Viscous limits for piecewise smooth solutions to systems of conservation laws

In this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity. It is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order as the viscosity coefficient goes to zero. The proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles.     

带冷却气流的亥姆霍兹共振器的声类比模型

亥姆霍兹共振器(HR)作为典型的被动消声装置,常被安装于航空发动机和燃气轮机的燃烧室上用以吸收噪声进而抑制燃烧热声振荡.在实际应用中,为防止燃烧室内高温气体损坏HR,常引入冷却气流从HR的背部空腔通过其颈部流入燃烧室,以保护HR.该冷却气流的温度一般显著低于燃烧室内的燃气温度.将这样的HR安装到燃烧室上时,该温差可能影...     

Structure of detonation waves,ionizing a gas,in the presence of an electromagnetic field. Cases of low magnetic viscosity

If behind a detonation wave, ionizing a gas, the magnetic Reynolds number is much greater than unity, then in order to describe such waves (just as for ionizing shock waves) complementary relations [1, 2] are necessary. These complementary relations are not the consequence of the basic integral laws, but can be found from a consideration of the wave structure. In [2], the structure of detonation waves, ionizing a gas, was investigated in an oblique magnetic field. It was supposed that the flow in a layer representing the structure is determined by the finite rate of the chemical reaction and the finite electrical conductivity. In the case when the characteristic length of the chemical reaction is much less than the characteristic dissipation length of the magnetic field, the complementary relations which ensure the existence of the structure are obtained in explicit form. The case is considered below when the characteristic length of the chemical reaction is much greater than the dissipation length of the magnetic field. In this case, the complementary relations are obtained in the explicit form.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 95–101, May–June, 1976.     

Non-Classical Shocks and Kinetic Relations: Scalar Conservation Laws

This paper analyzes the non-classical shock waves which arise as limits of certain diffusive-dispersive approximations to hyperbolic conservation laws. Such shocks occur for non-convex fluxes and connect regions of different convexity. They have negative entropy dissipation for a single convex entropy function, but not all convex entropies, and do not obey the classical Oleinik entropy criterion. We derive necessary conditions for the existence of non-classical shock waves, and construct them as limits of traveling-wave solutions for several diffusive-dispersive approximations. We introduce a “kinetic relation” to act as a selection principle for choosing a unique non-classical solution to the Riemann problem. The convergence to non-classical weak solutions for the Cauchy problem is investigated. Using numerical experiments, we demonstrate that, for the cubic flux-function, the Beam-Warming scheme produces non-classical shocks while no such shocks are observed with the Lax-Wendroff scheme. All of these results depend crucially on the sign of the dispersion coefficient. (Accepted February 8, 1996)     

The effects of time-dependent dissipation on the basins of attraction for the pendulum with oscillating support

We consider a pendulum with vertically oscillating support and time-dependent damping coefficient which varies until reaching a finite final value. Although it is the final value which determines which attractors eventually exist, the sizes of the corresponding basins of attraction are found to depend strongly on the full evolution of the dissipation. In particular, we investigate numerically how dissipation monotonically varying in time changes the sizes of the basins of attraction. It turns out that, in order to predict the behaviour of the system, it is essential to understand how the sizes of the basins of attraction for constant dissipation depend on the damping coefficient. For values of the parameters where the systems can be considered as a perturbation of the simple pendulum, which is integrable, we characterise analytically the conditions under which the attractors exist and study numerically how the sizes of their basins of attraction depend on the damping coefficient. Away from the perturbation regime, a numerical study of the attractors and the corresponding basins of attraction for different constant values of the damping coefficient produces a much more involved scenario: changing the magnitude of the dissipation causes some attractors to disappear either leaving no trace or producing new attractors by bifurcation, such as period doubling and saddle-node bifurcation. Finally, we pass to the case of an initially non-constant damping coefficient, both increasing and decreasing to some finite final value, and we numerically observe the resulting effects on the sizes of the basins of attraction: when the damping coefficient varies slowly from a finite initial value to a different final value, without changing the set of attractors, the slower the variation the closer the sizes of the basins of attraction are to those they have for constant damping coefficient fixed at the initial value. Furthermore, if during the variation of the damping coefficient attractors appear or disappear, remarkable additional phenomena may occur. For instance, it can happen that, in the limit of very large variation time, a fixed point asymptotically attracts the entire phase space, up to a zero-measure set, even though no attractor with such a property exists for any value of the damping coefficient between the extreme values.     

