Viscous Heat-conducting Fluid Research Articles

Extended Thermodynamics of Rarefied Polyatomic Gases: 15-Field Theory Incorporating Relaxation Processes of Molecular Rotation and Vibration.

After summarizing the present status of Rational Extended Thermodynamics (RET) of gases, which is an endeavor to generalize the Navier–Stokes and Fourier (NSF) theory of viscous heat-conducting fluids, we develop the molecular RET theory of rarefied polyatomic gases with 15 independent fields. The theory is justified, at mesoscopic level, by a generalized Boltzmann equation in which the distribution function depends on two internal variables that take into account the energy exchange among the different molecular modes of a gas, that is, translational, rotational, and vibrational modes. By adopting the generalized Bhatnagar, Gross and Krook (BGK)-type collision term, we derive explicitly the closed system of field equations with the use of the Maximum Entropy Principle (MEP). The NSF theory is derived from the RET theory as a limiting case of small relaxation times via the Maxwellian iteration. The relaxation times introduced in the theory are shown to be related to the shear and bulk viscosities and heat conductivity.

Global Existence of the Three Dimensional Heat-conductive Incompressible Viscous Fluids

In this paper, we consider the Cauchy problem of non-stationary motion of heatconducting incompressible viscous fluids in ℝ3. About the heat-conducting incompressible viscous fluids, there are many mathematical researchers study the variants systems when the viscosity and heat-conductivity coefficient are positive. For the heat-conductive system, it is difficulty to get the better regularity due to the gradient of velocity of fluid own the higher order term. It is hard to control it. In order to get its global solutions, we must obtain the a priori estimates at first, then using fixed point theorem, it need the mapping is contracted. We can get a local solution, then applying the criteria extension. We can extend the local solution to the global solutions. For the two dimensional case, the Gagliardo-Nirenberg interpolation inequality makes use of better than the three dimensional situation. Thus, our problem will become more difficulty to handle. In this paper, we assume the coefficient of viscosity is a constant and the coefficient of heat-conductivity satisfying some suitable conditions. We show that the Cauchy problem has a global-in-time strong solution (u,θ) on ℝ3 ×(0, ∞).

Numerical Investigation of the Influence of the Geometric Dimensions of a Thermosyphon on the Efficiency of Heat Transfer

A numerical analysis of heat and mass transfer in a closed two-phase thermosyphon is performed with the use of the ANSYS FLUENT software package. The analysis is carried out within the framework of a mathematical model of a viscous heat-conducting incompressible fluid for the steam and condensate of a heat-transfer agent based on the convective and conductive mechanisms of heat transfer. The effective heat conductivity in a test region as a function of the height of the thermosyphon and the density of the heat flux on the bottom cover (which together characterize the influence of the longitudinal dimension on the efficiency of heat transfer in the test device) is obtained as a result of numerical modeling. It is established that the density of the thermal load on the bottom cover exerts a significant influence on the temperature drop, speed of the steam, and the efficiency of the thermosyphon.

Problems on control for a steady-state magnetic-hydrodynamic model of a viscous heat-conducting fluid under mixed boundary conditions

The control problems for steady-state equations of magnetic hydrodynamics (MHD) for a viscous heat-conducting fluid considered under mixed boundary conditions for the magnetic field and temperature are investigated. Their solvability is proved, the optimality systems describing the necessary conditions of an extremum are derived, and the theorems of local uniqueness and stability of the optimum solutions for explicit quality functionals are formulated.

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.

On existence, regularity and uniqueness of thermally coupled incompressible flows in a system of three dimensional pipes

We study an initial–boundary-value problem for time-dependent flows of heat-conducting viscous incompressible fluids in a system of three-dimensional pipes on a time interval (0,T). We are motivated by the bounded domain approach with “do-nothing” boundary conditions. In terms of the velocity, pressure and temperature of the fluid, such flows are described by a coupled parabolic system with strong nonlinearities and including the natural boundary conditions for the velocity and temperature of the fluid on the part of the boundary where the fluid is supposed to leave the channel. The present analysis is devoted to the proof of the existence, regularity and uniqueness of the solution for the problem described above.

