Global existence of strong solutions to compressible Navier–Stokes system with degenerate heat conductivity in unbounded domains

In one-dimensional unbounded domains, we prove the global existence of strong solutions to the compressible Navier–Stokes system for a viscous and heat conducting ideal polytropic gas, when the viscosity is a constant and the heat conductivity is proportional to a positive power of the temperature. Note that the conditions imposed on the initial data are the same as those of the constant heat conductivity case (Kazhikhov, A. V. Siberian Math. J. 23 [1982], 44-49) and can be arbitrarily large. Therefore, our result generalizes Kazhikhov's result for the constant heat conductivity case to the degenerate and nonlinear one.     

非常量导热系数几例微分方程的推立

在本文里,曾先后假设物体的导热系数是依直线和指数函数空间地起改变,就这样来建立了六个二阶热传导微分方程;又对于变密度、变比热、变导热系数这样的更一般的情况也推立了六个二阶热传导的微分方程.     

Study of hydromagnetic heat and mass transfer flow over an inclined heated surface with variable viscosity and electric conductivity

The effects of variable electric conductivity and temperature dependent viscosity on hydromagnetic heat and mass transfer flow along a radiate isothermal inclined permeable surface in a stationary fluid in the presence of internal heat generation (or absorption) are analyzed numerically presenting local similarity solutions for various values of the physical parameters. The research shows that the difference in the results between variable Prandtl number and constant Prandtl number are significant when fluid viscosity strongly dependents on the temperature. The results also show that skin friction coefficient, Nusselt number and Sherwood number are lower for the fluids of constant electric conductivity than those of the variable electric conductivity.     

Melting heat and mass transfer in stagnation point micropolar fluid flow of temperature dependent fluid viscosity and thermal conductivity at constant vortex viscosity

Steady mixed convection micropolar fluid flow towards stagnation point formed on horizontal linearly stretchable melting surface is studied. The vortex viscosity of micropolar fluid along a melting surface is proposed as a constant function of temperature while dynamic viscosity and thermal conductivity are temperature dependent due to the influence of internal heat source on the fluid. Similarity transformations were used to convert the governing equation into non-linear ODE and solved numerically. A parametric study is conducted. An analysis of the results obtained shows that the flow-field is influenced appreciably by heat source, melting, velocity ratio, variable viscosity and thermal conductivity.     

On rotational gas heating

An exact solution of the Navier-Stokes equations for a normal state viscous heat conduct ing gas (with constant viscosity and heat conductivity) is obtained in the form of a stationary plane-parallel flow in a cylinder; the gas is heated by self-rotation at the angular velocity that monotonically increases (or decreases) along the central axis.     

The effect of variable properties on the unsteady Couette flow with heat transfer considering the Hall effect

The influence of temperature dependent viscosity and thermal conductivity on the transient Couette flow with heat transfer is studied. An external uniform magnetic field is applied perpendicular to the parallel plates and the Hall effect is taken into consideration. The fluid is acted upon by a constant pressure gradient. The two plates are kept at two constant but different temperatures and the viscous and Joule dissipations are considered in the energy equation. A numerical solution for the governing non-linear equations of motion and the energy equation is obtained. The effect of the Hall term and the temperature dependent viscosity and thermal conductivity on both the velocity and temperature distributions is examined.     

GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE

We consider the quantum Navier-Stokes equations for the viscous, compressible, heat conducting fluids on the three-dimensional torus T³. The model is based on a system which is derived by Jungel, Matthes and Milisic [15]. We made some adjustment about the relation of the viscosities of quantum terms. The viscosities and the heat conductivity coefficient are allowed to depend on the density, and may vanish on the vacuum. By several levels of approximation we prove the global-in-time existence of weak solutions for the large initial data.     

GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE THREE-DIMENSIONAL FULL COMPRESSIBLE QUANTUM EQUATIONS

We consider the quantum Navier-Stokes equations for the viscous, compressible, heat conducting fluids on the three-dimensional torus T~3. The model is based on a system which is derived by Jungel, Matthes and Milisic [15]. We made some adjustment about the relation of the viscosities of quantum terms.The viscosities and the heat conductivity coefficient are allowed to depend on the density, and may vanish on the vacuum. By several levels of approximation we prove the global-in-time existence of weak solutions for the large initial data.     

Local sensitivity analysis for the heat flux-temperature integral relationship in the half-space

This paper presents insight into the heat flux-temperature (q″ ? T) integral relationship based on constant thermophysical properties. This relationship is often used in one-dimensional, transient heat transfer studies involving null-point calorimetry and heat flux investigations. This study focuses on a short transient studies where energy has not fully penetrated the body as the result of an imposed surface heating condition. A full nonlinear heat transfer model is developed involving a half-space planar region. Temperature results are then introduced into the constant property integral relationship and a newly derived Kirchoff integral relationship for retrieving the local heat flux. Good agreement is observed between the fully nonlinear results and locally linearized system. Additionally, a sensitivity study is presented which involves perturbing the average thermophysical properties of thermal conductivity and heat capacity.     

Unsteady state heat flow in epidermis and dermis of a human body

We investigate the unsteady state temperature distribution in human skin where subcutaneous tissues are not present. The mathematical model is employed for a onedimensional unsteady state case, taking the blood mass flow rate and metabolic heat generation variable with respect to the position in the dermis. The metabolic heat generation depends on the tissue temperature. The thermal conductivity is taken constant but different in two layers. The problem has been solved using Laplace transform and Bessel functions. Numerical results for a simple case are discussed.     

