Continuum Fluid Flows

Consider the following, very generic pair of differential equations
that describe the dynamical motion of a particle (or a group
of particles) in the presence of some "external" force **F**,
which almost always is a function of the particle's position
and sometimes also depends on the velocity of the particle.

D_{t}**x** = **v**

It is this pair of equations that must be integrated forward in time in a self-consistent fashion when you're performing classical "N-body" (e.g., molecular dynamics or stellar dynamics) simulations. Rather than modeling a system that is composed of discrete particles (which is usually covered in the PHYS7411), here we will consider how to follow the time-evolution of a continuum fluid that moves under the influence of an external force.

First, let's discuss the difference between "Lagrangian" and "Eulerian"
representations of the equation of motion. Then let's walk through
a sampling of the wide variety of ways the equation of motion can be
written before making a tactical decision regarding the *form* of
the equation that we will choose to write in finite-difference form.

Useful on-line resources:

- Principal Governing Equations
- Various forms of the equation of motion
- Compare to Navier-Stokes equation listed in the original course Overview