Page 417 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 401

Analogously, the influence of viscosity, which is responsible for the generation of vorticity
on the boundary, depends on the speed U m far upstream. At low speed, the influence will be
deep into the field in all directions so that essentially the whole flow field is having vorticity.
On the other hand, at high speed, the effect of viscosity is confined in a thin layer ( known as
a boundary layer) near the body and behind it. Outside of the layer, the flow is essentially
irrotational. This concept enables one to solve a fluid flow problem by dividing the flow region
into an irrotational external flow region and a viscous boundary layer. Such a method simplifies
considerably the complexity of the mathematical problem involving the full Navier-Stokes
equations. We shall not go into the methods of solution and of the matching of the regions as
they belong to the boundary layer theory.

6.25 Compressible Newtonian Fluid

For a compressible fluid, to be consistent with the state of stress corresponding to the state
of rest and also to be consistent with the definition that/? is not to depend explicitly on any
kinematic quantities when in motion, we shall regard p as having the same value as the
thermodynamic equilibrium pressure. Therefore, for a particular density p and temperature
0, the pressure is determined by the equilibrium equation of state

For example, for an ideal gas/? = Rp®. Thus

Since

it is clear that the " pressure" p in this case does not have the meaning of mean normal
compressive stress. It does have the meaning if

which is known to be true for monatomic gases.

2
Written in terms of ju and k = A+-ju, the constitutive equation reads
«.?

With TIJ given by the above equation, the equations of motion become (assuming constant ju
andk)