Pressure tensor

In a simulation [19] the pressure tensor is obtained from the virial theorem [78]... 

Sinee our system in the slit is anisotropie and inhomogeneous, it makes sense to eonsider the loeal pressure tensor, Eq. (42), whieh depends on the distanee z from the adsorbing wall. If one defines Px = Pxx Py = Pyy ... 

Pz = Pzz as well as the total pressure Pjot = Px + Py + Pz, then for symmetry reasons Px = Py, and this symmetry holds with good aeeuraey in our data (Fig. 20(e)). Generally one finds from the simulation that the total pressure follows the density profile. The pressure tensor Pap(z) is very useful... 

FIG. 20 (a) Density profiles p(z) vs z for e = —2 and four average bulk densities (f> as indicated, (b) Surface excess vs density in the bulk for four choices of e. (c) Profiles for the diagonal components of the pressure tensor and of the total pressure for (p = l.O and e = —2. Insert in (c) shows the difference between P, and Px to show that isotropic behavior in the bulk of the film is nicely obtained, (d) Interfacial tension between the polymer film and the repulsive wall vs bulk density for all four choices of e. Curve is only a guide for the eye [18]. 

Bilayers have received even more attention. In the early studies, water has been replaced by a continuous medium as in the monolayer simulations [64-67]. Today s bilayers are usually fully hydrated , i.e., water is included exphcitly. Simulations have been done at constant volume [68-73] and at constant pressure or fixed surface tension [74-79]. In the latter case, the size of the simulation box automatically adjusts itself so as to optimize the area per molecule of the amphiphiles in the bilayer [33]. If the pressure tensor is chosen isotropic, bilayers with zero surface tension are obtained. Constant... 

Using this expression for A, we now calculate the first-order approximation to the pressure-tensor Pij ... 

The Navier-Stokes equations are recovered by substituting this first-order expression for the pressure tensor into the conservation theorem with n — mvi (i.e, into equation 9.55). 

Substituting this expression for Q and the first-order expression we found earlier for the pressure-tensor (equation 9.63) into equation 9.66 yields the general heat conduction equation ... 

By Eq. (1-55), we have px — m Jujaf dv. Since mut is the random momentum in the -direction (. e., the momentum associated with the -component of the random velocity), the (i,j) component of the pressure tensor is the average of the random flow in the -direction of the -directed momentum. From the definition of the temperature, Eq. (1-45), the hydrostatic pressure, defined as one-third of the trace of the pressure tensor, is... 

The components of the pressure tensor and of the heat-flow vector... 

Thus, the unknown forms for the pressure tensor and heat flow vector, which are required in the hydrodynamic equations, can be found when the coefficients are determined. 

If we refer to the relations between the expansion coefficients and the components of the pressure tensor, Eqs. (1-70), we see that Eqs. 

From Eqs. (1-76), the zero-order approximations to the pressure tensor and heat flow vector are ... 

These equations, with six farther equations for the other components of a20 and au, when solved by the Burnett iteration procedure, yield the Navier-Stokes equations when solved simultaneously, however, there is no longer the simple dependence of the pressure tensor upon the velocity gradients, and of the heat flow upon the temperature gradient, but, rather, an interdependence of these relations. 

Ponderomotive force, 382 Position operator, 492 in Dirac representation, 537 in Foldy-Wouthuysen representation, 537 spectrum of, 492 Power, average, 100 Power density spectrum, 183 Prather, J. L., 768 Predictability, 100 Pressure tensor, 21 Probabilities addition of, 267 conditional, 267 Probability, 106... 

In principle, one should solve the Boltzmann equation Eq. (65) in order to arrive at explicit expressions for the pressure tensor p and heat flux q, which proves not possible, not even for the simple BGK equation Eq. (11). However, one can arrive at an approximate expression via the Chapman Enskog expansion, in which the distribution function is expanded about the equilibrium distribution function fseq, where the expansion parameter is a measure of the variation of the hydrodynamic fields in time and space. To second order, one arrives at the familiar expression for p and q... 

