Navier-Stokes Equations
Introduction
The Navier-Stokes equations are a set of partial differential equations that describe the motion of viscous fluid substances. Named after Claude-Louis Navier and George Gabriel Stokes, these equations have wide applications in the field of fluid dynamics and are a fundamental tool in the mathematical modeling of physical phenomena.
Derivation
The Navier-Stokes equations can be derived from the principles of conservation of mass and momentum. The conservation of mass, also known as the continuity equation, states that the rate of change of mass in a fluid particle is equal to the negative divergence of its velocity. This gives the equation:
∂ρ/∂t + ∇.(ρu) = 0
where ρ is the fluid density, u is the fluid velocity vector, and ∇. denotes the divergence operator.
The conservation of momentum, also known as the Newton's second law, states that the rate of change of momentum in a fluid particle is equal to the sum of the forces acting on it. This gives the equation:
ρ(∂u/∂t + u.∇u) = -∇p + μ∇²u + ρg
where p is the pressure, μ is the dynamic viscosity, ∇² is the Laplacian operator, and g is the acceleration due to gravity.
Mathematical Formulation
The Navier-Stokes equations can be written in a number of different forms depending on the assumptions made about the fluid. The most general form of the equations is the compressible form, which allows for changes in density. This form of the equations is used in the study of compressible flows, such as those found in gas dynamics.
The incompressible form of the Navier-Stokes equations assumes that the fluid density is constant. This simplifies the equations and is a good approximation for many liquids and low-speed gases. The incompressible form of the equations is:
∂u/∂t + u.∇u = -1/ρ ∇p + ν∇²u
where ν = μ/ρ is the kinematic viscosity.
The Navier-Stokes equations can also be written in a dimensionless form using the Reynolds number, a dimensionless quantity that describes the ratio of inertial forces to viscous forces. This form of the equations is useful in the study of turbulent flows.
Solutions and Applications
The Navier-Stokes equations are used in a wide range of applications in fluid dynamics, including the design of aircraft and cars, the prediction of weather patterns, the modeling of blood flow in the human body, and the simulation of ocean currents.
Despite their wide use, finding solutions to the Navier-Stokes equations is a challenging task. In fact, one of the seven Millennium Prize Problems posed by the Clay Mathematics Institute in 2000 is to prove or disprove the existence and smoothness of solutions to the 3D incompressible Navier-Stokes equations.