## Newtonian Fluids

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) 