Viscosity and the generalised Navier-Stokes equations

The viscosity of a fluid may initially be defined as its resistance to shear. The shearing stress (t) between two thin sheets of fluid is given by  

There are several classes of fluid for which the viscosity does not remain constant at different rates of deformation (shear rates). This may be denoted as follows  

Polymer solutions virtually always show non-Newtonian flow behaviour at sufficiently high polymer concentrations. However, before discussing this type of fluid in more detail, an outline of how non-Newtonian behaviour fits into the general area of fluid mechanics is presented. This is done by considering the total stress tensor and the equation of momentum conservation. 

The momentum conservation equation for a viscous fluid in the absence of gravity is therefore  

This is an equation of motion and the total stress tensor, n, may be expressed as  

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