What are the Navier-Stokes Equations?

About Navier-Stokes Equations:

• The Navier-Stokes equations are a set of partial differential equations that describe the motion of viscous fluids.
• They are named after the French mathematician and physicist Claude-Louis Navier and the Irish mathematician and physicist George Gabriel Stokes, who developed them in the 19th century.
• The Navier-Stokes equations are based on the conservation of mass, momentum, and energy.
• They use Newton’s second law of motion (F = m x a) to describe how fluids move.
• Importantly, they treat a fluid as a continuous medium rather than tracking individual molecules.
• Equation:
  • The Navier-Stokes equation, in modern notation, is
    • ∂u/∂t + u·∇u = -∇P/ρ + v∇² u
  • where u is the fluid velocity vector, P is the fluid pressure, ρ is the fluid density, υ is the kinematic viscosity, and ∇² is the Laplacian operator.
• Navier-Stokes equations have been used to model a wide variety of fluid flows, including the flow of water in pipes, the flow of air around airplanes, and the flow of blood in the human body.
• The Navier-Stokes equation is used in a wide variety of applications, including:
  • Weather forecasting
  • Climate modeling
  • Ocean circulation
  • Aerodynamics
  • Fluid dynamics
  • Hydraulics
  • Lubrication
  • Combustion
  • Chemical engineering
  • Biomedical engineering
• There are a number of challenges associated with solving these equations, including:
  • The equations are nonlinear, which means that they cannot be solved using linear methods.
  • The equations are coupled, which means that they cannot be solved independently of each other.
  • Whether smooth solutions in three dimensions continue to exist for all time — the problem of global existence and smoothness — is still an unsolved mathematical question.
• Is the Navier-Stokes equation solved?
  • The Navier-Stokes equation is one of the most important unsolved problems in mathematics.
  • In 2000, whether smooth, reasonable solutions to the Navier-Stokes equation in three dimensions exist was designated a Millennium Problem, one of seven mathematical problems selected by the Clay Mathematics Institute of Cambridge, Massachusetts, U.S., for a special award.
  • The solution for each Millennium Problem is worth $1 million.
  • Despite these challenges, there has been significant progress in solving the Navier-Stokes equations in recent years.
    • This progress has been due in part to the development of new numerical methods and the use of high-performance computers.