Simulation of Traveling Solitary Plane Waves

The possibility of propagation of solitary plane waves with the Whittaker profile in materials with a microstructure (composites) is discussed. Solitary waves are defined as aperiodic smooth waves with an initial profile that is equal to zero everywhere except for some finite interval. Functions with indices 0.0, 0.1, –1/4, and 1/4 are chosen for computer simulation. It is observed that with some restrictions on the time or distance of propagation in the material, two modes of the traveling wave with the Whittaker profile and different phase-dependent phase velocities propagate simultaneously. The discussion section focuses attention on the conditions of blanking of the second mode for small values of the phase     

Large-scale instability of a fine-grained turbulent jet

The initial growth of a large scale perturbation on a fine-grained turbulent jet is studied via linear stability analysis. The turbulent jet is assumed to be homogeneous and isotropic with zero mean shear, and the inviscid stream outside the jet has a uniform velocity profile. The incremental Reynolds stress caused by the large scale perturbation is modeled by a viscoelastic constitutive equation, following the analysis of Crow (1968). It is found that the jet is always unstable to both sinuous and varicose types of perturbation, with the sinuous mode having a larger growth rate. In particular, short waves are always amplified, in contrast to the short wave stabilization at low speed found by Townsend (1966) for a purely elastic jet. The growth rates of these short waves are finite, and are smaller than those for the classical Kelvin-Helmholtz instability of an inviscid jet, but larger than those for the Hooper-Boyd (1983) instability of a viscous jet with continuous velocity profile.     

Steady mixed convection flow of Maxwell fluid over an exponentially stretching vertical surface with magnetic field and viscous dissipation

The steady mixed convection flow and heat transfer from an exponentially stretching vertical surface in a quiescent Maxwell fluid in the presence of magnetic field, viscous dissipation and Joule heating have been studied. The stretching velocity, surface temperature and magnetic field are assumed to have specific exponential function forms for the existence of the local similarity solution. The coupled nonlinear ordinary differential equations governing the local similarity flow and heat transfer have been solved numerically by Chebyshev finite difference method. The influence of the buoyancy parameter, viscous dissipation, relaxation parameter of Maxwell fluid, magnetic field and Prandtl number on the flow and heat transfer has been considered in detail. The Nusselt number increases significantly with the Prandtl number, but the skin friction coefficient decreases. The Nusselt number slightly decreases with increasing viscous dissipation parameter, but the skin friction coefficient slightly increases. Maxwell fluid reduces both skin friction coefficient and Nusselt number, whereas buoyancy force enhances them.     

Thermoelasticity without energy dissipation

This paper deals with thermoelastic material behavior without energy dissipation; it deals with both nonlinear and linear theories, although emphasis is placed on the latter. In particular, the linearized theory of thermoelasticity discussed possesses the following properties: (a) the heat flow, in contrast to that in classical thermoelasticity characterized by the Fourier law, does not involve energy dissipation; (b) a constitutive equation for an entropy flux vector is determined by the same potential function which also determines the stress; and (c) it permits the transmission of heat as thermal waves at finite speed. Also, a general uniqueness theorem is proved which is appropriate for linear thermoelasticity without energy dissipation.     

Entropy dissipation scheme and minimums‐increase‐and‐maximums‐decrease slope limiter

In this paper, we continue to study the entropy dissipation scheme developed in former. We start with a numerical study of the scheme without the entropy dissipation term on the linear advection equation, which shows that the scheme is stable and numerical dissipation and numerical dispersion free for smooth solutions. However, the numerical results for discontinuous solutions show nonlinear instabilities near jump discontinuities. This is because the scheme enforces two related conservation properties in the computation. With this study, we design a so‐called ‘minimums‐increase‐and‐maximums‐decrease’ slope limiter in the reconstruction step of the scheme and delete the entropy dissipation in the linear fields and reduce the entropy dissipation terms in the nonlinear fields. Numerical experiments show improvements of the designed scheme compared with the results presented in former. However, the minimums‐increase‐and‐maximums‐decrease limiter is still not perfect yet, and better slope limiters are still sought. Copyright © 2011 John Wiley & Sons, Ltd.     

基于涡粘性模型的翼型湍流流场熵产率计算

利用有限体积法实现了基于非正交同位网格的SIMPLE算法。基于熵分析方法,采用涡粘性模型求解湍流熵产方程,系统研究了湍流模型对二维翼型绕流流场熵产率的影响。通过计算NACA0012翼型在来流雷诺数为2.88×10⁶时,0°攻角~16.5°攻角范围内的翼型表面压力系数分布和升阻力特性,验证了算法及程序的正确性。结果表明,选择不同湍流模型时,翼型流场熵产的计算结果存在差异,湍流耗散是引起流场熵产的主要原因;翼型流场的熵产主要发生在翼型前缘区、壁面边界层和翼型尾流区域,流场熵产率与翼型阻力系数线性相关;当产生分离涡时,粘性耗散引起的熵产下降。     

两微极磁热弹性固体分界面上波的反射与折射

给出了磁场、热场和弹性场多场耦合作用下微极广义热弹性固体的一般控制方程.该方 程既包含了磁场、热场和弹性场的耦合作用,又在其广义热传导方程中涵盖了耦合热弹理论 (C-D)及其5类推广(L-S理论,G-L理论,G-N(II,III)理论和C-T理论).运用该微极广义磁热 弹性控制方程,研究了在定常磁场作用下, 具有均匀初始温度的两理想接触微极弹性介质平面分界面上磁热弹性波的反射和折射现象.给出了分别在缺少磁场、热场作用或不同广义热传 导理论下反射或折射热波、纵向位移波、耦合横向和微旋转波与入射纵向位移波的振幅比随 入射角变化的关系曲线.对缺少磁、热和微极性以及热松弛时间时对应的反射、折射系数进 行了对比.结果表明磁、热和微极性以及热松弛时间对振幅比均有不同程度的影 响,与磁、热和微极性一样,热松弛时间对不同类型波的影响能力差别明显,但对同 一类型的反射波和折射波的影响相似.     

