{dy}{{\rho }_0\left(y\right)}}},x\in ]0,1[,$$

(67)

where a is the radius of the smaller sphere defined in (40).

5. Faedo–Galerkin Approximations

In this section, we apply the Faedo–Galerkin approximation method to our problem in order to obtain a numerical solution. This approach is common in the numerical analysis of compressive micropolar fluid flow problems because it has a twofold benefit. On the one hand, we obtain a numerical solution to the problem, and on the other hand, these approximations can also be used to prove the local existence of the solution by studying the convergence of the approximations (see [2,24]). In this paper, for the sake of brevity, we focus on the derivation of the method and its experimental evaluation. In our further work, we will use the results obtained here for the existence proof.

We now introduce an approximate problem for our initial boundary value system. Let ${\left\{w_i\right\}}_{i=1}^{\infty }$ be an orthogonal basis for $H_0^1(0,1)$ and $H^1(0,1)$. For each $n\in \mathbf{N}$ and $u\in \{v,\omega ,\theta \}$, we define the approximate function $u^n$ as follows:

$$u^n(x,t)=\sum _{i=1}^n{u}_i^n\left(t\right)w_i\left(x\right),$$

(68)

whereby $u_i^n=u_i^n\left(t\right)$ are unknown functions that are determined by the projection of the system (61)–(64) and initial conditions (65) in the finite-dimensional space spanned by ${\left\{w_i\right\}}_{i=1}^n$. Taking into account (40), we define

$$r^n(x,t)=r_0\left(x\right)+{\int }_0^t{v}^n(x,\tau )d\tau .$$

(69)

On the other hand, we define ${\rho }^n$ as a solution to the following simple initial value problem stemming from (61) and (65):

$$\begin{array}{c}\hfill {\rho }_t^n+{\displaystyle \frac{1}{L}}{\left({\rho }^n\right)}^2{\left({\left(r^n\right)}^2{v}^n\right)}_x=0,{\rho }^n(x,0)={\rho }_0\left(x\right).\end{array}$$

(70)

Solving it yields the following closed-form formula:

$${\rho }^n(x,t)={\displaystyle \frac{L{\rho }_0\left(x\right)}{L+{\rho }_0\left(x\right){\left({\int }_0^t{\left(r^n\right)}^2{v}^{n}d\tau \right)}_x}}.$$

(71)

In this work, we use the following two bases for the Faedo–Galerkin method:

$${\{sin\left(\pi ix\right)\}}_{i=1}^{\infty }$$

(72)

for (62)–(63) and

$${\{cos\left(\pi jx\right)\}}_{j=0}^{\infty }$$

(73)

for (64). This ensures that the approximate functions (68) implicitly satisfy the boundary conditions (66). Via projection, we obtain the following approximate initial value problem for the unknown functions $v_i^n=v_i^n\left(t\right)$, ${\omega }_i^n={\omega }_i^n\left(t\right)$, and ${\theta }_j^n={\theta }_j^n\left(t\right)$:

$$\begin{array}{cc}\hfill & \dot{v_i^n}=2{\int }_0^1\left[-{\displaystyle \frac{R}{L}}{\left(r^n\right)}^2{\left({\left({\rho }^n\right)}^p{\theta }^n\right)}_x+{\displaystyle \frac{\lambda +2\mu }{L^2}}{\left(r^n\right)}^2{\left({\rho }^n{\left({\left(r^n\right)}^2{v}^n\right)}_x\right)}_x\right]sin\left(\pi ix\right)dx,\hfill \end{array}$$

(74)

$$\begin{array}{cc}\hfill & \dot{{\omega }_i^n}=2{\int }_0^1\left[-{\displaystyle \frac{4{\mu }_r}{j_I}}{\displaystyle \frac{{\omega }^n}{{\rho }^n}}+{\displaystyle \frac{c_0+2c_d}{j_I{L}^2}}{\left(r^n\right)}^2{\left({\rho }^n{\left({\left(r^n\right)}^2{\omega }^n\right)}_x\right)}_x\right]sin\left(\pi ix\right)dx,\hfill \end{array}$$

