Newton's law of viscosity
Viscosity is the property of a fluid that resists fluid motion. For example, in Figure 1, the blue fluid has a lower viscosity than the orange fluid. It can be said that the orange fluid has a greater resistance to flow and hence a higher viscosity.
Newtonian fluids are the simplest model of fluids that account for viscosity. Many common liquids and gases, such as water and air, can be assumed to be Newtonian for practical calculations under ordinary conditions. However, nonNewtonian fluids are also relatively common, which include paint, blood, polymers, and most personal care products as some examples.
Incompressible Newtonian fluid
A Newtonian fluid is a fluid in which the viscous stresses are proportional to the local strain rate. The corresponding constitutive law for a an incompressible Newtonian fluid is

(1) 
where is a 9 component, symmetric tensor and is the transpose of . Each component has the notation

(2) 
where the first subscript is the direction in which momentum is transferred due to the velocity gradient and the second subscript is the direction of the shear stress.
For instance, in Figure 2, a fluid is sandwiched between two plates. The top plate is moving at a constant speed while the bottom plate is stationary. Due to the noslip boundary condition, the velocity of the fluid touching the top plate is also while the velocity of the fluid touching the lower plate is . The only velocity gradient is the change of velocity in the direction Hence, shear stress in the direction is transferred in the direction (from the top plate to the lower plate).
Newton's law of viscosity in Cartesian coordinates for an incompressible fluid
The 9 components of the viscous stress tensor in Cartesian coordinates are
(2) 
Useful Conversions When Working With Viscosity
cp kg/m*s Pa*s N*s/m^{2}
cp poise g/cm*s cp lb_{m}/ft*s cp g/cm*s cp lb_{m}/ft*s cp lb_{f}*s/ft^{2} Pa*s N*s/m^{2} kg/m*s cp lb_{m}/ft*s 