The coefficient of viscosity ($η$) is defined as the ratio of shearing stress to the strain rate .

1. Shearing Stress: Defined as force per unit area, its dimensions are $[ML^{-1}T^{-2}]$ .
2. Strain Rate: Also known as velocity gradient ($v/l$), its dimensions are $[LT^{-1}]/[L] = [T^{-1}]$ .

By dividing the dimensions of stress by the dimensions of the strain rate, we get: $[η] = [ML^{-1}T^{-2}] / [T^{-1}] = [ML^{-1}T^{-1}]$ .

The SI unit for the coefficient of viscosity is $N \cdot s \cdot m^{-2}$ or $Pa \cdot s$ .