Global Existence of Heat-Conductive Incompressible Viscous Fluids

In this paper, we consider the Cauchy problem of non-stationary motion of heat-conducting incompressible viscous fluids in R2$\mathbb{R}^{2}$, where the viscosity and heat-conductivity coefficient vary with the temperature. It is shown that the Cauchy problem has a unique global-in-time strong solution (u,ź)(x,t)$(u, \theta)(x,t)$ on R2×(0,ź)$\mathbb{R}^{2}\times(0,\infty)$, provided the initial norm źźu0źL2$\|\nabla u_{0}\|_{L^{2}}$ is suitably small, or the lower-bound of the coefficient of heat conductivity (i.e. ź_$\underline{\kappa}$) is large enough, or the derivative of viscosity (i.e. |μź(ź)|$|\mu'(\theta)|$) is small enough.

Global Well-Posedness for the Heat-conductive Incompressible Viscous Fluids

This paper is concerned with the Cauchy problem derived from the non-stationary motion of heat-conducting incompressible viscous fluids in three-dimensional whole space, where the viscosity and heat-conductivity coefficient vary with the temperature. We establish blow-up criteria and existence of global strong solution provided that the initial data is small enough.

Mixed Boundary Value Problems for Stationary Magnetohydrodynamic Equations of a Viscous Heat-Conducting Fluid

We consider the boundary value problem for stationary magnetohydrodynamic equations of electrically and heat conducting fluid under inhomogeneous mixed boundary conditions for electromagnetic field and temperature and Dirichlet condition for the velocity. The problem describes the thermoelectromagnetic flow of a viscous fluid in 3D bounded domain with the boundary consisting of several parts with different thermo- and electrophysical properties. The global solvability of the boundary value problem is proved and the apriori estimates of the solution are derived. The sufficient conditions on the data are established which provide a local uniqueness of the solution.

Direct pore-scale flow simulation using quasi-hydrodynamic equations

The use of the set of quasi-hydrodynamic equations for mathematical simulation of flows of a viscous heat-conducting compressible fluid in digital samples of rocks is considered. On the basis of a computation experiment, the permeability coefficient is determined.

Physical characteristics of a lava flow determined from thermal measurements at the lava’s surface

We consider the problem about determination of characteristics of a lava flow from the physical parameters measured on its surface. The problem is formulated as an inverse boundary problem for the model simulating the dynamics of a viscous heat-conducting incompressible inhomogeneous fluid, where, on the basis of additional data at one part of the model boundary, the missing conditions at another part of the boundary have to be determined, and then the characteristics of fluid in the entire model domain have to be reconstructed. The considered problem is ill-posed. We develop a numerical approach to the solution of the problem in the case of a steady-state flow. Assuming that the temperature and the heat flow are known at the upper surface of the lava, we determine the flow characteristics inside the lava. We compute model examples and show that the lava temperature and flow velocity can be determined with a high precision when the initial data are smooth or slightly noisy.

Numerical heat fluxcontrol forunsteady viscous fluid flows

This paper is devoted to the boundary control problems for non-stationary equations of viscous heat-conducting fluid. In these problems we change the fluid flow by heating or cooling some parts of the boundary. Numerical algorithm based on the finite dimensional minimization approach is proposed. Some computational results for vortex suppression in a cavity are presented and discussed.