(75)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{{\theta }_j^n}& ={\lambda }_j{\int }_0^1\left[{\displaystyle \frac{k}{c_v{L}^2}}{\left({\left(r^n\right)}^4{\rho }^n{\theta }_x^n\right)}_x-{\displaystyle \frac{R}{c_{v}L}}{\left({\rho }^n\right)}^p{\theta }^n{\left({\left(r^n\right)}^2{v}^n\right)}_x+{\displaystyle \frac{\lambda +2\mu }{c_v{L}^2}}{\rho }^n{\left({\left(r^n\right)}^2{v}^n\right)}_x^2\right.\hfill \\ & -{\displaystyle \frac{4\mu }{c_{v}L}}{\left(r^n{\left(v^n\right)}^2\right)}_x+{\displaystyle \frac{c_0+2c_d}{c_v{L}^2}}{\rho }^n{\left({\left(r^n\right)}^2{\omega }^n\right)}_x^2-{\displaystyle \frac{4c_d}{c_{v}L}}{\left(r^n{\left({\omega }^n\right)}^2\right)}_x\hfill \\ & \left.+{\displaystyle \frac{4{\mu }_r}{c_v}}{\displaystyle \frac{{\left({\omega }^n\right)}^2}{{\rho }^n}}\right]cos\left(\pi jx\right)dx,\hfill \end{array}\end{array}$$

(76)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & v_i^n\left(0\right)=2{\int }_0^1{v}_0\left(x\right)sin\left(\pi ix\right)dx,{\omega }_i^n\left(0\right)=2{\int }_0^1{\omega }_0\left(x\right)sin\left(\pi ix\right)dx,\hfill \\ \hfill & {\theta }_i^n\left(0\right)=2{\int }_0^1{\theta }_0\left(x\right)cos\left(\pi ix\right)dx,{\theta }_0^n\left(0\right)={\int }_0^1{\theta }_0\left(x\right)dx,\hfill \end{array}\end{array}$$

(77)

for $i=1,\cdots ,n$, $j=0,1,\cdots ,n$, and where

$${\lambda }_j=\left\{\begin{array}{cc}1,\hfill & j=0\hfill \\ 2,\hfill & j=1,\cdots ,n\hfill \end{array}\right..$$

(78)

Note that the initial conditions are the Fourier coefficient of the initial functions $v_0$, ${\omega }_0$, and ${\theta }_0$.

So that the approximate problem (74)–(77) can be eligible for the ODE-solver, we need to rewrite it explicitly in terms of unknowns. To achieve that, we introduce the following auxiliary functions [2]:

$$\begin{array}{c}\hfill q_i^n\left(t\right)={\int }_0^t{v}_i^{n}d\tau ,{\lambda }_{ij}^n\left(t\right)={\int }_0^t{q}_i^n{v}_j^{n}d\tau ,{\mu }_{ijk}^n\left(t\right)={\int }_0^t{q}_i^n{q}_j^n{v}_k^{n}d\tau ,i,j,k=1,\cdots ,n.\end{array}$$

(79)

Using the introduced notation, we rewrite (69) and (71), and we obtain

$$\begin{array}{cc}\hfill & r^n(x,t)=r_0\left(x\right)+\sum _{i=1}^n{q}_i^n\left(t\right)sin\left(\pi ix\right),\hfill \end{array}$$

(80)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\rho }^n(x,t)& =L{\rho }_0\left(x\right)\left[L+{\rho }_0\left(x\right)\left(r_0^2\left(x\right)\sum _{i=1}^n{q}_i^n\left(t\right)sin\left(\pi ix\right)\right.\right.\hfill \\ \hfill & +2r_0\left(x\right)\sum _{i,j=1}^n{\lambda }_{ij}^n\left(t\right)sin\left(\pi ix\right)sin\left(\pi jx\right)\hfill \\ \hfill & {\left.{\left.+\sum _{i,j,k=1}^n{\mu }_{ijk}^n\left(t\right)sin\left(\pi ix\right)sin\left(\pi jx\right)sin\left(\pi kx\right)\right)}_x\right]}^{-1}.\hfill \end{array}\end{array}$$

