Transport coefficients shear viscosity

To close this Section we comment on two papers that do not fit under any neat heading. The first of these is by Xiao et al,261 who study the final stages of the collapse of an unstable bubble or cavity using MD simulations of an equilibrated Lennard-Jones fluid from which a sphere of molecules has been removed. They find that the temperature inside this bubble can reach up to an equivalent of 6000 K for water. It is at these temperatures that sonolumines-cence is observed experimentally. The mechanism of bubble collapse is found to be oscillatory in time, in agreement with classical hydrodynamics predictions and experimental observation. The second paper, by Lue,262 studies the collision statistics of hard hypersphere fluids by MD in 3, 4 and 5 dimensions. Equations of state, self-diffusion coefficients, shear viscosities and thermal conductivities are determined as functions of density. Exact expressions for the mean-free path in terms of the average collision time and the compressibility factor in terms of collision rate are also derived. Work such as this, abstract as it may appear, may be valuable in the development of microscopic theories of fluid transport as well as provide insight into transport processes in general. 

Equilibrium MD simulations of self-diffusion coefficients, shear viscosity, and electrical conductivity for C mim][Cl] at different temperatures were carried out [82] The Green-Kubo relations were employed to evaluate the transport coefficients. Compared to experiment, the model underestimated the conductivity and self-diffusion, whereas the viscosity was over-predicted. These discrepancies were explained on the basis of the rigidity and lack of polarizability of the model [82], Despite this, the experimental trends with temperature were remarkably well reproduced. The simulations reproduced remarkably well the slope of the Walden plots obtained from experimental data and confirmed that temperature does not alter appreciably the extent of ion pairing [82],... 

Transport properties, such as diffusion coefficients, shear viscosity, fhermal or electrical conductivity, can be determined from the time evolution of the autocorrelation function of a particular microscopic flux in a system in equilibrium based on the Green-Kubo formalism [217, 218] or the Einstein equations [219], Autocorrelation functions give an insight into the dynamics of a fluid and their Fourier transforms can be related to experimental spectra. The general Green-Kubo expression for an arbitrary transport coefficient y is given by ... 

These derivations yield general expressions for the transport coefficients that may be evaluated by simulating MPC dynamics or approximated to obtain analytical expressions for their values. The shear viscosity is one of the most important transport properties for studies of fluid flow and solute molecule... 

Hydrodynamic properties, such as the translational diffusion coefficient, or the shear viscosity, are very useful in the conformational study of chain molecules, and are routinely employed to characterize different types of polymers [15,20, 21]. One can consider the translational friction coefficient, fi, related to a transport property, the translational diffusion coefficient, D, through the Einstein equation, applicable for infinitely dilute solutions ... 

The work of Laufer (L3) indicates that eddy viscosity is not isotropic in shear flow. For this reason it is unlikely that eddy conductivity is isotropic in such flows. Therefore, uncertainties in the application of eddy conductivities must arise when it is assumed that this transport coefficient is isotropic. Until additional experimental information is available, it appears reasonable to consider eddy conductivity as isotropic except in circumstances when the vectorial nature (J4, R2) of the eddy viscosity may be estimated. Such an approximation appears acceptable since the measurements available described the conductivity normal to the axis of flow, which is the direction in which most detail is required in the prediction of temperature distribution in turbulently flowing streams. Throughout the remainder of this discussion all eddy properties will be treated as isotropic. Such a simplification is open to uncertainty, and further experimentation will be required in order to determine the error introduced by neglect of the vectorial characteristics of these macroscopic transport quantities. 

In a similar way the contribution for all the different modes to the three transport coefficients can be calculated. Equations (58) and (61) are the classic mode coupling theory expressions that provide general expressions for the shear viscosity and thermal conductivity, respectively. Using these general expressions and the ideas of static scaling laws, Kadanoff and Swift have calculated the transport coefficients near the critical point. 

As stated in the earlier section on Generalized Equations of Motion, we would ultimately like to find a set of equations of motion in the form of Eq. [91] to compute transport coefficients such as the shear viscosity and self diffusion... 

J. P. Ryckaert, A. Bellemans, G. Ciccotti, and G. V. Paolini, Phys. Rev. A, 39, 259 (1989). Evaluation of Transport Coefficients of Simple Fluids by Molecular Dynamics Comparison of Green-Kubo and Nonequilibrium Approaches for Shear Viscosity. 

Here (3Br(ij), Psll(i,j), and PDS(ij) are the transport coefficients for interparticle contacts between particles of diameters d, and dj by Brownian diffusion, fluid shear, and differential sedimentation, respectively kB is Boltzmanns constant T is the absolute temperature p, is the viscosity of the liquid G is the mean velocity gradient of the liquid g is the gravity acceleration and pp and p, are the densities of the particles and the liquid, respectively. 

