Glossary term
Navier-Stokes Equations
The momentum-conservation equations for viscous fluid flow, coupling velocity, pressure, body forces, and material properties.
Definition
modelThe Navier-Stokes equations are the governing momentum-balance equations for viscous fluid flow, relating acceleration, pressure gradients, viscous stresses, and body forces.
The Navier-Stokes equations express conservation of momentum for viscous fluids and, together with mass conservation and suitable boundary conditions, form the basis of most engineering fluid-flow analysis. They connect pressure, velocity, density, viscosity, body forces, and acceleration in a continuum model.
The Navier-Stokes equations describe how fluid momentum changes because of inertia, pressure gradients, viscous stresses, and external body forces. For an incompressible Newtonian fluid with constant viscosity, a common vector form is:
This equation is normally paired with the incompressible continuity equation:
where \rho is density, \mathbf{u} is velocity, p is pressure, \mu is dynamic viscosity, and \mathbf{g} represents body acceleration such as gravity. Compressible, multiphase, non-Newtonian, reacting, or heat-transfer problems require additional terms or coupled governing equations.
Engineering meaning
Each term has a physical role. The transient term captures local acceleration, the convective term captures transport of momentum by the flow itself, the pressure-gradient term drives or resists motion, and the viscous term diffuses momentum through shear. The relative size of inertial and viscous effects is summarized by Reynolds number, which helps distinguish laminar, transitional, and turbulent regimes.
Exact analytical solutions exist only for simplified cases such as fully developed pipe flow, Couette flow, or creeping flow around idealized geometries. Most practical engineering problems use approximations: Bernoulli models, boundary-layer theory, empirical correlations, reduced-order models, wind-tunnel measurements, or computational fluid dynamics.
Numerical use
In CFD, the equations are discretized over a mesh or set of control volumes, with boundary conditions for walls, inlets, outlets, symmetry planes, moving surfaces, or interfaces. Solver choices, turbulence models, near-wall treatment, mesh convergence, time step, and pressure-velocity coupling can strongly affect results. A converged residual plot alone is not proof that the solution is physically correct.
Common mistakes
A common mistake is to say a CFD model “solves Navier-Stokes” without stating the assumptions: incompressible or compressible, steady or transient, laminar or turbulent, Newtonian or non-Newtonian, single-phase or multiphase. Another is to use ideal no-slip wall and smooth-surface assumptions when roughness, separation, cavitation, vortex shedding, or thermal coupling controls the real system. A credible analysis documents governing assumptions, boundary conditions, mesh quality, validation data, and sensitivity to model choices.