Diffusion and Flow near a Fractal

As before, it is convenient to designate a home particle on the fractal adsorber. We now consider an arbitrary point at a distance r from the home point. Despite the absorption we expect some of the walker s tracks to remain at r, as shown in Fig. 8.4. The density of such walkers relative to the initial density is the probability that the walker at r has not been removed by adsorption. This probability is the probability that the random walk representing its past has not intersected the fractal. This is just the g(r) discussed in the last section. The fractals in this case are the adsorbing fractal, with dimension D, and the random walk, with dimension 2. As we saw above, this g r) depends on the fractal dimensions. If D 4- 2 is less than 3, the two are mutually transparent g r) 1. Virtually all the walkers in the pervaded volume of the fractal never touch the fractal. Their density u is thus virtually unaffected u r) Uq. But if D-l-2 is greater than 3, the two are mutually opaque, and g r) is substantially smaller than 1 for most points r within the absorbing fractal. All connected fractals have D 1 and thus show opaque behavior. For all such fractals, in- 

Hydrodynamic drag on an object also controls the Brownian motion of that object in the solution. Brownian motion is caused by random thermal forces in the surrounding solvent. These forces, like imposed forces, produce a propor- 

This opacity holds for other flow conditions, notably shear flow. In a shear flow, like that in a pipe or a glass of swirling water, the velocity v r) near the wall runs parallel to it and is proportional to the distance from it. In a shear flow a small cubical region of the fluid distorts into a rhomboidal box as shown in Fig. 8.5a. The fractional movement of the top of the box relative to the bottom is called the shear 7. This shear increases at a constant rate in time the shear rate is called 7. 

The Boltzmann constant fc is a conversion factor from conventional temperature units to energy units. At room temperature, kT is roughly 1/40 of an electron volt, a hundred times smaller than the energy needed to break a polymer chain. 

One consequence of the disturbed flow is an increased dissipation of the fluid s kinetic energy as heat. The power dissipated per unit volume in a small region is equal to the viscosity r) times the square of the shear rate, 7.  

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