(81)

The explicit form of the ODE initial value problem in terms of the unknown functions $v_i^n=v_i^n\left(t\right)$, ${\omega }_i^n={\omega }_i^n\left(t\right)$, ${\theta }_j^n={\theta }_j^n\left(t\right)$, $q_i^n=q_i^n\left(t\right)$, ${\lambda }_{ij}^n={\lambda }_{ij}^n\left(t\right)$, and ${\mu }_{ijk}^n={\mu }_{ijk}^n\left(t\right)$ is obtained by inserting (79) into (74)–(77) and taking the derivative of (79):

$$\begin{array}{cc}\hfill & \dot{v_i^n}={\int }_0^1\left[-{\displaystyle \frac{R}{L}}{\left(r^n\right)}^2{\left({\left({\rho }^n\right)}^p{\theta }^n\right)}_x+{\displaystyle \frac{\lambda +2\mu }{L^2}}{\left(r^n\right)}^2{\left({\rho }^n{\left({\left(r^n\right)}^2{v}^n\right)}_x\right)}_x\right]sin\left(\pi ix\right)dx,\hfill \end{array}$$

(82)

$$\begin{array}{cc}\hfill & \dot{{\omega }_i^n}={\int }_0^1\left[-{\displaystyle \frac{4{\mu }_r}{j_I}}{\displaystyle \frac{{\omega }^n}{{\rho }^n}}+{\displaystyle \frac{c_0+2c_d}{j_I{L}^2}}{\left(r^n\right)}^2{\left({\rho }^n{\left({\left(r^n\right)}^2{\omega }^n\right)}_x\right)}_x\right]sin\left(\pi ix\right)dx,\hfill \end{array}$$

(83)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{{\theta }_j^n}& ={\lambda }_j{\int }_0^1\left[{\displaystyle \frac{k}{c_v{L}^2}}{\left({\left(r^n\right)}^4{\rho }^n{\theta }_x^n\right)}_x-{\displaystyle \frac{R}{c_{v}L}}{\left({\rho }^n\right)}^p{\theta }^n{\left({\left(r^n\right)}^2{v}^n\right)}_x+{\displaystyle \frac{\lambda +2\mu }{c_v{L}^2}}{\rho }^n{\left({\left(r^n\right)}^2{v}^n\right)}_x^2\right.\hfill \\ & -{\displaystyle \frac{4\mu }{c_{v}L}}{\left(r^n{\left(v^n\right)}^2\right)}_x+{\displaystyle \frac{c_0+2c_d}{c_v{L}^2}}{\rho }^n{\left({\left(r^n\right)}^2{\omega }^n\right)}_x^2-{\displaystyle \frac{4c_d}{c_{v}L}}{\left(r^n{\left({\omega }^n\right)}^2\right)}_x\hfill \\ & \left.+{\displaystyle \frac{4{\mu }_r}{c_v}}{\displaystyle \frac{{\left({\omega }^n\right)}^2}{{\rho }^n}}\right]cos\left(\pi jx\right)dx,\hfill \end{array}\end{array}$$

(84)

$$\begin{array}{cc}\hfill & \dot{q_i^n}=v_i^n,\hfill \end{array}$$

(85)

$$\begin{array}{cc}\hfill & \dot{{\lambda }_{ik}^n}=q_i^n{q}_k^n,\hfill \end{array}$$

(86)

$$\begin{array}{cc}\hfill & \dot{{\mu }_{ikm}^n}=q_i^n{q}_k^n{v}_m^n,\hfill \end{array}$$

(87)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & v_i^n\left(0\right)=2{\int }_0^1{v}_0\left(x\right)sin\left(\pi ix\right)dx,{\omega }_i^n\left(0\right)=2{\int }_0^1{\omega }_0\left(x\right)sin\left(\pi ix\right)dx,\hfill \\ \hfill & {\theta }_i^n\left(0\right)=2{\int }_0^1{\theta }_0\left(x\right)cos\left(\pi ix\right)dx,{\theta }_0^n\left(0\right)={\int }_0^1{\theta }_0\left(x\right)dx,\hfill \\ \hfill & q_i^n\left(0\right)=0,{\lambda }_{ik}\left(0\right)=0,{\mu }_{ikm}\left(0\right)=0,\hfill \end{array}\end{array}$$

(88)

$i,k,m=1,\cdots ,n$, $j=0,1,\cdots ,n$, whereby we use (80)–(81) expressions for $r^n$ and ${\rho }^n$, but we omit this step here because of the unintelligibility and vastness of the resulting notation.

