Compressible Viscous Micropolar Fluid Research Articles

On the low Mach number limit of the compressible viscous micropolar fluid model

In this paper, we consider the low Mach number limit problem for the compressible viscous micropolar fluid model based on the concept of dissipative measure‐valued solutions. We prove that the dissipative measure‐valued solutions of the compressible micropolar fluid model converge to the smooth solution of the incompressible micropolar system in the case of well‐prepared initial data when the Mach number tends to zero.

Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry

<p style='text-indent:20px;'>This paper is concerned with three-dimensional compressible viscous and heat-conducting micropolar fluid in the domain to the subset of <inline-formula><tex-math id="M1">$ R^3 $</tex-math></inline-formula> bounded with two coaxial cylinders that present the solid thermoinsulated walls, being in a thermodynamical sense perfect and polytropic. We prove that the regularity and the exponential stability in <inline-formula><tex-math id="M2">$ H^2 $</tex-math></inline-formula>.</p>

A shear flow problem for compressible viscous micropolar fluid: Uniqueness of a generalized solution

In this paper, we consider the nonstationary shear flow of a compressible, viscous, and heat‐conducting micropolar fluid. The mathematical model is set up in Lagrangian description in the form of initial‐boundary problem with inhomogeneous boundary conditions for velocity and standard homogeneous boundary conditions for microrotation and heat flux. Under the assumptions that this problem has a generalized solution and that the initial mass density, temperature, the velocity, and microrotation vectors are smooth enough functions, we prove that this solution is unique.

A shear flow problem for compressible viscous micropolar fluid: Derivation of the model and numerical solution

In this paper we consider the nonstationary shear flow between two parallel solid and thermoinsulated horizontal plates with the upper one moving irrotationally. The fluid is compressible, micropolar, viscous and heat-conducting, as well as in the thermodynamical sense perfect and polytropic. We assume that, given a Cartesian coordinate system x, y and z, solutions of corresponding problem are x-dependent only. Mathematical model is derived in the Lagrangian description. By using the Faedo–Galerkin method, as well as homogenization of boundary conditions, we derive an approximate system, which we use to obtain a numerical solution to the given problem.

Global attractors for a nonlinear one-dimensional compressible viscous micropolar fluid model

This paper considers the dynamical behavior of solutions of constitutive systems for 1D compressible viscous and heat-conducting micropolar fluids. With proper constraints on initial data, we prove the existence of global attractors in generalized Sobolev spaces $$H_{\delta }^{(1)}$$ and $$H_{\delta }^{(2)}$$ . These attractors are unique in corresponding phase spaces.

Large‐time behavior of solutions to the 3‐D flow of a compressible viscous micropolar fluid with cylindrical symmetry

In this paper, we study the asymptotic behavior of global weak solutions in H1 to the problem that describes compressible viscous and heat‐conducting micropolar fluid flow in a three‐dimensional domain bounded by two circular, coaxial, and infinite cylinders that present the solid thermoinsulated walls. In the thermodynamical sense, the fluid is perfect and polytropic. We prove that the global weak solution exists and converges to a steady state as time goes to infinity. We have been working under the assumption that the initial data are cylindrically symmetric and the initial total energy is sufficiently small.

3-D flow of a compressible viscous micropolar fluid model with spherical symmetry: A brief survey and recent progress

We consider the non-stationary 3-D flow of a compressible viscous and heat-conducting micropolar fluid with the assumption of spherical symmetry. We analyze the flow between two concentric spheres that present solid thermo-insulated walls. The fluid is perfect and polytropic in the thermodynamical sense and the initial density and temperature are strictly positive. The corresponding problem has homogeneous boundary data. In this work, we present the described model and provide a brief overview of the progress in the mathematical analysis of the associated initial-boundary problem. We consider existence and uniqueness of the generalized solution, asymptotic behavior of the solution and regularity of the solution.

