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Diffusion and viscosity

Page history last edited by Joe Redish 9 years, 8 months ago

4.3.3.P2

 

 

Our molecular model of matter describes a fluid as consisting of lots of little particles (atoms or molecules) moving around very fast. Collisions between the molecules makes the particles of the fluid change directions and speeds often and randomly. We've talked about two different phenomena that depend on these random molecular interactions: viscosity and diffusion.  Viscosity is the way collisions with other parts of the fluid slow down faster moving bits of fluid and diffusion is the way collisions spread out concentrations of molecules. Could they be related?

 

Let's consider how the viscosity and diffusion coefficients might depend on the properties of the fluid by dimensional analysis.

 

(a) First, determine the dimensionality (in terms of M, L, and T) of the viscosity coefficient, μ, and the diffusion coefficient, D. Recall that they are defined by the equation for viscous force

and Fick's law for diffusion

where J is the number of molecules crossing a unit area per unit time and n is the concentration. Use dimensional analysis of these equations to determine the dimensionality of μ and D.

 

(b) Now let's consider what they might depend on. Since what is controlling both of them are the molecules of the fluid colliding with each other, here are some parameters it might depend on:

  • the mass density of the fluid (ρ)
  • the average speed of the molecules of the fluid (v0)
  • how far the molecules travel between collisions (the mean free path, λ).

If we only are going to use these three parameters, find combinations of these three parameters that give you the correct dimensionality for μ and D.


(c) From the dimensionality of each, construct a plausible relation between μ and D (i.e., an equation relating μ and D).

 

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