In these equations, r/ is the shear viscosity of the fluid and P the pressure tensor moreover, as they are only valid at large distances, the incompressibility condition has been used (see footnote to page 255). 

Notation-. T is the temperature, Vi the fluid velocity, II,j the viscous pressure tensor, Jg the heat current density, p its chemical potential, the current density of molecular species a, v J the stoichiometric coefficient (13), and Wp the speed of reaction p. 

The diffusion coefficient D is one-third of the time integral over the velocity autocorrelation function CvJJ). The second identity is the so-called Einstein relation, which relates the self-diffusion coefficient to the particle mean square displacement (i.e., the ensemble-averaged square of the distance between the particle position at time r and at time r + f). Similar relationships exist between conductivity and the current autocorrelation function, and between viscosity and the autocorrelation function of elements of the pressure tensor. 

The Laplace-Young equation refers to a spherical phase boundary known as the surface of tension which is located a distance from the center of the drop. Here the surface tension is a minimum and additional, curvature dependent, terms vanish (j ). The molecular origin of the difficulties, discussed in the introduction, associated with R can be seen in the definition of the local pressure. The pressure tensor of a spherically symmetric inhomogeneous fluid may be computed through an integration of the one and two particle density distributions. 

Both the surface tension and the surface of tension may be expressed in terms of integrals of radial moments of the components of the pressure tensor (17-20)... 

Equations 12 and 13 are two different expressions for the surface tension ( ) which at the present time are not known to be consistent. Although Equation 12, and Equation 9, are independent of the form of the localization of the pressure tensor. Equation 13 is not (22). With the assumption of the equivalence of Equations 12 and 13 an expression for the surface of tension is obtained... 

Ve have obtained numerical solutions of Equations 1-5, with boundary values which satisfy Equations 7-9 for a wide range of state points. At each state the pressure tensor was computed from Equation 10 and used to evaluate Equations 12 and 13 and Equations 14-17. 

Bulk phase fluid structure was obtained by solution of the Percus-Yevick equation (W) which is highly accurate for the Lennard-Jones model and is not expected to introduce significant error. This allows the pressure tensors to return bulk phase pressures, computed from the virial route to the equation of state, at the center of a drop of sufficiently large size. Further numerical details are provided in reference 4. 

Finally, we compare the results of our calculations with the computer simulation results of Thompson et al. (7) in Figure 6. In these studies the surface tension was determined by a direct evaluation of the pressure tensor. Equations 12 and 13 were combined, replacing R with Equation 9 to obtain... 

The statistical quality of simulation estimates of the pressure tensor was poor near the origin resulting in large uncertainties in the estimate of the surface tension through Equation 24. By combining a truncated Taylor series expansion of Equation 20 with Equation 9 they obtained... 

In the case of the flux of mass, the result is the normal component of pua. But for the flux of momentum and energy, in general the flux density is not the normal component of a vector or tensor function of (t, x), since it will depend on the extended shapes of if and Y. But in the case of short-range forces and slowly varying p, ua, E, it can be shown to have this form with sufficient approximation. Thus one is led to the familiar pressure tensor and heat flow vector Qa, both as functions of (t, x). It is to be emphasized that the general expression of these quantities involves not only expected values of products of momenta (or velocities), but the effect of intermolecular forces. 

To show that the pV V term emerges, we write the pressure tensor as... 

The interfacial layer is the inhomogeneous space region intermediate between two bulk phases in contact, and where properties are notably different from, but related to, the properties of the bulk phases (see Figure 6.1). Some of these properties are composition, molecular density, orientation or conformation, charge density, pressure tensor, and electron density [2], The interfacial properties change in the direction normal to the surface (see Figure 6.1). Complex profiles of interfacial properties take place in the case of multicomponent systems with coexisting bulk phases where attractive/repulsive molecular interactions involve adsorption or depletion of one or several components. 

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