The Cluster's Transport Phenomena

A page about transport phenomena — the three parallel transport processes: viscosity, thermal conductivity, and electrical conductivity.

The three transport coefficients

Transport phenomena study the flow of extensive quantities (momentum, energy, charge) in response to gradients:

• Viscosity (eta): Flow of momentum in response to a velocity gradient (shear stress = eta du/dy)
• Thermal conductivity (k): Flow of heat in response to a temperature gradient (Fourier's law: j_q = -k nabla T)
• Electrical conductivity (sigma): Flow of charge in response to an electric field (Ohm's law: j = sigma E)

In the cluster, the edit three transport coefficients give an edit momentum flow.

The kinetic theory results

From kinetic theory of gases (assuming Maxwell-Boltzmann distribution, mean free path lambda):

• eta = (1/3) n m v_bar lambda (independent of density!)
• k = (1/3) n c_v v_bar lambda (where c_v is the specific heat per molecule)
• sigma = n e^2 tau / m (Drude model, where tau = lambda / v_bar)

The striking prediction: eta ~ T^{1/2} (increases with temperature) and is independent of density at moderate pressures. This is experimentally verified for gases.

In the cluster, the edit kinetic theory results give an edit density prediction.

The Wiedemann-Franz law

For metals, the ratio of thermal to electrical conductivity is: k / (sigma T) = (pi^2 / 3) (k_B / e)^2 = L_0 = 2.44 x 10^{-8} W Ohm / K^2 This universal ratio L_0 is the Lorenz number. The Wiedemann-Franz law follows because both heat and charge are carried by free electrons.

In the cluster, the edit Wiedemann-Franz law gives an edit universal ratio.

The Einstein relation

The Einstein relation connects the diffusion coefficient D to the mobility mu: D / mu = k_B T / q where q is the charge of the diffusing particle. For an electric charge carrier: D / mu = k_B T / e. This relates the random walk (diffusion) to the directed motion (mobility).

In the cluster, the edit Einstein relation connects an edit diffusion coefficient.

Applications

• Engineering: Design of heat exchangers, lubrication systems, electrical circuits
• Materials science: Thermal management of electronic devices, designing thermoelectric materials (low k / high sigma)
• Geophysics: Mantle convection, magma transport
• Astrophysics: Stellar interiors, accretion disks
• Biophysics: Nerve impulse propagation (action potentials)

In the cluster, edit applications include:

• edit Engineering
• edit Materials science
• edit Geophysics
• edit Astrophysics
• edit Biophysics

This phenomena

This page is about transport phenomena. eta = (1/3) n m v_bar lambda. k / (sigma T) = L_0. D / mu = k_B T / q. The phenomena is real.