6. Numerical Solution

In this section, we give a detailed description of the method for obtaining a numerical solution to the approximate problem (82)–(88). The method consists of two discretization steps. This approach is characteristic to the problems in the compressible micropolar flow research area. The first step deals with the right-hand side of the system, and the second step is discretization with respect to time. In the following two subsections, we describe each step.

6.1. Numerical Approximation for the Integrals

A numerical integration method needs to be applied to the right-hand side of Equations (82)–(84) in order to be calculated in any practical sense. To this end, we chose a quadrature formula from the family of Gauss integration formulas, specifically the Gauss–Legendre formula:

$${\int }_0^{1}f\left(x\right)dx\approx \sum _{i=1}^N{w}_{i}f\left(x_i\right),$$

(89)

where $x_i$ are nodes that correspond to roots of the N-th Legendre polynomial, and $w_i$ denotes the corresponding weights. After experimenting with various orders of the formula, we concluded that the optimal results are obtained using a formula of the order of 20. Programming languages with numerical libraries, such as Python 3.11 and its libraries SciPy and NumPy, enable easy implementation of this discretization step.

6.2. Temporal Discretization

After we learned how to numerically calculate the right-hand side of the system in question, it remained to solve the initial value problem. Numerical libraries are equipped with a substantial amount of ready-to-use algorithms for solving this class of problems that are optimized, reliable, and offer various feedback. For this purpose, we elected to employ a method based on a backward differentiation formula because it proved to be suitable for similar problems (see [4]).

6.3. Numerical Experiments

We implemented the method described in previous (sub)sections and empirically validated it.

We set up the parameters of the system as follows:

$$\begin{array}{c}\hfill a=1,b=2,R=1,c_d=1,c_0=1,c_v=1,j_I=1,\\ \hfill \lambda =-2,\mu =3,{\mu }_r=1,\kappa =0.024.\end{array}$$

(90)

These parameter values were chosen to test the numerical scheme under controlled conditions. Since the micropolar fluid model is relatively new and there are limited empirical data on the exact parameter values that would represent real-world conditions, these values were artificially selected to demonstrate the stability and effectiveness of the proposed numerical method. The goal of the numerical experiments was to verify the model’s behavior and robustness, rather than to replicate a specific physical scenario. Future work will focus on refining these parameters based on experimental data as it becomes available.

We analyzed two numerical examples with different initial conditions to demonstrate the applicability and robustness of the Faedo–Galerkin method.

Example 1. 

This case is relatively simpler from the standpoint of the Faedo–Galerkin method. The chosen initial functions can be exactly represented by a finite-order trigonometric polynomial. This exact representation allows for an efficient and straightforward implementation of the numerical method, facilitating the analysis of the solution’s behavior under ideal conditions. The initial conditions for this example are given by the following functions:

$$\begin{array}{c}\hfill {\rho }_0\left(x\right)=1,v_0\left(x\right)=sin\left(\pi x\right),{\omega }_0\left(x\right)=sin\left(2\pi x\right),{\theta }_0\left(x\right)=2+cos\left(\pi x\right).\end{array}$$

(91)

Example 2. 

In contrast, this example involves more general initial conditions whose representations do not permit a finite Fourier expansion. The initial functions cannot be expressed exactly as a finite sum of trigonometric polynomials, making the problem more complex. This example tests the method’s ability to handle more intricate scenarios where the initial conditions are not ideally suited for the method’s basis functions. In this case, the initial conditions are described by the following functions:

$$\begin{array}{c}\hfill {\rho }_0\left(x\right)=1,v_0\left(x\right)=x-x^2,{\omega }_0\left(x\right)=x^2(1-x^2),{\theta }_0\left(x\right)=2x^3-3x^2+2.\end{array}$$

(92)

By considering both examples, we showcase the versatility of the Faedo–Galerkin method in handling both simple and complex initial conditions, highlighting its potential for broader applications in micropolar fluid dynamics.

