Newton's law of viscosity

Newton's law of viscosity: the shear stress on a fluid layer is directly proportional to the rate of shear strain (velocity gradient) normal to the layer:

$$\tau = \mu \frac{du}{dy}$$

where $\mu$ is the coefficient of dynamic viscosity (N·s/m²). Fluids that obey this law are Newtonian fluids.

Importance of viscosity in fluid motion

1. Resistance to flow and head loss: viscosity causes shear between layers and friction at the walls. This produces head loss in pipes, penstocks and canals ($h_f = \frac{fLV^2}{2gD}$). In hydropower, friction loss in the penstock reduces the net head and power.
2. Type of flow: viscosity appears in the Reynolds number $Re = \rho VD/\mu$, which decides whether flow is laminar or turbulent.
3. No-slip condition and velocity profile: because of viscosity, fluid touching a wall has zero velocity, giving the parabolic (laminar) or flatter (turbulent) velocity profile and the boundary layer.
4. Drag and power needs: viscous drag on bodies and the power needed to pump fluids depend on viscosity.
5. Lubrication: a thin oil film separates moving surfaces (bearings of turbines and generators), reducing wear; correct viscosity is essential.
6. Energy dissipation: viscosity converts kinetic energy into heat, for example in hydraulic jumps below spillways.
7. Temperature effect: viscosity of liquids falls with rise in temperature (weaker cohesion) while that of gases rises (more molecular momentum exchange); this must be considered in design.