I-Shih Liu

Relativistic Thermodynamics of Gases

Abstract Relativistic thermodynamics of degenerate gases is presented here as a field theory of the 14 fields of particle density—particle flux, and stress—energy—momentum. The field equations are based on the conservation laws of particle numbers, and energy-momentum and on a balance of fluxes. The necessary constitutive equations are strongly restricted by the principle of relativity, entropy principle, requirement of hyperbolicity. It turns out that the resulting field equations contain only viscosity, bulk viscosity and heat conductivity as unknown functions. All other constitutive coefficients may be calculated from the equilibrium equations of state that are known from statistical arguments. The paper offers a more systematic version of relativistic thermodynamics of gases than the earlier papers by Muller and Israel. At the same time the present version contains less unknown functions than those earlier papers. All speeds of propagation are finite. The relation between the present theory and the classical one formulated by Eckart is described.

Extended Thermodynamics of Classical and Degenerate Ideal Gases

Extended thermodynamics is a field theory with the primary objective of determining the thirteen fields of mass, momentum and energy densities, stress deviator and heat flux. This distinguishes it from ordinary thermodynamics which knows only five basic fields, viz. the densities of mass, momentum and energy.

Effect of moving boundaries on the vibrating elastic string

A new derivation of a wave equation for small vibrations of elastic strings fastened at ends varying with time is presented. The model takes into account the change of length during the vibration and the nonlinear behavior of elastic strings in general. This model is a generalization of the Kirchhoff equation which contains a nonlinear term involving the displacement gradient. Numerical simulations of the model are based on finite difference approximations. Differences between linear and nonlinear aspects and the assumptions of numerical and theoretical analysis are briefly discussed and comparisons are made for linear and nonlinear elastic strings as well as the Kirchhoff model and the linear model without the term containing the displacement gradient.

On the entropy supply in a classical and a relativistic fluid

As long as one assumes that constitutive functions are independent of external supplies, the investigation of thermodynamic restrictions on such functions may as well be based on the consideration of supply-free bodies. In fact, it is desirable to base a constitutive theory upon the equations of balance for supply-free bodies, because thus one is spared the necessity of making a speci f ic assumpt ion on the form of the entropy supply. Therefore MiJLLER [1], [2], L1u [3] and ALTS & MOLLER [4] considered only supply-free bodies in their recent papers on the constitutive theory of simple heat-conducting bodies. However, once their results are known, it is possible to derive the expression of the entropy supply, if only this quantity is assumed to be linear in the supplies of momentum and energy. The purpose of this paper is to substantiate this last remark and to derive the form of the entropy supply for simple heat-conducting fluids, both in a classical and in a relativistic theory. In the latter case the entropy supply will be seen to depend on the external body force in general.

On Fourier's law of heat conduction

A linear theory of fluid is considered in which the gradients of density, internal energy and velocity are among the constitutive variables. Thus the heat flux may be a linear combination of the gradients of density and internal energy. It is proved that this linear combination may be written as the gradient of temperature so that Fouriers law of heat conduction holds.

On the transformation property of the deformation gradient under a change of frame

If the deformation gradients are denoted by F and F* respectively before and after a change of frame, they are related by the transformation formula, F*=QF, where Q is the orthogonal transformation associated with the change of frame. Although it has been pointed out that this relation is valid “provided that the reference configuration be unaffected by the change of frame” (see p. 308 of [1]), this formula is found in most textbook of Continuum Mechanics, and is used, without further justification, in deriving the condition of material frame-indifference, ℋ(QF)=Qℋ(F)QT for the constitutive function ℋ of the stress tensor of an elastic body. In this note, we shall analyze the effect of change of frame on the transformation property of the deformation gradient, and show that the above transformation formula is not valid in general. However, we shall confirm the validity of the above well-known condition of material frame-indifference without the assumption that the reference configuration be unaffected by the change of frame.

Hyperbolic system for viscous fluids and simulation of shock tube flows

A system of field equations for viscous fluids with heat conduction is formulated. Unlike the theory of Navier-Stokes fluids, a hyperbolic system of field equations in conservative form is obtained. Numerical simulation of a viscous shock tube flow is presented as an application.

A non-simple heat-conducting fluid

The constitutive quantities in thermodynamic theories of fluids are usually assumed to be given by functionals of the histories of density O, velocity v~ and temperature & In simple theories one tends to regard the constitutive quantities at a place x~ and time t as functions of O, v~, ~ and of the derivatives of v~ and at that place and time. Derivatives of O are never considered as variables although there does not seem to be an a priori reason for their omission. After this remark, it becomes an interesting problem to develop a theory where the constitutive quantities depend on the rate of density and on its gradient as well as on the other variables mentioned above. Such a theory is developed in this paper where, to make things simple, I disregard the derivatives of v~ as variables. The simple assumption that the heat flux vanish, if the temperature gradient does, will turn out to be sufficient to remove derivatives of e from all constitutive relations except the one for the pressure. Thus, apart from the expressions involving the pressure, the theory reduces to the theory of simple heat-conducting fluids presented in [1 ], and, in particular, the concept of the coldness is still valid in this more complex theory.

Successive linear approximation for finite elasticity

A method of successive Lagrangian formulation of linear approximation for solving boundary value problems of large deformation in finite elasticity is proposed. Instead of solving the nonlinear problem, by assuming time steps small enough and the reference configuration updated at every step, we can linearize the constitutive equation and reduce it to linear boundary value problems to be solved successively with incremental boundary data. Moreover, nearly incompressible elastic body is considered as an approximation to account for the condition of incompressibility. For the proposed method, numerical computations of pure shear of a square block for Mooney-Rivlin material are considered and the results are compared with the exact solutions. Mathematical subject classification: Primary: 65C20; Secondary: 74B20.

A note on material symmetry

In formulating mathematical models for materials, we describe the symmetry of a material usually in the following way: Choose a reference configuration for the material and determine the response functions of the material relative to the reference configuration. Now change the reference configuration into a new one and determine the new response functions. Suppose that they turn out to be exactly the same as those before the change. Then the change of reference configuration is called a material symmetry transformation, and the set of all material symmetry transformations, which form a group in a natural way, is called the symmetry group of the material relative to the chosen reference configuration. The preceding description of material symmetry is reasonable from the physical standpoint, since if the response functions remain unchanged under a change of reference configuration, then as far as the response of the material is concerned, the reference configurations before and after the change are indistinguishable. In other words a material symmetry transformation is a transformation that preserves the response of the material. We may apply the preceding concept of symmetry separately to varying response of the material. For instance it is quite possible for a material to have the same mechanical response but different optical response or thermal response relative to two reference configurations. Then the change of reference configuration involved is a material symmetry transformation with respect to mechanical response but is not one with respect to optical response or to thermal response. In continuum mechanics the (mechanical) response of a material point is usually taken to be the stress tensor at that point in the deformed configuration. For example an elastic material point is defined by the constitutive equation