In both examples, the initial conditions are specified in mass Lagrangian coordinates. For each case, we employ $n=8$ basis functions in the Faedo–Galerkin method. Here, n represents the number of elements in the chosen basis, determining the dimensionality of the approximation space and playing a crucial role in the accuracy and convergence of our numerical solutions.

In the analysis of the examples, among other aspects, we focused on the influence of the pressure exponent on the behavior of the solution. The generalized equation of state, specifically the corresponding pressure exponent, is a distinctive feature of our model. Selecting the exponent $p=1$ reduces the model to that of an ideal micropolar gas, which has been previously studied (see, for example, [2]). Exponents strictly greater than 1 characterize a true real micropolar gas. By comparing different values of the pressure exponent, we aim to investigate how a real gas behaves in comparison to an ideal gas and to examine the effects of various choices of the pressure exponent. The selection of the pressure exponent is a crucial part of setting up the model parameters, and it should be chosen to best represent the actual behavior of the gas in practice.

6.3.1. Example 1

In Figure 1, Figure 2, Figure 3 and Figure 4, we show the calculated approximate solutions for $\rho$, v, $\omega$, and $\theta$ for different values of the pressure exponent p, $p\in \{1,1.5,4,10\}$ for initial conditions (91).

Figure 1 illustrates the evolution of mass density $\rho$ over time for different values of the pressure exponent p. Starting from an initial homogeneous distribution of ${\rho }_0=1$, the density changes as time progresses and eventually stabilizes at a value of ${\rho }^{*}=1$, which aligns with theoretical predictions for similar models (see, for example, [25,26]). Throughout the simulation period of $[0,10]$, the density remains positive, which is consistent with physical assumptions and indicates the stability of the model. Regarding the influence of the pressure exponent p, we observe that stabilization occurs significantly faster for higher values of p. Specifically, the cases with $p=4$ and $p=10$ show progressively quicker convergence to the stable density value. This behavior is consistent with previous studies (e.g., [26]), which suggest that a higher-pressure exponent enhances the rate at which the system reaches equilibrium. The faster stabilization observed for higher-pressure exponents, p, can be attributed to the increased sensitivity of pressure to changes in density in the generalized equation of state. With a larger exponent p, even minimal variations in density result in significant changes in pressure, leading to stronger pressure gradients. These enhanced gradients act as driving forces that accelerate the redistribution of mass within the fluid, enabling the system to reach equilibrium more quickly.

Figure 2 illustrates the evolution of the velocity, v, over time for various values of the pressure exponent, p. The velocity changes sign throughout the simulation, reflecting the oscillatory motion of gas particles responding to initial perturbations. Additionally, the velocity progressively approaches zero, signifying that the particles are coming to rest relative to the reference frame. This trend is expected due to the parabolic nature of the governing equations, associated with diffusion processes and viscous dissipation that attenuate motion over time. Theoretical results on stabilization for similar models support this convergence toward a stationary state [25,26]. Importantly, the stabilization of velocity occurs more rapidly for higher-pressure exponents, p, and this effect is even more pronounced than that observed for density. Higher values of p enhance the pressure’s responsiveness to density fluctuations, leading to stronger pressure gradients in response to any density inhomogeneities. These intensified pressure gradients exert greater damping forces on the particles, accelerating the decay of velocity fluctuations. As a result, the system reaches equilibrium more quickly with higher-pressure exponents, demonstrating the significant impact of p on the dynamics of velocity stabilization.

