Navier-Stokes Equations

The Navier-Stokes equations represent the conservation of momentum. They arise from applying Newton’s second law to fluid motion (along with some other assumptions).

ns

These equations are always solved together with the continuity equation:

ce

The continuity equation represents the conservation of mass.

Energy conservation equation represents the first law of thermodynamics.

ee

Together with the equation of state such as the ideal gas law (p V = n R T), the flow and temperature field is solved.

Some books view all three (continuity, momentum and energy equations) combined as Navier-Stokes equations.

Assumption

Fluid is referred to as a continuum (representative physical length scale of the system is much larger than the mean free path of the molecules that make up the fluid). The ratio of the mean free path, λ, and the representative length scale, L, is called the Knudsen number, Kn=λ/L. The NS equations are valid for Kn<0.01. For 0.01<Kn<0.1, these equations can still be used, but they require special boundary conditions. For Kn>0.1, they are not valid. At the ambient pressure of 1 atm for instance, the mean free path of air molecules is 68 nm. The characteristic length of the model should therefore be larger than 6.8 μm for the Navier-Stokes equations to be valid.

General form

ns

For Compressible Newtonian fluid (constant viscosity)
ns1

 For Incompressible Newtonian fluid (constant viscosity and density)
ns2

Assuming inviscid flow (ideal fluid – no viscosity), we get Euler equation from Navier Stokes equation. Euler’s equation in the streamwise direction with the z-axis directed vertically upward becomes

ee

Bernoulli Equation arises from integration of Euler’s Equation along a streamline for steady flow

be

Integration of above equation (along s) gives

be2

Hence the restrictions for the Bernoulli equation are steady, inviscid, incompressible and irrotational flow (flow along a streamline).