Internal pressure tensor

In the matrix model (Jongschaap, 1990), the global thermodynamic system is composed of two separate physical parts, which are called the environment and the internal variables. For the polymer solutions, for example, the pressure tensor Pv may be the internal variables, and the classical variables density, velocity, and internal energy are the environment variables. 

Here, the first equation is the usual Fourier law, the second relates the viscous pressure tensor to the internal variable W, and the last is the evolution of the internal variable. The matrix of the transport coefficients /.(/ is positive definite... 

Winkler, Morawitz, and Yoon reported MD simulations at constant pressure in a recent paper.The volume of the (periodic) box changed while the external pressure was kept constant the two parameters were related by a stress tensor. The authors derived a new instantaneous external stress tensor that allowed both volume and internal pressure to fluctuate during the course of the simulation. The technique did not require the use of artificial parameters or the extension of the phase space sampled. Time-averaged ensembles of various system properties were shown to be comparable to averages from isoenthalpic-isobaric ensembles. 

To start with, let us determine the stress and the deformation of a hollow sphere (outer radius J 2, inner radius R ) under a sudden increase in internal pressure if the material is elastic in compression but a standard solid (spring in series with a Kelvin-Voigt element) in shear (Fig. 16.1). As a consequence of the radial symmetry of the problem, spherical coordinates with the origin in the center of the sphere will be used. The displacement, obviously radial, is a function of r alone as a consequence of the fact that the components of the strain and stress tensors are also dependent only on r. As a consequence, the Navier equations, Eq. (4.108), predict that rot u = 0. Hence, grad div u = 0. This implies that... 

Case 3. Consider an infinite medium with a spherical cavity of radius R subjected to internal pressurization. If the boundary conditions P = p and = 0 re assumed, then the components of the stress tensor are... 

Equation III. 19 leads to an interpretation of cr+o as the pressure tensor of the medium with a contribution to the pressure tensor due to the internal degrees of freedom of the atoms. 

The stress tensor, as was done by Einstein, is obtained from the kinetic theory of galactic clusters, assumed to behave like dust in a fluid of average density p, internal pressure p and rms velocity v, ... 

In Ihe extended nonequilibrium Ihermodynamics for a binary liquid mixlure, Ihe viscous pressure tensor P and Ihe diffusion flux J are considered as addilional independenl variables. The viscous pressure tensor P, by Ihe simplesl Maxwell model, is defined by Eqn (14.30). In extended nonequilibrium Ihermodynamics of polymer solulions, Ihe generalized extended Gibbs equalion for a fluid characterized by internal energy u and viscous pressure P is... 

Equations (80a)-(80c) are called conservation equations, since their form is a direct consequence of the conservation of number of particles, momentum, and energy in the binary collisions taking place in the gas. In the phenomenological theories of fluid dynamics, equations in the form of Eqs. (80a)-(80c) are derived from the fact that mass, momentum and energy are conserved in the fluid, but in these theories one does not express the heat flow vector Jj- and the pressure tensor P in terms of a microscopic quantity, like the distribution function f(r, v, t). Instead, one relates Jr and P to n, u, and T by means of the so-called linear laws. ° For a one-component fluid with no internal structure, these linear laws are Fourier s law of heat conduction... 

The coupUng is implemented by allowing the matrix h, made up from the basis vectors, a, b and c which determine the shape of the primary dynanaics cell, to respond to imbalances between the internally measured pressure tensor and an externally applied pressure tensor. The equation for the rate of change of the h matrix with time is then defined as... 

Here Pa is the respective component of an external force (per unit mass) acting on the volume element. The second term on the right is the same force density component due to internal forces, where the stress tensor introduced in Eq.(1.3) is replaced by its negative—the pressure tensor. Multiplication with Va (including summation over a) and application of Eq. (7.83) yields... 

First we want to compute the internal entropy production diS/dt using Eqs.(7.78) and (7.105). There are no external forces, the interior of the system is homogeneous in the concentrations as well as temperature, and viscosity effects (off diagonal elements of the pressure tensor) are negligible. This means that only the last two terms in Eq. (7.105) must be included in the calculation. We begin with the activities for the two reactions ... 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

We notice that stress tensors are not a priori symmetric for (16) and that c)J. symmetric tensors. Further, the 3rd order microstress tensor Ss is normally related to boundary micro tractions, even if, in some cases, it could express weakly non-local internal effects % is interpreted as an externally controlled pore pressure (s includes interactive forces between the gross and fine structures. 

The pressure p includes both the partial pressure of the gas of Brownian particles n(N +1 )T and the partial pressure of the carrier monomer liquid. We shall assume that the viscosity of the monomer liquid can be neglected. The variables xt k in equation (9.19) characterise the mean size and shape of the macromolecular coils in a deformed system. The other variables ut k are associated mainly with orientation of small rigid parts of macromolecules (Kuhn segments). As a consequence of the mesoscopic approach, the stress tensor (9.19) of a system is determined as a sum of the contributions of all the macromolecules, which in this case can be expressed by simple multiplication by the number of macromolecules n. The macroscopic internal variables x -k and u"k can be found as solutions of relaxation equations which have been established in Chapter 7. However, there are two distinctive cases, which have to be considered separately. 

Anyhow, (1.255) is not properly closed yet as the pressure drop variable is still undetermined. Therefore, before we can apply (1.255) we need to parameterize the losses in terms of known flow parameters in pipes, valves, fittings, and other internal flow devices. Assuming that the viscous stress tensor reduces to a single shear stress component per unit wall surface (e.g., [102] ]185]), Apf per unit cross sectional area can be related directly to the friction drag force on the tube wall surface —Cwt DL. That is, since the friction drag force in a horizontal tube with a constant rate of flow is given by Z p/(- )D, the wall shear stress 3uelds ... 

The stress tensor can be expressed as the sum of an isotropic diagonal part, a normal stress described by a scalar pressure p, and a deviatory component or shear stress that appears as a result of internal friction between fluid layers moving relative to each other. In the simplest case of an incompressible Newtonian fluid the stress tensor is of the form... 

Tensor fields C. Z and 0 are symmetric the first one can be interpreted as an externally controlled pore pressure the second one, which does not necessarily sum to zero as internal forces do, includes interactive forces between the gross and fine structures as well as internal dissipative contributions due to the stir of the pores surface for the third one, one should notice that micromomentum is not necessarily conserved for the mixture. 

Yi = pjp is the mass fraction of species i and j, is its diffusirm mass flux, where the subscript /, i = 1,..N, denotes species i. The specific enthalpy is h = e + pip, where e denotes the specific internal energy and p is the pressure q is the mass specific heat flux, a is the extra stress tensor and d/dt denotes the material or substantial derivative. 

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