Figure 3 depicts the evolution of microrotation $\omega$ over time for various values of the pressure exponent, p. The microrotation displays behavior similar to that of the velocity, v, which is expected due to the intrinsic coupling between translational and rotational motions in micropolar fluids. This coupling arises because the microrotation is directly influenced by the gradients of velocity and the microstructural interactions within the fluid. The microrotation exhibits sign changes, indicating oscillations in the direction of particle spin as the fluid responds to initial disturbances. These oscillations reflect the dynamic adjustments of the fluid’s microelements as they interact and exchange angular momentum. As time progresses, the microrotation stabilizes at around zero, demonstrating that the rotational motions are damped out, and the particles’ spin is approaching a stationary state. This stabilization aligns with the dissipative properties of the system, where viscous effects and internal friction contribute to the attenuation of both translational and rotational motions. The influence of the pressure exponent is somewhat less pronounced here due to the small scale and the presence of rounding errors.

Figure 4 presents the evolution of temperature $\theta$ over time for various values of the pressure exponent, p. The temperature behaves similarly to the density, $\rho$, remaining positive throughout the simulation and showing a tendency to stabilize toward a constant value, in accordance with thermodynamic principles and energy conservation, as observed in similar models [25,26]. Interestingly, the influence of the pressure exponent, p, on the temperature is less pronounced than its effect on the velocity, v, and microrotation, $\omega$. In fact, the stabilization of temperature occurs more slowly for higher values of p. This can be attributed to the interplay between thermal diffusion and the pressure–density relationship. While higher p values enhance pressure sensitivity to density changes, leading to stronger mechanical interactions, temperature evolution is primarily governed by heat conduction, a process characterized by thermal diffusivity and inherently slower time scales. Consequently, the increased mechanical work associated with higher-pressure exponents introduces additional thermal energy, temporarily impeding the diffusion-driven stabilization of temperature. As a result, the temperature reaches equilibrium more slowly when the pressure exponent is higher.

6.3.2. Example 2

Figure 5, Figure 6, Figure 7 and Figure 8 correspond to Example 2, where we analyze the system using more general initial conditions that cannot be exactly represented by a finite Fourier expansion. This example tests the numerical method’s capability to handle increased complexity in the initial data. Qualitatively, we observe the same findings as in the first example regarding stabilization, the maintenance of positivity, and the influence of the pressure exponent, p.

Figure 5 illustrates the evolution of mass density, $\rho$, over time. The density remained positive throughout the simulation, and stabilization toward a consistent value was observed. This behavior confirms the preservation of mass and aligns with physical expectations, indicating the stability of the numerical method even with more complex initial conditions.

Figure 6 depicts the evolution of velocity v. The velocity changes sign over time, reflecting the oscillatory motion of fluid particles as they adjust to the initial perturbations. The velocity gradually converges to zero, demonstrating stabilization to a stationary state due to dissipative effects inherent to the system. Higher-pressure exponents, p, accelerate this stabilization, consistent with the results from Example 1.

Figure 7 shows the evolution of microrotation, $\omega$. Similar to the velocity, the microrotation exhibits oscillatory behavior with diminishing amplitude over time. The rotational motions stabilize at around zero, and the influence of the pressure exponent, p, on the rate of stabilization is even more pronounced for microrotation than for velocity. Higher values of p enhance the pressure’s responsiveness to density fluctuations, leading to stronger pressure gradients. These increased pressure gradients intensify the couple stresses acting on the fluid’s microstructure, thereby accelerating the damping of microrotational motions. Consequently, systems with higher-pressure exponents achieve rotational equilibrium more rapidly. This observation highlights the significant impact of the pressure exponent on the rotational behavior of micropolar fluids and underscores its importance in controlling microstructural dynamics.

Figure 8 presents the evolution of temperature $\theta$. The temperature maintains positivity and stabilizes toward a specific value, mirroring the behavior observed in the density. The influence of the pressure exponent on temperature remains less significant, and higher p values result in slower stabilization, as previously noted.

These consistent findings between Examples 1 and 2 demonstrate that the Faedo–Galerkin approximations yield a stable and reliable numerical method, capable of handling both simple and complex initial conditions. The method’s robustness is evident, as it effectively captures the essential dynamics of the micropolar fluid model without being sensitive to the increased complexity of the initial data. This reinforces the applicability of the numerical scheme to a wide range of problems in micropolar fluid dynamics.

