Viscosity microscopic length scale

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

Despite the relative simplicity of most ionomers, questions about them remain. One of the key questions is how structure and dynamics on different length scales connect. Specifically, how does the metal coordination to the neutralized acid groups (which is on an angstrom level) correlate with the size, shape, and distribution of the ion-iich aggregates in the hydrophobic matrix (which is on a nanometer length scale) Furthermore, how does the microscopic structure control the macroscopic properties like melt viscosity or elastic modulus ... 

Since both b and d are measurable on the molecular scale, this expression does not differ in a macroscopically thick layer from the standard lubrication mobility given by Eq. (47), and the correction becomes significant only in theimmediate vicinity of the contact line. The three microscopic scales d, 6, and A are about of the same order of magnitude. The natural choice for d is the nominal molecular diameter, identified with the cut-off distance in the van der Waals theory. The standard value is about 0.2nm. The slip length is likely to be of the same order of magnitude. The approximate relation between viscosity / and self-diffusivity D,n in a liquid yields DniV lO T/37rd. The surface diffusivity should be somewhat lower than the diffusivity in the bulk liquid, and with D/Dm 0.1 we have 6 d. 

The previous section outlines several multi-particle algorithms. A detailed discussion of the link between the microscopic dynamics described by (1) and (2) or (3) and the macroscopic hydrodynamic equations, which describe the behavior at large length and time scales, requires a more careful analysis of the corresponding Liou-ville operator C. Before describing this approach in more detail, we provide a more heuristic discussion of the equation of state and of one of the transport coefficients, the shear viscosity, using more familiar approaches for analyzing the behavior of dynamical systems. 

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