TLDR:
I was reading up on the Navier Stokes equations today and it is so elegant that it might be my favourite math equation.
Equation 1: ∇u = 0 (conservation of mass) states that the divergence of the velocity vector u is zero, meaning there is no net change in fluid mass.
Equation 2: ρ Du/Dt = -∇p + μ∇^2 u + ρF (conservation of momentum) expresses Newton's second law for fluid flow. It balances the acceleration of fluid particles (LHS) with internal forces (pressure and viscosity) and external forces (gravity or other external influences) on the RHS.
This equation is foundational for modelling various fluid dynamics scenarios, from celestial bodies like stars and galaxies to F1 cars.
Long Version:
Here's how it works:
Equation 1:
∇u = 0 (conservation of mass)
So, u is velocity that can be represented as (u,v,w) vector, where u,v,w are x,y,z components of the vector.
∇u tells us that we need to do a partial derivative on u.
So,
∇u = ∂u/∂x + ∂v/∂y + ∂w/∂z = 0
or, the partial derivative of every component wrt corresponding direction is 0.
Equation 2:
ρ Du/Dt = -∇p + μ∇^2 u + ρ F (conservation of momentum)
LHS:
Since, u is velocity, then Du/Dt is acceleration and ρ is density. Newton's second law, F = m x a, applies here. Wherein, Du/Dt is acceleration of fluid particles and m is the density of the fluid.
RHS:
-∇p + μ∇^2 are the internal forces of particles hitting into each other while F represents the external force.
F in most cases is gravity, so one can replace it with g. However, if you put in electromagnetism then, you can combine Navier-Stokes with Maxwell's equations.
This has over time led to the development of magnetohydrodynamics, ie how stars and galaxies form. You can model the growth of our sun with this.
∇p is our pressure gradient and represents the change in pressure. Essentially, fluids move from high pressure to low pressure.
μ∇^2 represents viscous forces yielding from viscosity.
Imagine this can model aerodynamics of F1 cars.
