Newton's law of viscosity

From Chemepedia
Figure 1. Viscosities

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, non-Newtonian 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)
Figure 2. Directions of momentum transfer and shear stress. The top plate is moving with a velocity and the bottom plate is stationary. The direction of the shear stress is in the x-direction, which gets transferred in the y-direction

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 no-slip 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/m2

cp poise g/cm*s

cp lbm/ft*s

cp g/cm*s

cp lbm/ft*s

cp lbf*s/ft2

Pa*s N*s/m2 kg/m*s cp lbm/ft*s