有限长Timoshenko梁弹性碰撞接触瞬间的动态特性

给出了质点与有限长Timoshenko梁横向弹性碰撞接触问题的半解析解,分析了该碰撞问题在碰撞接触瞬间的动态响应特性：揭示了其中的波传播现象．     

多孔材料中声波的传播与演化

采用两相多孔介质的拉格朗日模型来描述一种理论流体充填的多孔弹性固体材料,其中孔隙度的变化满足一个附加的平衡方程。     

Entropy analysis in electrical magnetohydrodynamic(MHD) flow of nanofluid with effects of thermal radiation,viscous dissipation,and chemical reaction

The unsteady mixed convection flow of electrical conducting nanofluid and heat transfer due to a permeable linear stretching sheet with the combined effects of an electric field, magnetic field, thermal radiation, viscous dissipation, and chemical reaction have been investigated. A similarity transformation is used to transform the constitutive equations into a system of nonlinear ordinary differential equations.The resultant system of equations is then solved numerically using implicit finite difference method.The velocity, temperature, concentration, entropy generation, and Bejan number are obtained with the dependence of different emerging parameters examined. It is noticed that the velocity is more sensible with high values of electric field and diminished with a magnetic field. The radiative heat transfer and viscous dissipation enhance the heat conduction in the system. Moreover, the impact of mixed convection parameter and Buoyancy ratio parameter on Bejan number profile has reverse effects. A chemical reaction reduced the nanoparticle concentration for higher values.     

Entropy and Global Existence for Hyperbolic Balance Laws

This paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables. We propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state. In addition, we show that a system of balance laws satisfies a Kawashima condition if and only if its first-order approximation, namely the hyperbolic-parabolic system derived through the Chapman-Enskog expansion, satisfies the corresponding Kawashima condition. The result is then applied to Bouchuts discrete velocity BGK models approximating hyperbolic systems of conservation laws.     

Thomson and rotation effects during photothermal excitation process in magnetic semiconductor medium using variable thermal conductivity

This study investigates a strong magnetic field acting over an elastic rotator semiconductor medium. The Thomson effect due to the magnetic field during the photothermal transport process is studied, and the thermoelectricity theory is used to explain the behavior of waves in the homogenous and isotropic medium under the effect of variable thermal conductivity. The variable thermal conductivity is considered as a linear function of the temperature. The two-dimensional deformation equations are u...     

Spatially polarized structure of detonation waves ionizing a gas

Additional relationships must be used [1–3], in addition to those following from the main integral laws, in describing ionizing detonation waves, exactly as for ionizing shocks. These additional relationships are obtained from the requirement for the existence of wave structure. The structure of detonation waves ionizing a gas in an oblique magnetic field was investigated in [1, 2]. The case of a plane-polarized structure was considered, when the velocity vector and the magnetic field lie in a plane passing through the normal to the front. The structure of ionizing detonation waves is studied in this paper for the case when the wave is spatially polarized and both transverse magnetic field components vary in the structure. It is considered that the magnetic viscosity and a quantity reciprocal to the chemical reaction rate are much greater than the remaining dissipative coefficients in the layer representing the structure. Conditions for the existence of such a spatial structure are clarified. Plane-polarized ionizing detonation waves whose structure is not planar are also considered. When the characteristic length of magnetic field dissipation is much greater or much less than the characteristic length of the chemical reaction, the additional relationships assuring the existence of structure are written down explicitly or are investigated qualitatively.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 166–169, November–December, 1976.     

Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy

We consider the Cauchy problem for a general one-dimensional n×n hyperbolic symmetrizable system of balance laws. It is well known that, in many physical examples, for instance for the isentropic Euler system with damping, the dissipation due to the source term may prevent the shock formation, at least for smooth and small initial data. Our main goal is to find a set of general and realistic sufficient conditions to guarantee the global existence of smooth solutions, and possibly to investigate their asymptotic behavior. For systems which are entropy dissipative, a quite natural generalization of the Kawashima condition for hyperbolic-parabolic systems can be given. In this paper, we first propose a general framework for this kind of problem, by using the so-called entropy variables. Then we go on to prove some general statements about the global existence of smooth solutions, under different sets of conditions. In particular, the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension. Our main tools will be some refined energy estimates and the use of a suitable version of the Kawashima condition.