Compressible Heat-conducting Fluid Research Articles

Homogenization of a Thermoelastic Bristly Structure Immersed in a Thermofluid

The article considers the mathematical model describing the joint motion of a viscous compressible heat-conducting fluid and a thermoelastic plate with a fine two-level thermoelastic bristly microstructure attached to it. The bristly microstructure consists of a great amount of taller and shorter bristles, which are periodically located on the surface of the plate, and the model under consideration incorporates a small parameter, which is the ratio of the characteristic lengths of the microstructure and the entire plate. Using classical methods in the theory of partial differential equations, we prove that the initial-boundary value problem for the considered model is well-posed. After this, we fulfill the homogenization procedure, i.e., we pass to the limit as the small parameter tends to zero, and, as a result, we derive the effective macroscopic model in which the dynamics of the interaction of the ‘liquid–bristly structure’ is described by equations of two homogeneous thermoviscoelastic layers with memory effects. The homogenization procedure is rigorously justified by means of the Allaire–Briane three-scale convergence method. The developed effective macroscopic model can potentially find application in further mathematical modeling in biotechnology and bionics taking account of heat transfer.

Navier-Stokes-Fourier fluids interacting with elastic shells

We study the motion of a compressible heat-conducting fluid in three dimensions interacting with a non-linear flexible shell. The fluid is described by the full Navier--Stokes--Fourier system. The shell constitutes an unknown part of the boundary of the physical domain of the fluid and is changing in time. The solid is described as an elastic non-linear shell of Koiter type; in particular it possesses a non-convex elastic energy. % Its deformation is modelled by a (non-linear) elastic Koiter shell. We show the existence of a weak solution to the corresponding system of PDEs which exists until the moving boundary approaches a self-intersection or the non-linear elastic energy of the shell degenerates. It is achieved by compactness results (in highest order spaces) for the solid-deformation and fluid-density. Our solutions comply with the first and second law of thermodynamics: the total energy is preserved and the entropy balance is understood as a variational inequality.

Compressible Flows with Oscillating Boundaries

The flow of a viscous compressible heat-conducting fluid in a plane channel with rigid oscillating walls is considered. It is shown that even a vanishingly small compressibility can be the reason for the onset of a resonance, i.e., sharp increase in the amplitude of oscillations of the flow parameters at a properly selected wall oscillation frequency. An exact analytical expression for the leading resonant frequency is given. Numerical calculations have demonstrated the possibility of a cumulative effect, namely, sharp increase in the mass flow rate even at constant pressure gradient.

Stability with respect to domain of the low Mach number limit of compressible heat-conducting viscous fluid

We investigate the asymptotic limit of solutions to the Navier-Stokes-Fourier system with the Mach number proportional to a small parameter $\rightarrow 0$, the Froude number proportional to $\sqrt{}$ and when the fluid occupies large domain with spatial obstacle of rough surface varying when $\rightarrow 0$. The limit velocity field is solenoidal and satisfies the incompressible Oberbeck–Boussinesq approximation. Our studies are based on weak solutions approach and in order to pass to the limit in a convective term we apply the spectral analysis of the associated wave propagator (Neumann Laplacian) governing the motion of acoustic waves.

BICOMPACT SCHEMES FOR COMPRESSIBLE NAVIER–STOKES EQUATIONS

For the first time, bicompact schemes are generalized to non-stationary Navier–Stokes equations for a compressible heat-conducting fluid. The proposed schemes have an approximation of the fourth order in space and the second order in time, are absolutely stable (in the frozen-coefficients sense), conservative, and efficient. One of the new schemes is tested on several two-dimensional problems. It is shown that when the mesh is refined, the scheme converges with an increased third order. A comparison is made with the WENO5-MR scheme. The superiority of the chosen bicompact scheme in resolving vortices and shock waves, as well as their interaction, is demonstrated.