High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids

This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics recently proposed by Peshkov & Romenski, denoted as HPR model. In that framework, the viscous stresses are computed from the so-called distortion tensor A, which is one of the primary state variables. A very important key feature of the model is its ability to describe at the same time the behavior of inviscid and viscous compressible Newtonian and non-Newtonian fluids with heat conduction, as well as the behavior of elastic and visco-plastic solids. This is achieved via a stiff source term that accounts for strain relaxation in the evolution equations of A. Also heat conduction is included via a first order hyperbolic evolution equation of the thermal impulse, from which the heat flux is computed. The governing PDE system is hyperbolic and fully consistent with the principles of thermodynamics. It is also fundamentally different from first order Maxwell-Cattaneo-type relaxation models based on extended irreversible thermodynamics. The connection between the HPR model and the classical hyperbolic-parabolic Navier-Stokes-Fourier theory is established via a formal asymptotic analysis in the stiff relaxation limit. From a numerical point of view, the governing partial differential equations are very challenging, since they form a large nonlinear hyperbolic PDE system that includes stiff source terms and non-conservative products. We apply the successful family of one-step ADER-WENO finite volume and ADER discontinuous Galerkin finite element schemes in the stiff relaxation limit, and compare the numerical results with exact or numerical reference solutions obtained for the Euler and Navier-Stokes equations. To show the universality of the model, the paper is rounded-off with an application to wave propagation in elastic solids.

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ТЕЧЕНИЯ ВЯЗКОЙ ЖИДКОСТИ ПО ТЕПЛОВЫМ ИЗМЕРЕНИЯМ НА ЕЕ ПОВЕРХНОСТИ

The viscous heat-conducting fluid flow characteristics are determined based on temperature measurements and heat flow on its daylight surface. The desired characteristics are temperature and fluid velocity in the whole model area. The problem is considered in a stationary setting and formalized as an inverse boundary problem for the model of high-viscosity incompressible fluid. A mathematical model of this fluid flow is described with the help of the Navier–Stokes equations for a Newtonian fluid in the Boussinesq approximation in a gravity field, incompressible fluid equation, and equation of the energy conservation with the appropriate boundary conditions. The fluid density and viscosity depend on the temperature. The considered problem is incorrect and does not possess the property of stability. Therefore, a small perturbation of the initial data on the accessible part of the border leads to uncontrolled errors in the determination of the unknown quantities in the model area. Conventional classical numerical methods are not suitable for solving the problem, which is why a variation method is used for its numerical solution, which reduces the solution of the original inverse problem to a series of solutions for stable problems. The Polak–Ribiere conjugate gradient method is used to minimize a merit functional in a variation method. This method steadily solves a corresponding extremal problem. The gradient of merit functional is defined analytically as a sequential solution of the direct and conjugate boundary problems. Direct and conjugate problems are numerically solved by the classical method of finite volumes. Constructed algorithms of numerical simulation are implemented in OpenFOAM software. The calculations of model problems are done

Mixed boundary value problems for the stationary magnetohydrodynamics model of a viscous heat-conducting fluid

The boundary value problem for the stationary magnetohydrodynamics model of a viscous heatconducting fluid considered under inhomogeneous mixed boundary conditions for an electromagnetic field and the temperature and Dirichlet condition for the velocity is investigated. This problem describes the flow of an electricaland heat-conducting liquid in a bounded three-dimensional domain the boundary of which consists of several parts with different thermoand electrophysical properties. Sufficient conditions imposed on the initial data to provide for global solvability of the problem and local uniqueness of its solution are established.

Weak solutions to the three-dimensional steady full compressible Navier–Stokes system

We study the motion of the steady compressible viscous heat-conductive fluid in a bounded three dimensional domain governed by the full Navier–Stokes system. For the pressure law of the form p(ρ,θ)=(γ−1)ρe(ρ,θ) the existence of weak solutions for any γ>54 is obtained. Our method relies upon the weighted estimates of both pressure and kinetic energy for the approximate system. Other tools such as effective viscous flux and oscillation defect measure developed by Lions and Feireisl respectively are also needed.

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.

3-D flow of a compressible viscous micropolar fluid with spherical symmetry: uniqueness of a generalized solution

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.

The global wellposedness of the 3D heat-conducting viscous incompressible fluids with bounded density

We consider the global wellposedness of the inhomogeneous incompressible heat-conducting viscous fluids in three dimension space. We generalize the result of Fujita & Kato for Navier–Stokes to the heat-conducting inhomogeneous incompressible viscous fluids. The key point is that we get the global wellposedness under the assumption that the initial density has positive lower and upper bound and the initial temperature can be arbitrarily large.