Viscosity in the incompressible Newtonian model

For an incompressible Newtonian fluid, dynamic viscosity $\mu>0$ relates the viscous stress to the rate of strain by

\tau=2\mu D(u).

At constant density $\rho>0$ , the kinematic viscosity is $\nu=\mu/\rho$ . With constant $\mu$ and divergence-free velocity, the viscous force per unit mass is $\nu\Delta u$ .

Units and normalization

Kinematic viscosity has units of length squared divided by time. Mathematical formulations often take density equal to one and use $\nu$ directly as the coefficient of the velocity Laplacian. Setting $\nu=1$ requires a corresponding choice of units or rescaling; it is not an additional physical identity.

The inviscid Euler model omits this viscous term.

References

Lenya Ryzhik, Lecture Notes for Math 256B (2024), Sections 1–3.