Well-posedness and exponential decay for the Navier–Stokes equations of viscous compressible heat-conductive fluids with vacuum

This paper is concerned with the Cauchy problem of Navier–Stokes equations for compressible viscous heat-conductive fluids with far-field vacuum at infinity in [Formula: see text]. For less regular data and weaker compatibility condition than those proposed by Cho–Kim [Existence results for viscous polytropic fluids with vacuum, J. Differ. Equ. 228 (2006) 377–411], we first prove the existence of local-in-time solutions belonging to a larger class of functions in which the uniqueness can be shown to hold. The local solution is in fact a classical one away from the initial time, provided the initial density is more regular. We also establish the global well-posedness of classical solutions with large oscillations and vacuum in the case when the initial total energy is suitably small. The exponential decay estimates of the global solutions are obtained.

Разрешимость нестационарных уравнений трехмерного движения теплопроводных вязких сжимаемых двухкомпонентных жидкостей

Рассматриваются уравнения, описывающие трехмерные нестационарные движения смесей теплопроводных вязких сжимаемых жидкостей в рамках многоскоростного подхода. Доказана теорема существования, в целом по времени и входным данным, обобщенного (диссипативного) решения начально-краевой задачи, соответствующей течениям в ограниченной области. Библиография: 36 наименований.

Solubility of unsteady equations of the three-dimensional motion of two-component viscous compressible heat-conducting fluids

We consider equations for the three-dimensional unsteady motion of mixtures of viscous compressible heat-conducting fluids in the multi-velocity approach. We prove the existence, globally in time and the input data, of a generalized (dissipative) solution of the initial-boundary value problem corresponding to flows in a bounded domain.

On the global unique solvability of a problem for one-dimensional equations of dynamics of mixtures of viscous compressible heat-conducting fluids

One-dimensional equations of the multi-velocity model of the dynamics of mixtures of viscous compressible heat-conducting fluids and their approximation are considered. The existence and uniqueness of a strong solution to the initial-boundary value problem is proved globally in time.

Solvability of a regularized boundary-value problem for the system of equations of dynamics of mixtures of viscous compressible heat-conducting fluids

Solvability of a regularized boundary-value problem for the system of equations of dynamics of mixtures of viscous compressible heat-conducting fluids

Dissipation Limit for the Compressible Navier-Stokes to Euler Equations in One-Dimensional Domains

We prove that as the viscosity and heat-conductivity coefficients tend to zero, respectively, the global solution of the Navier-Stokes equations for one-dimensional compressible heat-conducting fluids with centered rarefaction data of small strength converges to the centered rarefaction wave solution of the corresponding Euler equations uniformly away from the initial discontinuity.

Steady compressible heat-conductive fluid with inflow boundary condition

In this paper, we study strong solutions to the steady compressible heat-conductive fluid near a non-zero constant flow with the Dirichlet boundary condition for the velocity on the inflow and outflow part of the boundary. We also consider the Dirichlet boundary condition for the temperature, and we do not need the thermal conductivity coefficient κ to be large. The existence of strong solutions is established for any Reynolds number and Mach number in the framework of perturbation.

Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficients

We consider the Navier–Stokes equations for compressible heat-conducting ideal polytropic gases in a bounded annular domain when the viscosity and thermal conductivity coefficients are general smooth functions of temperature. A global-in-time, spherically or cylindrically symmetric, classical solution to the initial boundary value problem is shown to exist uniquely and converge exponentially to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent γ. The initial data can be large if γ is sufficiently close to 1. These results are of Nishida–Smoller type and extend the work (Liu et al. (2014) [16]) restricted to the one-dimensional flows.

One-dimensional model and solutions for creeping gas flows in the approximation of uniform pressure.

A model, along with analytical and numerical solutions, is presented to describe a wide variety of one-dimensional slow flows of compressible heat-conductive fluids. The model is based on the approximation of uniform pressure valid for the flows, in which the sound propagation time is much shorter than the duration of any meaningful density variation in the system. The energy balance is described by the heat equation that is solved independently. This approach enables the explicit solution for the fluid velocity to be obtained. Interfacial and volumetric heat and mass sources as well as boundary motion are considered as possible sources of density variation in the fluid. A set of particular tasks is analyzed for different motion sources in planar, axial, and central symmetries in the quasistationary limit of heat conduction (i.e., for large Fourier number). The analytical solutions are in excellent agreement with corresponding numerical solutions of the whole system of the Navier-Stokes equations. This work deals with the ideal gas. The approach is also valid for other equations of state.