As discussed in Sect. 4, in the fluid, MCT-ITT flnds a linear or Newtonian regime in the limit y 0, where it recovers the standard MCT approximation for Newtonian viscosity rio of a viscoelastic fluid [2, 38]. Hence a yrio holds for Pe 1, as shown in Fig. 13, where Pe calculated with the structural relaxation time T is included. As discussed, the growth of T (asymptotically) dominates all transport coefficients of the colloidal suspension and causes a proportional increase in the viscosity j]. For Pe > 1, the non-linear viscosity shear thins, and a increases sublin-early with y. The stress vs strain rate plot in Fig. 13 clearly exhibits a broad crossover between the linear Newtonian and a much weaker (asymptotically) y-independent variation of the stress. In the fluid, the flow curve takes a S-shape in double logarithmic representation, while in the glass it is bent upward only. 

An integration over all orientations of the vector R has been effected in these equations. The first term arises from momentum transport from mass motion and can be neglected in comparison with the second term. The indicated integration of Eq. 42 has been carried out numerically. That of Eq. 43 proved to be too sensitive to variation in the radial distribution function g0(2) to yield reliable results. In order to obtain the shear viscosity r), it is now necessary to evaluate the frictional coefficient f. A solution in series was obtained by Kirkwood25 but its numerical evaluation is too unwieldy for practical calculations. The analysis, however, stiggests the following estimate for the frictional coefficient ... 

The kinetic theory of gases attempts to explain the macroscopic nonequilibrium properties of gases in terms of the microscopic properties of the individual gas molecules and the forces between them. A central aim of this theory is to provide a microscopic explanation for the fact that a wide variety of gas flows can be described by the Navier-Stokes hydrodynamic equations and to provide expressions for the transport coefficients appearing in these equations, such as the coefficients of shear viscosity and thermal conductivity, in terms of the microscopic prop>erties of the molecules. We devote most of our attention in this article to this problem. 

Here we will not go through the detailed calculations that lead to the Enskog theory values for the transport coeflicients of shear viscosity, bulk viscosity, and thermal conductivity appearing in the Navier-Stokes hydro-dynamic equations. Instead we shall merely cite the results obtained and refer the reader to the literature for more details. One finds that the coefficient of shear viscosity 17 is given by ... 

The states a < 5, may be called the hydrodynamic states since they are associated with the conserved variables of number density, longitudinal and transverse components of the current, and kinetic energy. The other two states, correspond to the stress tensor and heat current, respectively. Therefore, the diagonal matrix elements involving these states must be related to the transport coefficients of shear viscosity and thermal conductivity as is well known in conventional transport theory. We will see below that these elements are important in formulating kinetic models. Besides the matrix elements shown in Table 1, we will include one additional element, namely. 

The transport coefficients that we want to investigate are the shear viscosity, the thermal conductivity, and the sound attenuation coefficient. It has been shown recently that the shear viscosity can be expressed directly in terms of certain matrix elements of... 

Thus we see that the shear viscosity of phenomenological hydrodynamics is directly proportional to the area under the ordinary time correlation function of Txz and inversely proportional to Vk T where V is the volume of the system. Equation (109) is called a Green-Kubo relation. Such relations can always be derived for transport coefficients and are useful in conjunction with molecular dynamics calculations in the determination of these transport coefficients. Projection operator techniques can thus be used to derive Green-Kubo relations although these relations were first derived in other ways. We have... 

In fact, two expressions of this type are very well known, the Stokes-Einstein law for the self-diffusion coefficient, and the Debye law for the rotational diffusion coefficient. Both of these laws for the appropriate diffusion coefficient D are derived by hydrodynamics and have Darj, where 17 is the coefficient of shear viscosity, a transport coefficient. The Stokes-Einstein and Debye laws were reconciled with formal theory with the use of mode-mode... 

Let us first consider nonequilibrium properties of dense fluids. Linear response theory relates transport coefficients to the decay of position and velocity correlations among the particles in an equilibrium fluid. For example, the shear viscosity ti can be expressed in Green-Kubo formalism as a time integral of a particular correlation function ... 

According to the second law of thermodynamics the transport coefficients for F = G (that is, the thermal conductivity, shear and bulk viscosities, and coefficients of diffusion) cannot be negative. To show that this is actually the case, consider the inequality... 

Solids are characterized by a nonzero static shear elastic constant, whereas a liquid will not support static shear. Conventional viscometric techniques (i.e., capillary flow and falling balls), which now seem rather crude considering the fragile nature of the blue phase lattice, initially showed a large viscosity peak at the helical-isotropic transition [89], [90], [91]. Viscosities and other transport coefficients associated with the pretransitional region of the isotropic phase were addressed by light scattering [92], [93], [94], [95]. 

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