6.4. Back to Eulerian Coordinates

The numerical method is formulated to solve the problem (61)–(64) in mass Lagrangian coordinates that were chosen because the transformed problem domain is simpler and the resulting system of PDE-s is purely parabolic. On the other hand, the system written in Eulerian coordinates (34)–(37) is a mix of parabolic and hyperbolic, which makes the analysis harder, but it is more suitable for interpretation and graphical representation. For these reasons, we transformed the calculated numerical solution into Eulerian coordinates using relations of the respective variables explained in Section 4.

For example, for ${\rho }_0\left(x\right)=1$, we have

$$\eta \left(\xi \right)={\displaystyle \frac{{\xi }^3}{3}}-{\displaystyle \frac{a^3}{3}}\mathrm{and}L={\displaystyle \frac{b^3}{3}}-{\displaystyle \frac{a^3}{3}},$$

(93)

that is

$$\xi ={\eta }^{-1}\left(Lx\right)=\sqrt[3]{3Lx+a^3}.$$

(94)

The function $r=r(\xi ,t)=r({\eta }^{-1}\left(Lx\right),t)$ can be explicitly calculated from the obtained solution of the problem because relation (40) holds or, to be more precise, the approximate relation (80) holds. Now, for a given Eulerian coordinate, $r\in [a,b]$, the corresponding mass Lagrangian coordinate, $x\in [0,1]$, can be obtained using any method for solving non-linear equations.

Example 2 in Eulerian Coordinates

In Figure 9, Figure 10, Figure 11 and Figure 12, the obtained approximations for the solutions $\rho$, v, $\omega$, and $\theta$ in Eulerian coordinates are displayed in 2D for the chosen times and for the initial conditions (92) with $p=4$.

Figure 9 illustrates the evolution of density $\rho$ over time in the Eulerian description. The change in the particle distribution is clearly observable. At the initial time, particles are more concentrated near the outer boundary of the domain, while their concentration near the inner boundary is lower. As time progresses, these disparities diminish due to diffusion and internal interactions within the fluid, leading to a more uniform distribution of particles. The system exhibits a clear tendency toward homogenization, which aligns with the natural drive toward equilibrium in thermodynamic systems. This behavior was anticipated based on the results obtained in the Lagrangian description, and it is consistent with theoretical predictions for micropolar fluids.

Figure 10 illustrates the evolution of velocity, v, over time in the Eulerian description, within the spherical domain bounded by the inner radius $r=1$ and the outer radius $r=2$. An exceptionally rapid stabilization of velocity towards zero is observed, occurring within the initial time frame from $t=0$ to $t=10$ s. Due to this swift change, we dynamically adjusted the colormap scale at each time step to effectively visualize the diminishing velocity variations; otherwise, these subtle changes would not be perceptible. Stabilization occurs more quickly near the boundaries, which is expected, given the boundary conditions applied to the system. The no-slip conditions at the walls enforce zero velocity at the boundaries, resulting in a greater damping of the fluid motion in these regions. The increased viscous effects and frictional forces near the walls accelerate the attenuation of velocity fluctuations compared to the central regions of the domain. This rapid stabilization aligns with theoretical expectations for dissipative systems described by parabolic partial differential equations. The inherent diffusion-like processes promote the swift smoothing of velocity gradients, especially in proximity to the boundaries where the effects of viscosity and friction are the most pronounced.

Figure 11 presents the evolution of microrotation $\omega$ over time in the Eulerian description. Similar to the velocity v depicted in Figure 11, the microrotation exhibits rapid stabilization, but with an even more pronounced rate. This indicates that the rotational motions inherent to the micropolar fluid dissipate more swiftly than the translational motions, likely due to the higher rotational viscosity that characterizes such fluids. The extremely rapid stabilization necessitated the adjustment of the color map scale at each time step to effectively capture and visualize the transient changes in microrotation. Without rescaling, the rapid decrease in microrotation values would make it challenging to discern the dynamics occurring within the fluid. Consistent with expectations, stabilization occurs more quickly near the boundaries of the domain. The boundary conditions impose constraints that enhance the damping of rotational motions, leading to a faster attenuation of microrotation in these regions. This spatial variation underscores the influence of boundary effects on the behavior of micropolar fluids.

