Consider a pipe of length L and radius R. In class we argued that it’s diffusive conductance should
scale like g
dif f ∼ DR2/L while its hydrodynamic conductance should scale like g
hydro ∼ R4/ηL with
D the diffusion constant and η the diffusive conductance. Here we will calculate the prefactors more
carefully, and compare the two more mathematically. For this problem consider z to parameterize the
length of the pipe (0 < z < L) while r, θ parametrize the interior of the pipe (r < R).
In class we argued that both the hydrostatic pressure P and the particle density ρ obey Laplace’s
equation. Thus for diffusive dynamics in steady state ∇2ρ = 0 while for hydrodynamics ∇2P = 0.
Diffusive dynamics are defined by J~ = −D∇ρ while hydrodynamics are defined by ∇2V~ = (1/η)∇P
(a) To calculate the diffusive conductance we consider ρ(z = 0) = ρo while ρ(z = L) = ρ1. What
Boundary conditions are appropriate at the edge of the cylinder, that is when r = R?
(b) To calculate the hydrodynamic conductance we consider P(z = 0) = Po while P(z = L) = P1. What
boundary conditions are appropriate at the edge of the cylinder, that is when r = R?
(c) Boundary conditions on all of these surfaces uniquely specify the value of P and ρ in the interior.
What satisfy these boundary conditions?
(d) For the case of diffusion, if your answer to the previous question is correct, you should be able to
easily solve for the current density J~ and resulting conductance.
(e) For the case of hydrodynamics, you still have a complicated equation to solve! You should have
something of the form ∇2V~ = (1/η)∇P where P is taken from your solution above. In cylindrical
coordinates, this equation must be supplemented by boundary conditions for V~ at r = 0 and at
r = R. What are they?
(f) You can look up (or derive!) the formula for a vector Laplacian in cylindrical coordinates. Use this
to solve for the velocity field V~ (r, θ, z) in the pipe.
(g) Integrate this over r, θ to find the conductance g
hydro. (here I =
R
Vz(r, θ, z = 0)rdrdθ over the
surface.)

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