Viscosity may be thought of as a fluids resistance to the shearing forces acting on the fluid. We may picture a fluid between two plates, one stationary and one moving parallel to the first.
The relative motion of the two plates cause shearing forces on the liquid, deforming it. More viscous fluids offer more resistance to the shearing forces.
It the shear stress is proportional to the rate of deformation (1), the fluid is said to be Newtonian.
To derive an expression for the coefficient of viscosity, we assume the fluid above is incompressible.
The fluid moves in the x direction only so the continuity equation for an incompressible fluid,
gives
For infinite plates the will be no
dependence so
so the speed in the
– direction is a function of
only. Let P and Q have coordinates
and 
let the speed at P be
and the the speed at Q be
In time
the fluid particles at points P and Q have moved to P' and Q' respectively so that P'Q' is PQ deformed and the angle
measures the deformation. To a first approximation
The rate of deformation of the fluid is then
(2). If we label the stress by
we can use (1) to write
where
is the constant of proportionality called the coefficient of viscosity. Then from (2)