General Viscosity Law

Newton's viscosity law is

(24.1)

where,

  = Shear Stress,
 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

(24.2a)       (24.2b)      (24.2c)

has two subscripts---first subscript : denotes the direction of the normal to the plane on which the stress acts, while the second subscript : denotes direction of the force which causes the stress.

The expressions of Stokes' law of viscosity for normal stresses are

(24.3a)           (24.3b)           (24.3c)

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,

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 for  in 24.3a, 24.3b, 24.3c we obtain

(24.5a)           (24.5b)          (24.5c)

In deriving the above stress-strain rate relationship, it was assumed that a fluid has the following properties

• 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.