Navier Stokes Equation

Introduction:

Navier stokes equation was remarkable effort of Claude Louis Navier and George Gabriel Stokes. It refers to fluids motion. Navier Stokes equation is derived from Newton’s second law of fluid motion. Major assumption of this equation is fluid stress which includes both diffusing viscous term sum and pressure term. It helps in prediction of modeling ocean currents, weather, air and water flow, blood flow study, aircraft designing, automobile designing, power station designing, stars motion in galaxy and other related. The understanding of magneto hydrodynamics is made better by coupling Navier Stokes equation with Maxwell equation. This equation sounds velocity rather than location. The velocity at particular time can be identified by velocity field or flow field of Navier Stokes equation. When the velocity is known through this equation, then drag force and flow rate can be easily found. Knowledge of velocity is essential than position in case of fluid study.

Properties of Navier Stokes equation

• Navier Stokes equation of real fluids is non linear partial differential equations. It may be because of convective acceleration. Convective acceleration refers to acceleration that is coupled with velocity.
• Navier Stokes equation incorporates turbulence because of inertia of fluid and is time dependent. This equation defines turbulence in systematic way. Time averaged equations such as Reynolds averaged Navier Stokes equation (RANS) is a turbulence model. It finds its own application in computational fluid dynamics. Similar to this Large Eddy Simulation (LES) model was developed. LES is more cost effective than RANS. LES provide users with good quality results.
• Navier Stokes equation formulates fluid motion accurately. It is assumed that substances are in continuum without relative velocity. Statistical mechanics and molecular dynamics can be substantiated by this equation.
• Navier Stokes equations provide exact solution to statistical problems by non linear equation, for instance Taylor Green vortex.

Questions:

• What are applications of Navier Stokes equation?
• Derive Navier Stoke equation.