## 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,givesFor infinite plates the will be nodependence soso the speed in the– direction is a function ofonly. Let P and Q have coordinatesand let the speed at P beand the the speed at Q beIn timethe fluid particles at points P and Q have moved to P' and Q' respectively so that P'Q' is PQ deformed and the anglemeasures the deformation. To a first approximation

The rate of deformation of the fluid is then(2). If we label the stress bywe can use (1) to writewhereis the constant of proportionality called the coefficient of viscosity. Then from (2)