Solution of nonstationary heat-conduction problems for thermosensitive bodies under convective heat exchange

We propose a technique for solving nonstationary heat-conduction problems for thermosensitive bodies with simple nonlinearity (the coefficients of thermal conductivity and the heat capacity per unit volume depend on temperature, but the coefficient of thermal diffusivity is constant) heated by convective heat exchange from the surrounding medium. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 148–152.     

Self-similar convergence of a shock wave in a heat conducting gas

The problem of the convergence of a spherical shock wave (SW) to the centre, taking into account the thermal conductivity of the gas in front of the SW, is considered within the limits of a proposed approximate model of a heat conducting gas with an infinitely high thermal conductivity and a small temperature gradient, such that the heat flux is finite in a small region in front of the converging SW. In this model, there is a phase transition in the surface of the SW from a perfect gas to another gas with different constant specific heat and the heat outflow. The gas is polytropic and perfect behind the SW. Constraints are derived which are imposed on the self-similarity indices as a function of the adiabatic exponents on the two sides of the SW. In front of the SW, the temperature and density increase without limit. In the general case, a set of self-similar solutions with two self-similarity indices exists but, in the case of strong SW close to the limiting compression, there are two solutions, each of which is completely determined by the motion of the spherical piston causing the self-similar convergence of the SW.     

Analytical solutions for movement of cold water thermal front in a heterogeneous geothermal reservoir

In the present study an analytical model has been presented to describe the transient temperature distribution and advancement of the thermal front generated due to the reinjection of heat depleted water in a heterogeneous geothermal reservoir. One dimensional heat transport equation in porous media with advection and longitudinal heat conduction has been solved analytically using Laplace transform technique in a semi infinite medium. The heterogeneity of the porous medium is expressed by the spatial variation of the flow velocity and the longitudinal effective thermal conductivity of the medium. A simpler solution is also derived afterwards neglecting the longitudinal conduction depending on the situation where the contribution to the transient heat transport phenomenon in the porous media is negligible. Solution for a homogeneous aquifer with constant values of the rock and fluid parameters is also derived with an aim to compare the results with that of the heterogeneous one. The effect of some of the parameters involved, on the transient heat transport phenomenon is assessed by observing the variation of the results with different magnitudes of those parameters. Results prove the heterogeneity of the medium, the flow velocity and the longitudinal conductivity to have great influence and porosity to have negligible effect on the transient temperature distribution.     

Universal Attractor for a Penrose-Fife System with Special Heat Flux Law

A Penrose-Fife system for non isothermal phase transitions with non conserved order parameter is introduced. A linear growth of the latent heat density with respect to the phase field is allowed. Continous dependence on data and the existence of the universal attractor for the associated nonlinear semigroup are shown. These properties hold with respect to a strong metric accounting for the nonlinear and even singular terms characterizing the system. The present analysis extends a former result by the same authors, holding in the case of a constant latent heat.     

Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations

The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space R³ and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a different idea from [X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. (2008), in press] to show the compactness of solutions of viscous, compressible, heat conducting magnetohydrodynamic flows, derives a new entropy identity, and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations.     

L∞ continuation principle to the non‐baratropic non‐resistive magnetohydrodynamic equations without heat conductivity

The aim of this paper is to establish a continuation principle for strong solutions to the full compressible magnetohydrodynamic system without resistivity and heat conductivity. We prove that if the solution loses its regularity in finite time, the dominated part is due to the hyperbolic effect. More precisely, it is essentially shown that the strong solution exists globally if the density, temperature, and magnetic field are bounded from above, where vacuum is allowed to exist. This verifies a problem proposed by D.Serre. Copyright © 2016 John Wiley & Sons, Ltd.     

GLOBAL EXISTENCE AND CONVERGENCE RATES OF SMOOTH SOLUTIONS FOR THE 3-D COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITHOUT HEAT CONDUCTIVITY

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H3-norm of the initial perturbation around a constant states is small enough and its L1-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.     

Numerical study of dynamic thermal conductivity of nanofluid in the forced convective heat transfer

In this work, forced convective heat transfer of nanofluid in the developing laminar flow (entrance region) in a circular tube is considered. The nanofluid thermal conductivity, as an important parameter, is considered as two parts: static and dynamic part. Simulated results show that the dynamic part of nanofluid thermal conductivity due to the Brownian motion has a minor effect on the heat transfer coefficients, on the other hand, static part of thermal conductivity including nanolayer around nanoparticle has an important role in heat transfer.     

Solvability of the initial-boundary-value problem for the equations of motion of a viscous compressible fluid

It is proved that the initial-boundary-value problem for the system of equations describing the motion of a compressible fluid with a constant viscosity is locally solvable with respect to time. The heat conductivity is not taken into account. The solution is found in the class W _q ^2.1 , q>3.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 128–142, 1976.     

Global solutions of a radiative and reactive gas with self‐gravitation for higher‐order kinetics

The existence of global solutions is established for compressible Navier–Stokes equations by taking into account the radiative and reactive processes, when the heat conductivity κ (κ₁(1 + θ^q) ≤ κ ≤ κ₂(1 + θ^q),q ≥ 0), where θ is the temperature. This improves the previous results by enlarging the scope of q including the constant heat conductivity. Copyright © 2012 John Wiley & Sons, Ltd.