The fourth constitutive relation hypothesis states that the properties of a fluid are isotropic, i.e. independent of direction.

For example, an object moving in a fluid encounters the same resistance regardless of the direction of movement. From eq11, the properties of a fluid are described by *β _{ijkl }*, which must be isotropic.

As described in the articles on tensors, the general form of a fourth-order isotropic tensor is:

From eq11 of the previous article, *τ _{ij }*is symmetric. Therefore,

*τ*=

_{ij }*τ*and

_{ji }Substitute eq12 in the above equation, we have

Substitute eq13 in eq12,

Substitute eq14 in eq11,

Substitute eq15 in eq3,

The values of *μ* and *λ* can only be determined through experiments. *μ* is known as the shear viscosity of the fluid while *λ *is the volume viscosity of the fluid, which is zero for an incompressible fluid. Eq16 becomes:

###### Question

Show that *μ* in eq16 and eq7 are the same.

###### Answer

Consider the flow of an incompressible fluid where the flow velocity components are *u* = *u*(*y*), *v* = 0 and *w* = 0. Using eq16, the stress components are:

with the remaining components equal to zero.

Clearly, the components of stress in this case include the pressure *p* and the shear stress , which is the result of the derivation of Newton’s law of viscosity, eq7. Hence, *μ* in eq16 and eq7 are the same.