Three-dimensional compressible viscous micropolar fluid with cylindrical symmetry: Derivation of the model and a numerical solution

In this paper we consider the nonstationary 3D flow of a compressible viscous and heat-conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic. The fluid domain is the subset of R3 bounded with two coaxial cylinders that present solid thermoinsulated walls. We assume that the initial mass density, temperature, as well as the velocity and microrotation vectors are radially dependent only. The corresponding solution is also spatially radially dependent. We derive the mathematical model in the Lagrangian description and by using the Faedo–Galerkin method we introduce a system of approximate equations and construct its solutions. We also analyze two numerical examples.

Stability of rarefaction waves for 1-D compressible viscous micropolar fluid model

In this paper, we study the large time behavior of the Cauchy problem of the compressible micropolar fluids in one dimensional space. According to Darcy's law, we see from the equation describing the evolution of the microrotation ω that ω→0 as t→∞. Actually, if the microstructure of the fluid is not taken into account, that is to say the effect of the angular velocity fields of the particle's rotation is omitted, i.e., ω=0, then the micropolar fluids equations reduce to the classical Navier–Stokes equations. Therefore the system should tend time-asymptotically to the corresponding classical full Navier–Stokes equations. We consider the case that the far field of the initial data for the microrotation velocity ω is zero, and the far fields of the initial data for other variables, such as the specific volume v, velocity u and entropy s, are connected by rarefaction waves to the corresponding Euler equations. In this case, we prove the stability of rarefaction waves for this compressible micropolar fluids model. Compared with the classical Navier–Stokes equations, the angular velocity ω in this model brings both benefit and trouble. The benefit lies in the fact that the term −vω is a damping term which provides extra regularity of ω, while the trouble is brought by the term vω2 which increases the nonlinearity of the system.

Global solution to 3D problem of a compressible viscous micropolar fluid with spherical symmetry and a free boundary

We consider a nonstationary 3D flow of a compressible viscous heat-conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic, in the domain bounded with two concentric spheres in R3. In this paper we establish the existence of a global solution to the free boundary problem defined with the homogeneous boundary conditions for velocity, microrotation, heat flux on the fixed border and homogeneous boundary conditions for strain, microrotation, and heat flux on the free boundary. We suppose that the initial data are spherically symmetric, with positive initial density and temperature, having the zero density on the free boundary. Because of the spherical symmetry, the starting three-dimensional problem is transformed to the one-dimensional problem in Lagrangian coordinates in the domain that is a segment. The solution to our problem is obtained as a limit of the sequence of approximate solutions derived from suitable semi-discrete finite difference approximate systems. By using the derived bounded estimates of the approximate solutions and the results of the weak and strong compactness, we establish the convergence to the generalized solution of our problem globally in time.

Three-dimensional flow of a compressible viscous micropolar fluid with cylindrical symmetry: a global existence theorem

Three-dimensional flow of a compressible viscous micropolar fluid with cylindrical symmetry: a global existence theorem

Stability of contact discontinuity for 1-D compressible viscous micropolar fluid model

This paper is concerned with a one-dimensional compressible micropolar fluid model with initial data whose behaviors at far fields x→±∞ are different. Motivated by the relationship between micropolar fluid model and Navier–Stokes system, we can prove that the solutions to the one-dimensional compressible micropolar fluid model tend time-asymptotically to a viscous contact wave which is constructed from a contact discontinuity solution of the Riemann problem on Euler system. This result is proved by an elaborate elementary energy estimates.

3‐D flow of a compressible viscous micropolar fluid with cylindrical symmetry: uniqueness of a generalized solution

In this paper, we consider a nonstationary 3‐D flow of a compressible viscous and heat‐conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic. The fluid domain is a subset of R3 bounded with two coaxial cylinders that present solid thermoinsulated walls. The mathematical model is set up in Lagrangian description. If we assume that the initial mass density, temperature, as well as the velocity and microrotation vectors are smooth enough cylindrically symmetric functions, then our problem has a generalized cylindrically symmetric solution for a sufficiently small time interval. Here, we prove the uniqueness of this solution. Copyright © 2016 John Wiley & Sons, Ltd.