Figure 12 presents the evolution of temperature $\theta$ over time in the Eulerian description. In contrast to previous figures depicting velocity and microrotation, adapting the color map scale was unnecessary, indicating that temperature variations remained within a consistent and manageable range throughout the simulation. This suggests that temperature changes are more gradual and less abrupt than those observed in other variables. The figure demonstrates that temperature stabilization occurs over time, with the distribution becoming uniform across the domain. Importantly, this stabilization proceeds uniformly from both the inner and outer boundaries, reflecting symmetrical thermal behavior within the fluid. This uniformity is likely due to the boundary conditions imposing equal thermal properties at both edges and the effective thermal diffusivity of the fluid, which promotes even heat conduction throughout the domain. The gradual stabilization of temperature highlights the diffusive nature of thermal processes in the fluid. Unlike the rapid stabilization observed in velocity and microrotation, temperature evolution is governed by heat conduction, which operates over longer time scales. The absence of significant temperature gradients indicates that thermal equilibrium is steadily achieved, reinforcing the model’s capability to accurately capture thermal dynamics in micropolar fluids.

7. Conclusions

In this work, we have significantly advanced the modeling of viscous, micropolar, polytropic, and heat-conducting real gases by transitioning from a one-dimensional framework to a fully three-dimensional, spherically symmetric model. This advancement allows for a more accurate and comprehensive representation of fluid behavior under spherical symmetry, marking the first detailed exploration of this type of fluid model.

We incorporated a generalized equation of state by modeling the pressure as a power-law function of mass density, moving beyond the limitations of the ideal gas equation of state used in previous studies. This generalization enables a more realistic depiction of real gas behavior, capturing the nuances of fluid dynamics that emerge under various flow conditions.

A primary objective of this study was to demonstrate a functional numerical method for modeling the flow of such complex fluids. We established the applicability and robustness of the Faedo–Galerkin method to solve this complex system, demonstrating its effectiveness for both simple and complex initial conditions. Our initial conditions were designed to represent two scenarios based on the number of terms in the Fourier expansion, illustrating the method’s robustness. This contribution is significant because previous studies have not applied this numerical method in such a context—specifically, the combination of spherical symmetry and real fluids presents unique challenges that our method successfully addresses.

While we selected specific parameter values and initial conditions to test the effectiveness of our numerical scheme, we recognize that exploring various combinations of parameters could provide deeper insights into the model’s applicability. However, such an exploration would involve extensive detail, and it is thus recommended for future research. Drawing from previous studies and theoretical results on similar problems, we anticipate that variations in constants, such as (micro)viscosity coefficients, would not significantly impact the construction or stability of the numerical solution. Since the existence of solutions to similar initial boundary value problems has been theoretically established through the limit of Faedo–Galerkin approximations under physical conditions like Duhamel’s inequalities, we do not expect parameter variations to affect the numerical solution’s stability.

Furthermore, we performed simulations in the Eulerian description for the spherically symmetric problem, offering novel insights into fluid behavior and enabling a new form of solution visualization. This approach differs from earlier works that have primarily relied on the Lagrangian description, and to the best of our knowledge, it has not been previously employed for similar micropolar fluid flows. By utilizing the Eulerian framework, we were able to observe the spatial distribution and temporal evolution of fluid properties more directly, providing a deeper understanding of the flow characteristics. By comparing simulations using the generalized equation of state with those based on the ideal gas law, we investigated the influence of the pressure exponent on the solutions. Higher-pressure exponents lead to the faster stabilization of variables such as density, velocity, and microrotation, while the impact on temperature stabilization is less pronounced.

In conclusion, our work offers a significant advancement in the modeling and numerical analysis of micropolar real gases under spherical symmetry. The proposed methodologies and obtained results lay the groundwork for future research, including extending the model to cases with varying parameters and external forces—such as gravitational fields or electromagnetic forces—and applying it to specific real-world situations. The established numerical method demonstrates robustness and reliability, providing a valuable tool for further investigations into the complex behaviors of compressible micropolar fluids.