Navier-Stokes Equation

The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations that describe the motion of fluid substances such as liquids and gases. These equations establish that changes in momentum (acceleration) of fluid particles are simply the product of changes in pressure and dissipative viscous forces (similar to friction) acting inside the fluid. These viscous forces originate in molecular interactions and dictate how sticky (viscous) a fluid is. Thus, the Navier-Stokes equations are a dynamical statement of the balance of forces acting at any given region of the fluid.

They are one of the most useful sets of equations because they describe the physics of a large number of phenomena of academic and economic interest. They are used to model weather, ocean currents, water flow in a pipe, motion of stars inside a galaxy, and flow around an airfoil (wing). They are also used in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of the effects of pollution, etc. Coupled with Maxwell's Equations they can be used to model and study magnetohydrodynamics.

The Navier-Stokes equations are partial differential equations which describe the motion of a fluid. These equations, unlike algebraic equations, do not establish relations among the variables of interest (e.g. velocity and pressure), rather they establish relations among the rates of change or fluxes of these quantities. In mathematical terms these rates correspond to their derivatives. Thus, the Navier-Stokes equations for the most simple case of an ideal fluid with zero viscosity states that acceleration (the rate of change of velocity) is proportional to the derivative of internal pressure.

This means that solutions of the Navier-Stokes equations for a given physical problem must be sought with the help of calculus. In practical terms only the simplest cases can be solved in this way and their exact solution is known. These cases often involve non turbulent flow in steady state (flow does not change with time) in which the viscosity of the fluid is large or its velocity is small (small Reynolds number).

For more complex situations, such as global weather systems like El Niño or lift in a wing, solutions of the Navier-Stokes equations must be found with the help of computers. This is a field of sciences by its own called computational fluid dynamics.

Letting $u_i$ be the $i$ th component of the velocity field(which is a vector field), $p$ be the pressure field(which is a scalar), $\rho$ be the density of the fluid, $\nu$ be the kinematic viscosity, and $f_i$ be the $i$ th component of the force field, the equations are:

$\frac{\partial v_i}{\partial t} + \sum_{j=1}^n u_j \frac{\partial u_i}{\partial x_j} = - \frac{1}{\rho}\frac{\partial p}{\partial x_i} + \nu\sum_{j=1}^n\frac{\partial^2 v_i}{\partial x_j^2} + f_i$ .

Even though turbulence is an everyday experience it is extremely difficult to find solutions for this class of problems. A $1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute to whoever makes substantial progress toward a mathematical theory which will help in the understanding of this phenomenon.

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Navier-Stokes Equation