text: In books on elementary kinetic theory one can find results for dilute gas modeling that are used in many fields. Derivation of the kinetic model for shear viscosity usually starts by considering a Couette flow where two parallel plates are separated by a gas layer. The upper plate is moving at a constant velocity to the right due to a force F. The lower plate is stationary, and an equal and opposite force must therefore be acting on it to keep it at rest. The molecules in the gas layer have a forward velocity component  which increase uniformly with distance  above the lower plate. The non-equilibrium flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.
  Inside a dilute gas in a Couette flow setup, let  be the forward velocity of the gas at a horizontal flat layer (labeled as );  is along the horizontal direction. The number of molecules arriving at the area  on one side of the gas layer, with speed  at angle  from the normal, in time interval  is 
  These molecules made their last collision at , where  is the mean free path. Each molecule will contribute a forward momentum of
  where plus sign applies to molecules from above, and minus sign below. Note that the forward velocity gradient  can be considered to be constant over a distance of mean free path.
  Integrating over all appropriate velocities within the constraint 
  yields the forward momentum transfer per unit time per unit area (also known as shear stress):
  The net rate of momentum per unit area that is transported across the imaginary surface is thus
  Combining the above kinetic equation with Newton's law of viscosity
  gives the equation for shear viscosity, which is usually denoted  when it is a dilute gas:
  Combining this equation with the equation for mean free path gives
  Maxwell-Boltzmann distribution gives the average (equilibrium) molecular speed as
  where  is the most probable speed. We note that
  and insert the velocity in the viscosity equation above. This gives the well known equation  (with  subsequently estimated below) for shear viscosity for dilute gases:
  and  is the molar mass. The equation above presupposes that the gas density is low (i.e. the pressure is low). This implies that the transport of momentum through the gas due to the translational motion of molecules is much larger than the transport due to momentum being transferred between molecules during collisions. The transfer of momentum between molecules is explicitly accounted for in Revised Enskog theory, which relaxes the requirement of a gas being dilute. The viscosity equation further presupposes that there is only one type of gas molecules, and that the gas molecules are perfect elastic and hard core particles of spherical shape. This assumption of elastic, hard core spherical molecules, like billiard balls, implies that the collision cross section of one molecule can be estimated by
  The radius  is called collision cross section radius or kinetic radius, and the diameter  is called collision cross section diameter or kinetic diameter of a molecule in a monomolecular gas. There are no simple general relation between the collision cross section and the hard core size of the (fairly spherical) molecule. The relation depends on shape of the potential energy of the molecule. For a real spherical molecule (i.e. a noble gas atom or a reasonably spherical molecule) the interaction potential is more like the Lennard-Jones potential or Morse potential which have a negative part that attracts the other molecule from distances longer than the hard core radius. The radius for zero  Lennard-Jones potential may then be used as a rough estimate for the kinetic radius. However, using this estimate will typically lead to an erroneous temperature dependency of the viscosity. For such interaction potentials, significantly more accurate results are obtained by numerical evaluation of the required collision integrals.
  The expression for viscosity obtained from Revised Enskog Theory reduces to the above expression in the limit of infinite dilution, and can be written as
  where  is a term that tends to zero in the limit of infinite dilution that accounts for excluded volume, and  is a term accounting for the transfer of momentum over a non-zero distance between particles during a collision.