For an incompressible Newtonian fluid with constant density and viscosity, conservation of momentum becomes the Navier–Stokes equation

$$\rho\left(\frac{\partial\mathbf v}{\partial t}+(\mathbf v\cdot\nabla)\mathbf v\right) =-\nabla p+\mu\nabla^2\mathbf v+\rho\mathbf g,$$

combined with incompressibility,

$$\nabla\cdot\mathbf v=0.$$

Acceleration

The left side is mass density times the material acceleration of the fluid.

Pressure force

The pressure-gradient term

$$-\nabla p$$

accelerates fluid from higher toward lower pressure.

Viscous diffusion

For a Newtonian fluid of constant viscosity, the term

$$\mu\nabla^2\mathbf v$$

represents the diffusion of momentum by viscosity.

Body forces

The term $\rho\mathbf g$ represents gravity; other body forces can be added when relevant.

The equations are compact, but their nonlinear advection term couples the velocity field to itself. Except for special cases, useful solutions require approximation or numerical computation.