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The diagram
shows a velocity distribution in a gas moving over a fixed plate. The
velocity gradient, (du/dy) decreases with distance, y, from the plate. Layers
of gas in this near plate region have different velocities, VA
and VB for example, and will experience a viscous shear stress, t
= m(du/dy), due to the velocity gradient
at their location. This shear stress arises from the momentum exchange between
gas molecules in these layers as illustrated in the insert in the diagram.
The molecules can be considered to have two components of momentum, one
due to the average molecular speed, c, which is a constant for the gas, and
the other due to the local drift velocity, u, of the gas which is position dependent.
Normally, c >> u as c is the speed of sound in the gas. Molecules
move some average distance,
l a
"mean free path," before colliding with another molecule. The momentum
exchange due to the drift velocity takes place in regions of this dimension.
For molecules of mass, m, the net rate of momentum transfer per unit area
and time is: nmcl(du/dy)/3, and is equal to the viscous
force per unit area. From Newton's second law, the coefficient of viscosity
is then: m = nmcl/3. |
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The temperature
dependence of the gas viscosity comes from the temperature dependence
of c. Kinetic theory indicates that c = (8kT/pm)0.5,
and so the gas viscosity increases as the square root of
the absolute temperature, T. Experiments show that this model is correct provided
the gas thickness is many mean free paths. |
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