2.4 Transport processes

In particular, in equilibrium the local **energy density**, i.e. the energy contained
in a volume element divided by that , as well as the
**momentum density** and the **particle** or **mass density** remain
constant. These densities, which refer to the three conserved quantities of
mechanics, play a special role in what follows.

Again considering the dilute gas, and assuming there are no external fields,
the densities are

(2.41) |

If a physical quantity - such as mass, or momentum - is conserved,
any local change can only be achieved by a flow into or out of the
space region under consideration; we are then speaking of
**transport processes** whose speeds are governed by the
respective **transport coefficients** - such as the viscosity ,
the heat conductivity (for the energy transport) and the
diffusion constant (for mass transport).

In real experiments aimed at determining these transport coefficients
the respective gradient is artificially maintained. Thus there will be
a continuing flow of momentum, energy or matter in the (reverse) direction
of the gradient - which is obviously a non-equilibrium situation.
However, by a careful setup we can keep at least these flows and the
local densities and gradients constant in time. This is called a
**stationary non-equilibrium situation**.

**DEFINITION OF TRANSPORT COEFFICIENTS**

In order to define the transport coefficients , and
we consider the basic experimental setup.

**Viscosity:**To measure the coefficient of momentum transport we generate a**laminary flow**by placing a gas or fluid layer between two horizontal plates and moving the upper plate with constant velocity to the right. In this manner we superimpose a systematic -velocity onto the random thermal motion of the molecules.The magnitude of the thermal speed is of the order ; by adding a ``shear velocity'' of some centimeters per second the local equilibrium is not considerably disturbed. Thus we may assume that we have still a Maxwell-Boltzmann distribution of velocities at any point in the fluid, with the same value of (bzw. ) everywhere.

Yet by imposing a velocity gradient we have slightly perturbed the equilibrium; a certain amount of -momentum will flow against the gradient - in our case downwards - so as to reestablish equilibrium. The amount of momentum flowing down through a unit of area per unit of time is called

**flow density**. In a first (linear) approximation this flow will be proportional to the imposed velocity gradient, and the coefficient of proportionality is, by definition, the transport coefficient :

(2.42) **Newton's law**of viscous flow. The parameter is known as viscosity.

**Thermal conductivity:**To generate a gradient of energy density we may place the substance between two heat reservoirs with different temperatures. if we use the same simple geometry as for the viscosity then the temperature (or energy) gradient will have only the one non-vanishing component which accordingly produces a flow of energy in the counter direction. The coefficient of thermal conduction is defined by (**Fourier's law**),

(2.43) **Diffusion constant:**Even in a homogeneous gas (or a fluid, or a solid, for that matter) the individual molecules will alter their positions in the course of time. This process is known as**self diffusion**. It is possible to ``tag'' some of the particles - for instance, by giving them a radioactive nucleus. The density of the marked species - call it - may then once more have a gradient that will be balanced by a particle flow (**Fick's law**):

(2.44)

We will presently attempt to calculate the above defined transport coefficients from the microscopic dynamics of the molecules. In a simple kinetic treatment the elementary process determining the various flows is the free flight of a gas particle between two collisions with other molecules. Thus the first step should be an estimation of the average path length covered by a particle between two such collisions.

Let be the - as yet unknown - probability that a particle meets
another one **before** it has passed a distance . Then is
the probability for the molecule to fly a distance without undergoing
a collision. But the probability of an encounter within an infinitesimal
path element is given by
, where
is the diameter of the particles. ( equals the fraction
of the ``target area'' covered by other particles.)
Thus the differential probability for a collision between and
is

(2.45) |

(2.46) |

(2.47) |

(2.48) |

EXAMPLE:The diameter of a molecule is ( Ångstrøm). According to Loschmidt (and the later but more often quoted Avogadro) a volume of contains, at standard conditions, particles; the particle density is thus , and the mean free path is .

Applet Hspheres: Start |

In the case of a dilute gas we can actually find expressions for the transport coefficients. Assuming once more the basic geometric setting used in most experiments, we suppose that the gradient of the respective conserved quantity has only a component, leading to a flow that is also directed along the axis.

**Viscosity:**In the course of the**random**(thermal) motion of the particles the small**systematic**term is carried along. On the average, an equal number of particles will cross the interface between two horizontal layers from below and above. However, since the upper layer has a larger (assuming , it will gradually slow down, while the lower layer will take up speed. The flow of momentum due to this mechanism can be estimated as follows:

The mean number of particles crossing the unit of area from below and from above, respectively, is . Each of the molecules carries that systematic which pertains to the layer of its previous collision, i.e. . The momentum flow density is therefore

(2.49)

It is a somewhat surprising consequence of this formula that, since , the viscosity is apparently independent of density! Maxwell, who was the first to derive this result, had to convince himself of its validity by accurate experiments on dilute gases.**Thermal conductivity:**A similar consideration as in the case of momentum transport yields, with ,

(2.51) **Diffusion:**Again we assume a gradient of the relevant density - in this case, the particle density of some species - along the direction. Writing and using the same reasoning as before we find

(2.52)

2005-01-25