External force in a normalized fluid equation

An external force per unit mass is a prescribed vector field $f(t,x)$ entering the momentum equation as an additive acceleration source. If a physical force density $F$ is given and the constant mass density is $\rho$ , the normalized forcing is $f=F/\rho$ .

Forced and unforced equations

A problem with $f=0$ is unforced. Nonzero forcing is separate from the initial velocity and can generate motion from zero initial data. Smoothness, support, and time dependence of $f$ must be stated when they matter.

Gradient forces

If $f=\nabla\phi$ , it can be absorbed algebraically into normalized pressure: $-\nabla p+\nabla\phi=-\nabla(p-\phi)$ . Boundary or pressure-normalization conditions may still affect how this representation is used.

References

Charles L. Fefferman, Existence and Smoothness of the Navier–Stokes Equation.