The Navier-Stokes Equations

They are a set of partial differential equations that describe the motion of fluids. They are named after Claude-Louis Navier and George Gabriel Stokes, who derived them in the 19th century.

The equations describe the conservation of mass, momentum, and energy in a fluid. They are used to model a wide range of physical phenomena, from the flow of air over an airplane wing to the behaviour of blood flowing through the human body.

The Navier-Stokes equations are expressed mathematically as follows: ∂ρ/∂t + ∇·(ρv) = 0 ρ(∂v/∂t + v·∇v) = -∇p + μ∇²v + f ∂(ρe)/∂t + ∇·(ρev) = ∇·(k∇T) + Q

Where, ρ=density of the fluid v=velocity p=pressure μ=dynamic viscosity f=external force per unit volume e=internal energy per unit mass k=thermal conductivity T=temperature Q=heat source term