Boundary-layer phenomena for the cylindrically symmetric Navier–Stokes equations of compressible heat-conducting fluids with large data at vanishing shear viscosity

This paper concerns the asymptotic behavior of the solution to an initial-boundary value problem of the cylindrically symmetric Navier–Stokes equations with large data for compressible heat-conducting ideal fluids, as the shear viscosity μ goes to zero. A suitable corrector function (the so-called boundary-layer type function) is constructed to eliminate the disparity of boundary values. As by-products, the convergence rates of the derivatives in L2 are obtained and the boundary-layer thickness (BL-thickness) of the value with is shown by an alternative method, compared with the results proved in Jiang and Zhang (2009 SIAM J. Math. Anal. 41 237–68) and Qin et al (2015 Arch. Ration. Mech. Anal. 216 1049–86).

Mixing rules and the Navier-Stokes-Cahn-Hilliard equations for compressible heat-conductive fluids

The framework of this article is the compressible Navier-Stokes-Cahn- Hilliard system for the dynamics of a fluid whose two phases are macroscopically immiscible; partial mixing is permitted leading to narrow transition layers. This so-called NSCH model was originally derived by Lowengrub and Truskinowsky [7], but only for the isothermal case. The purpose of this work is to present the non-isothermal version as well as a well-posedness result. The PDEs constitute a strongly coupled hyperbolicparabolic system.

Time-periodic solutions to the full Navier–Stokes–Fourier system with radiation on the boundary

The Navier–Stokes–Fourier system is a well established model to describe motion of viscous compressible heat-conducting fluids. We study the existence of time-periodic weak solutions and improve the result from [4] in the following sense: we extend the class of pressure functions (id est, consider lower exponent γ) and include also the effect of radiation on the boundary.

Existence of strong solutions to the steady Navier–Stokes equations for a compressible heat-conductive fluid with large forces

We prove that there exists a strong solution to the Dirichlet boundary value problem for the steady Navier–Stokes equations of a compressible heat-conductive fluid with large external forces in a bounded domain Ω⊂Rd (d=2,3), provided that the Mach number is appropriately small. At the same time, the low Mach number limit is rigorously verified. The basic idea in the proof is to split the equations into two parts, one of which is similar to the steady incompressible Navier–Stokes equations with large forces, while another part corresponds to the steady compressible heat-conductive Navier–Stokes equations with small forces. The existence is then established by dealing with these two parts separately, establishing uniform in the Mach number a priori estimates and exploiting the known results on the steady incompressible Navier–Stokes equations.

Solubility of a stationary boundary-value problem for the equations of motion of a one-temperature mixture of viscous compressible heat-conducting fluids

We consider a boundary-value problem describing the stationary motion of a two-component mixture of viscous compressible heat-conducting fluids in a bounded domain. We make no simplifying assumptions except for postulating the coincidence of phase temperatures (which is physically justified in certain situations), that is, we retain all summands in equations that are a natural generalization of the Navier–Stokes–Fourier model of the motion of a one-component medium. We prove the existence of weak generalized solutions of the problem.

Global Smooth Solutions in $\mathbb{R}^3$ to Short Wave-Long Wave Interactions Systems for Viscous Compressible Fluids

The short wave-long wave interactions for viscous compressible heat conductive fluids is modeled, following Dias and Frid [SIAM J. Math. Anal., 43 (2011), pp. 764--787], by a Benney-type system coupling Navier--Stokes equations with a nonlinear Schrödinger equation along particle paths. We study the global existence of smooth solutions to the Cauchy problem in $\mathbb{R}^3$ when the initial data are small smooth perturbations of an equilibrium state.