Newton's viscosity law is
(24.1) |
where,
n is the coordinate direction normal to the solid-fluid interface,
μ is the coefficient of viscosity, and
V is velocity.
The above law is valid for parallel flows.
Considering Stokes' viscosity law: shear stress is proportional to rate of shear strain so that
second subscript : denotes direction of the force which causes the stress. |
The expressions of Stokes' law of viscosity for normal stresses are
|
where
is a proportionality factor and it is related to the second coefficient of viscosity μ1 by the relationship
.
We have already seen that the thermodyamic pressure is 
Now if we add the three equations 24.3(a),(b) and (c) , we obtain,
Now if we add the three equations 24.3(a),(b) and (c) , we obtain,
or
(24.4)
|
- For incompressible fluids,
So,is satisfied eventually. This is known as Thermodynamic pressure.
- For compressible fluids, Stokes' hypothesis is
.
- Invoking this to Eq. (24.4), will finally result in
(same as for incompressible fluid).
- Interesting historical aspects of the Stoke's assumption
can be found in Truesdell
.
- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------† Truesdell , C.A. "Stoke's Principle of Viscosity", Journal of Rational Mechanics and Analysis, Vol.1, pp.228-231,1952.
- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------
- Generally, fluids obeying the ideal gas equation follow this hypothesis and they are called Stokesian fluids .
- The second coefficient of viscosity, μ1 has been verified to be negligibly small.
Substituting μ forin 24.3a, 24.3b, 24.3c we obtain
- Fluid is homogeneous and isotropic, i.e. the relation between components of stress and those of rate of strain is the same in all directions.
- Stress is a linear function of strain rate.
- The stress-strain relationship will hold good irrespective of the orientation of the reference coordinate system.
The stress components must reduce to the hydrostatic pressure "p" (typically thermodynamic pressure = hydrostatic pressure ) when all the gradients of velocities are zero.