Global solution to 1D model of a compressible viscous micropolar heat-conducting fluid with a free boundary

In this paper we consider the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamically sense perfect and polytropic. The fluid is between a static solid wall and a free boundary connected to a vacuum state. We take the homogeneous boundary conditions for velocity, microrotation and heat flux on the solid border and that the normal stress, heat flux and microrotation are equal to zero on the free boundary. The proof of the global existence of the solution is based on a limit procedure. We define the finite difference approximate equations system and construct the sequence of approximate solutions that converges to the solution of our problem globally in time.

Global behavior for compressible viscous micropolar fluid with spherical symmetry

In this paper, we prove the regularity and exponential stability of solutions for a viscous micropolar fluid in the domain to be subset of R3 bounded with two concentric spheres that present the solid thermoinsulated walls. Some new ideas and more delicate estimates are introduced to prove these results.

Numerical analysis of the solutions for 1d compressible viscous micropolar fluid flow with different boundary conditions

The intention of this work is to concern the numerical solutions to the model of the nonstationary 1d micropolar compressible viscous and heat conducting fluid flow that is in the thermodynamical sense perfect and polytropic. The mathematical model consists of four partial differential equations, transformed from the Eulerian to the Lagrangian description, and which are associated with different boundary conditions. By using the finite difference scheme and the Faedo–Galerkin method we make different numerical simulations to the results of our problems. The properties of both numerical schemes are analyzed and numerical results are compared on the chosen test examples. The comparison of the numerical results on problems that have the homogeneous or the non-homogeneous boundary conditions for velocity and microrotation show good agreement of both approaches. However, the advantage of the used finite difference method over the Faedo–Galerkin method lies in the simple implementation of the non-homogeneous boundary conditions and in the possibility of approximation of the free boundary problem on which the Faedo–Galerkin method is not applicable.

3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Regularity of the solution

In this paper we consider the nonstationary 3-D flow of a compressible viscous and heat-conducting micropolar fluid, that is in the thermodynamical sense perfect and polytropic. The fluid domain is a subset of R3 bounded with two concentric spheres that present the solid thermoinsulated walls. The corresponding mathematical model is set up in the Lagrangian description. We assume that the initial data are spherically symmetric functions, and that the initial density and temperature are strictly positive. This problem has a unique spherically symmetric generalized solution globally in time. Here we introduce the Hölder continuous initial functions and prove that, for any T>0, the state function is also Hölder continuous.

Exponential stability of spherically symmetric solutions for compressible viscous micropolar fluid

This paper is concerned with the dynamics for the Navier-Stokes for a viscous micropolar fluid in bounded annular domains in R3. We prove that spherically symmetric solutions decay to a constant state exponentially as time goes to infinity.

3-D flow of a compressible viscous micropolar fluid with spherical symmetry: a global existence theorem

We consider the nonstationary 3-D flow of a compressible viscous heat-conducting micropolar fluid in the domain to be the subset of \(\mathbf{R}^{3}\) bounded with two concentric spheres that present the solid thermo-insulated walls. In the thermodynamical sense the fluid is perfect and polytropic. We assume that the initial density and temperature are bounded from below with a positive constant and that the initial data are sufficiently smooth spherically symmetric functions. The starting problem is transformed into the Lagrangian description on the spatial domain \(]0,1[\). In this work we prove that our problem has a generalized solution for any time interval \([0,T ]\), \(T\in\mathbf{R}^{+}\). The proof is based on the local existence theorem and the extension principle.

3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Large time behavior of the solution

We consider the non-stationary 3-D flow of a compressible viscous heat-conducting micropolar fluid in the domain that is the subset of R3 bounded with two concentric spheres that present solid thermo-insulated walls. Under the assumption that the fluid is perfect and polytropic in the thermodynamical sense as well as that the initial density and temperature are strictly positive and that the initial data are sufficiently smooth spherically symmetric functions, the corresponding problem with homogeneous boundary data has a unique generalized solution for any time interval [0,T], T∈R+. In this work we analyze large time behavior of the solution and prove its stabilization.