Stress tensor molecular contributions

Sarvestani and Picu [2004] proposed a molecular network model of nonentangled polymer matrix with less than 10 wt% of nanoflller particles. The model assumed the creation and destruction of loops, tails, and bridges between nanoparticles. The stress tensor contains contributions pertaining to bridging and dangling segments B andZ), respectively) ... 

Figure 11.23—Comparison of theo-retical and experimental first and second normal stress differences N and N2. The theoretical results (a) were calculated from the Smoluchowski equation (11-3) using the Onsager potential with U = 10.67, the minimum value for a fully nematic state, y/ >r is the dimensionless shear rate (or Deborah number), where Dr is the rotary diffusivity of a hypothetical isotropic fluid at the same concentration. Only the molecular-elastic contribution to the stress tensor was considered. The experimental results (b) are for 12.5% (by weight) PBLG (molecular weight = 238,000) in w-cresol. (Reprinted with permission from Magda et al., Macromolecules 24 4460. Copyright 1991, American Chemical Society.)...

This completes the discussion of the various contnbutions to the stress tensor. Table 1 gives a summary of the expressions for the flux expressions that are obtained by taking the first term in the Taylor-senes expansions of the fluxes, and Table 2 summarizes the complete expressions (except for the inter-molecular contributions). 

Now let s consider the molecular transport of momentum. The molecular mechanism is given by the stress tensor or molecular momentum flux tensor, r. Each element Ty can be interpreted as the component of momentum flux transfer in the direction. We are therefore interested in the terms tix- The rate at which the x component of momentum enters the volume element at face x is XxxAyAx Ij, the rate at which it leaves at face x + Ax is XxxAyAx i+ax, and the rate at which it enters at face y is TyxAxAz y. The net molecular contribution is therefore... 

As mentioned earlier, the applicability of the SOR does not rest on fundamental physical laws. However, there are two important statements to emphasize. First, the observation of a breakdown of the SOR can be voy informative about the molecular level contributions to the stress tensor, and second, in principle rtieo-optical measurements can be at least as informative in regions where the SOR does not apply. Apparently, the SOR is successful when the... 

Stress enters in a development of hydrodynamics when one considers the equation of conservation of momentum. The rate of change of momentum in some volume element at point r is written as the acceleration produced by external forces on that element and a (negative) flux of momentum across the surface. The flux of momentum has two parts. The first is the momentum associated with the average velocity, u(r), of the fluid at r. Thus momentum density in the a direction (with a x,y, or z) is p(r)t/ (r), where p(r) is the mass density at r. This momentum is transported in the direction at a rate u ir). Therefore this contribution to the flux of a momentum in the /S direction is p(r)M (r)M (r). Additional observed momentum transfer is called minus the stress tensor. The stress tensor can be separated into contributions from two molecular sources. One is also kinetic, and arises from the fact that the particles have a distribution of velocities about the average fluid flow velocity. We can write this term as a statistical average... 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

There are two proper explanations, one based on physical intuition and the other based on the principle of material objectivity. The latter is discussed in many books on continuum mechanics.19 Here, we content ourselves with the intuitive physical explanation. The basis of this is that contributions to the deviatoric stress cannot arise from rigid-body motions -whether solid-body translation or rotation. Only if adjacent fluid elements are in relative (nonrigid-body) motion can random molecular motions lead to a net transport of momentum. We shall see in the next paragraph that the rate-of-strain tensor relates to the rate of change of the length of a line element connecting two material points of the fluid (that is, to relative displacements of the material points), whereas the antisymmetric part of Vu, known as the vorticity tensor 12, is related to its rate of (rigid-body) rotation. Thus it follows that t must depend explicitly on E, but not on 12 ... 

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