The Navier-Stokes equations are among the most important equations in physics and engineering. They describe how fluids like water and air move and behave. These equations connect the velocity, pressure, temperature, and density of moving fluids through mathematical relationships. When we see fluid flowing around objects like this cylinder, the Navier-Stokes equations govern every aspect of that motion, from the smooth streamlines to the complex patterns that can emerge.
The Navier-Stokes equations are founded on three fundamental physical principles. First is conservation of mass, shown by the continuity equation - the amount of fluid entering a region must equal the amount leaving, unless the fluid is compressible. Second is conservation of momentum, which applies Newton's second law to fluid elements - the net force on a fluid particle equals its mass times acceleration. Third is conservation of energy, accounting for kinetic energy, potential energy, and internal energy changes. These three principles work together to completely describe fluid behavior.
Here we see the complete mathematical formulation of the Navier-Stokes equations. The main equation contains four key terms. The time derivative represents local acceleration - how velocity changes at a fixed point in space. The convective term represents advection - how fluid carries momentum as it moves. The pressure gradient term shows how pressure differences drive fluid motion. Finally, the viscous term accounts for internal friction that smooths out velocity variations. The continuity equation ensures mass conservation by requiring that the divergence of velocity equals zero for incompressible flow.
Viscosity represents internal friction within fluids - it's what makes honey flow slowly compared to water. The Reynolds number is a dimensionless parameter that characterizes flow regimes. It's defined as the ratio of inertial forces to viscous forces. When Reynolds number is low, viscous forces dominate and we get smooth, laminar flow like blood moving through small arteries. When Reynolds number is high, inertial forces dominate and we get chaotic, turbulent flow like air rushing over an aircraft wing. This transition from laminar to turbulent flow fundamentally changes how fluids behave.
Boundary conditions are crucial for solving the Navier-Stokes equations. The no-slip condition means fluid velocity equals the wall velocity at solid boundaries. Free-slip conditions apply at free surfaces where there's no tangential stress. Two classic analytical solutions demonstrate these principles. Poiseuille flow describes flow in a pipe, where the no-slip condition at the walls creates a parabolic velocity profile with maximum flow at the center. Couette flow occurs between parallel plates where one plate moves, creating a linear velocity profile. These solutions provide fundamental insights into how boundary conditions shape fluid behavior.