Navier-Stokes Equation (Incompressible Flow)

The Navier-Stokes equations describe the motion of viscous fluid substances. For an incompressible flow, assuming no external forces, the equations can capture how the velocity field of such fluids evolves over time within a domain. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term and a pressure term, hence describing the balance of momentum.

The formula given below represents a simplified version of the momentum equation for incompressible flows in one dimension, where ν is the kinematic viscosity, u is the velocity, and x is the spatial coordinate. Time-dependent solutions of the equation require advanced numerical methods and cannot be simplified to a single general formula.