Search

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.

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 R 3 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.

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.

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.

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.

We consider the non-stationary 3-D flow of a compressible viscous heat-conducting micropolar fluid in the domain that is the subset of $\mathbf{; ; ; ; R}; ; ; ; ^3$ 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 $\left[0, T\right]$, $T\in\mathbf{; ; ; ; R}; ; ; ; ^+$. In this work we analyze large time behavior of the solution and prove its stabilization.

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 R 3 . 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.

Gaussove formule za numeričku integraciju se temelje na izračunu vrijednosti podintegralne funkcije u Gaussovim čvorovima i množenju tih vrijednosti s težinskim faktorima pri čemu su čvorovi neuniformno distribuirani. Tako se postiže veća točnost u odnosu na uniformnu distribuciju čvorova. U radu su detaljnije opisane Gauss-Legendre-Radauva i Gauss- Legendre-Lobattova integracijska formula, koje se koriste za numeričku integraciju funkcija kod kojih je barem jedan rub intervala integracije ujedno i čvor integracije. Navedene formule testiraju se na primjeru Runge funkcije.

We are dealing with the nonstationary 3D flow of a compressible viscous heat-conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic. It is assumed that the domain is a subset of R3 and that the fluid is bounded with two concentric spheres. The homogeneous boundary conditions for velocity, microrotation, heat flux, and spherical symmetry of the initial data are proposed. By using the assumption of the spherical symmetry, the problem reduces to the one-dimensional problem. The finite difference formulation of the considered problem is obtained by defining the finite difference approximate equation system. The corresponding approximate solutions converge to the generalized solution of our problem globally in time, which means that the defined numerical scheme is convergent. Numerical experiments are performed by applying the proposed finite difference formulation. We compare the numerical results obtained by using the finite difference and the Faedo–Galerkin approach and analyze the properties of the numerical solutions.

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 R 3 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 Holder continuous initial functions and prove that, for any T > 0 , the state function is also Holder continuous.

We consider the nonstationary 3-D flow of a compressible viscous and heat-conducting micropolar fluid bounded with two concentric spheres that present solid thermoinsulated walls. We assume that the fluid is perfect and polytropic in the thermodynamical sense, as well as that the initial density and temperature are strictly positive. We take sufficiently smooth spherically symmetric initial functions and analyze the corresponding problem with homogeneous boundary data. In this work we give the overview of the current progress in mathematical analysis of the described problem with particular emphasis on the existence theorems and the large time behavior of the solution.

Rješavanje Poissonove parcijalne diferencijalne jednadžbe primjenom Greenove funkcije analitički je postupak određivanja rješenja za dane rubne uvjete. U ovom radu dana je teoretska baza određivanja Greenove funkcije za zadanu parcijalnu diferencijalnu jednadžbu koja je definirana u domeni D, pri čemu mora zadovoljiti određene rubne uvjete na granici domene C. Za navedeni Dirichletov problem prikazano je rješenje Poissonove parcijalne diferencijalne jednadžbe, a na kraju su prikazana dva primjera., Obtaining a solution of the Poisson partial differential equation using Green’s function represents an analytic method for determining the solution for the defined boundary value problems. In this paper the theoretical basis for the Green’s function has been elaborated for the Poisson partial differential equation which is defined in domain D with boundary values on boundary C. The solution for the Poisson partial differential equation for the Dirichlet problem with two examples is shown.

We consider non-stationary 1-D flow of a compressible viscous and heat-conducting micropolar fluid, assuming that it is in thermodynamical sense perfect and polytropic. The homogeneous boundary conditions for velocity and microrotation, as well as non-homogeneous boundary conditions for temperature are assumed. Using the Faedo–Galerkin method we prove a local-in-time existence of a generalized solution.

An initial-boundary value problem for 1-D flow of a compressible viscous heat-conducting micropolar fluid is considered ; the fluid is assumed thermodynamically perfect and polytropic. By transforming the original problem into homogeneous one we prove a global-in-time existence theorem.

An initial-boundary value problem for 1-D flow of a compressible viscous heat-conducting micropolar fluid is considered ; the fluid is assumed thermodynamically perfect and polytropic. The original problem is transformed into homogeneous one and studied the Faedo-Galerkin method. A local-in-time existence of generalized solution is proved.

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.

We consider non-stationary 1-D flow of a compressible viscous heat-conducting micropolar fluid, assuming that it is in the thermodynamical sense perfect and polytropic. The homogeneous boundary conditions for velocity and microrotation, as well as non-homogeneous boundary conditions for temperature are introduced. This problem has a unique generalized solution locally in time. With the help of this result and using the principle of extension we prove a global-in-time existence theorem.

We consider 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 thermoinsulated walls. In thermodynamical sense fluid is perfect and polytropic. Assuming that the initial density and temperature are strictly positive we will prove that for smooth enough spherically symmetric initial data there exists a spherically symmetric generalized solution locally in time.

We consider the Cauchy problem for non-stationary 1-D flow of a compressible viscous and heat-conducting fluid, being in a thermodynamical sense perfect and polytropic. This problem has a unique generalized solution on Rx]0, T[, for each T > 0. Assuming that the initial functions are small perturbations of the constants and using some estimates for the solution independent of T, we analyze the behavior of the solution as time variable tends to infinity.

An initial-boundary value problem for 1D flow of a compressible viscous heat-conducting micropolar fluid is considered; the fluid is thermodynamically perfect and polytropic. Assuming that the initial data are Holder continuous on and transforming the original problem into homogeneous one, we prove that the state function is Holder continuous on , for each . The proof is based on a global-in-time existence theorem obtained in the previous research paper and on a theory of parabolic equations.

The Cauchy problem for one-dimensional flow of a compressible viscous heatconducting micropolar fluid is considered. It is assumed that the fluid is thermodynamically perfect and polytropic. This problem has a strong solution on R×]0, T[ for each T>0. We prove that the solution of the Cauchy problem is unique.

We consider nonstationary 1-D flow of a compressible and heat-conducting micropolar fluid, being in a thermodynamical sense perfect and polytropic. A corresponding initial-boundary value problem has a unique strong solution on ]0, 1[×]0, T[, for each T>0. By using this result we obtain vsome a priori estimates of the solution not depending on T and we prove that the solution of the nonstationary problem converges to stationary in the norm of (H^1(]0, 1[))^4 when T tends to infinity.

In this work we consider unsteady one-dimensional flow of compressible micropolar fluid whiich is in the termodynamical sense perfect and polytropic. Associated initial boundary value problem has unique solution on (0, 1)x(0, T), for all T>0. In considering the problem of stabilization the main difficulty is to obtian the estimate for the solution independent of T. Using some of ideas of Antontsev, Kazhykhov and Monakhov we derive the estimate.

We consider nonstationary 3-D flow of a compressible viscous heat-conducting micropolar fluid in the domain that is the subset of bounded with two concentric spheres that present the solid thermoinsulated walls. In the thermodynamical sense the fluid is perfect and polytropic. If we assume that the initial density and temperature are strictly positive and that the initial data are sufficiently smooth spherically symmetric functions then our problem has a generalized solution for a sufficiently small time interval. We study the problem in the Lagrangian description and prove the uniqueness of its generalized solution.