As an object moves through a gas, the gas molecules near the object
are disturbed and move around the object.
Aerodynamic
forces are generated between the gas and the object. The
magnitude of these forces depend on the shape of the object, the
speed of the object,
the mass
of the gas going by the object and on two other important properties
of the gas; the viscosity, or stickiness, of the gas and the
compressibility, or springiness, of the gas. To properly model
these effects, aerodynamicists use
similarity parameters
which are
ratios
of these effects to other forces present in the
problem. If two experiments have the same values for the similarity
parameters, then the relative importance of the forces are being
correctly modeled. Representative values for the properties of
air
are given on another page,
but the actual value of the parameter depends
on the
state of the gas
and on the
altitude.
On this page, we examine the viscosity of a gas.

As an object moves through a gas, the gas molecules stick to
the surface.
If we have two surfaces, as shown in the figure, with one surface
fixed and the other surface moving parallel to the fixed surface,
a shearing force is generated in the fluid between the
surfaces. An external force F must be applied to the moving
surface to overcome the resistance of the fluid. If we denote the
magnitude of the shearing force by the Greek letter tau.
Then:

tau = F / A

where A is the area of the moving surface. It is experimentally
observed that, for most gases, the shear stress is directly proportional to the
gradient of the velocity between the surfaces:

tau is linearly proportional to dV/dy

where y is the distance between the surfaces and V is the
velocity. The proportionality constant is called the coefficient of
dynamic viscosity and assigned the Greek letter mu

tau = (mu) * (dV/dy)

The value of the dynamic viscosity coefficient is found to be a constant with
pressure
but the value depends on the
temperature
of the gas. For air, D. M. Sutherland provides an equation for the
dependence on temperature T:

where mu0 and T0 are reference values
given at sea level stanfard conditions.
The temperature is specified in degrees
Rankine:

mu0 = 3.62 x 10^-7 lb-sec/ft^2

T0 = 518.7 R

The shearing of the flow of air creates a
layer of air near the surface, called a
boundary layer,
which, in effect, changes the shape of the
object. The flow of gas reacts to the edge of the boundary layer as if
it was the physical surface of the object. To make things more
confusing, the boundary layer may
separate
from the
body and create an effective shape much different from the physical
shape. And to make it even more confusing, the flow conditions in and
near the boundary layer are often unsteady (changing in time).
The boundary layer is very important in determining the
drag
of an object. To determine and predict these conditions,
aerodynamicists rely on
wind tunnel
testing and very sophisticated computer analysis.

The
similarity parameter
for viscosity is the
Reynolds number.
The Reynolds number expresses the ratio of
inertial forces to viscous
forces. From a detailed
analysis of the
momentum conservation equation,
the inertial forces are characterized by the product of the
density r times the velocity V times the
gradient of the velocity dV/dx. The viscous forces
are characterized by the viscosity coefficient mu times
the second gradient of the velocity d^2V/dx^2. The
Reynolds number Re then becomes:

Re = (r * V * dV/dx) / (mu * d^2V/dx^2)

Re = (r * V * L) / mu

where L is some characteristic length of the problem. If
the Reynolds number of the experiment and flight are close, then we
properly model the effects of the viscous forces relative to the
inertial forces. If they are very different, we do not correctly
model the physics of the real problem and predict incorrect levels of
the aerodynamic forces.

The dynamic viscosity coefficient divided by the density is called the
kinematic viscosity and given the Greek symbol nu

nu = mu / r

Re = V * L / nu

The units of nu are length^2/sec.

Here's a Java program to calculate the coefficient of viscosity
and the Reynolds number
for different altitude, length, and speed.

To change input values, click on the input box (black on white),
backspace over the input value, type in your new value, and
hit the Enter key on the keyboard (this sends your new value to the program).
You will see the output boxes (yellow on black)
change value. You can use either Imperial or Metric units and you can input either the Mach number
or the speed by using the menu buttons. Just click on the menu button and click
on your
selection.
The non-dimensional Mach number and Reynolds number are displayed in
white on blue boxes.
If you are an experienced user of this calculator, you can use a
sleek version
of the program which loads faster on your computer and does not include these instructions.
You can also download your own copy of the program to run off-line by clicking on this button:

The Similarity Parameter Calculator was modified in May, 2009, by Anthony Vila,
a student at Vanderbilt University,
during a summer intern session at NASA Glenn.

Note: there are some classes of fluids for which the shear force is
not linearly proportional to the velocity gradient. Some long chain polymers,
like latex paint, behave as "thick" liquids without shear, and very "thin"
liquids when subjected to shear. Other liquids are known to "thicken" when
a shear stress is applied. But for most gases, like air, the shear stress
is linearly proportional to the velocity gradient.