Section 4: Examples Using the Navier-Stokes Equation
In general, these equations are handy to have as they establish a starting point for going
about modeling fluid flow. When it comes to analytically deriving models (as in using
pen and paper), it is orders of magnitude more diffucult when you deal with fluid that
move in more than one direction. By that, I mean that if fluid flow has direction partially
in the x-direction and the y-direction, you get a partial differential equation (PDE),
as opposed to just in either x or y or z, which would result in an ordinary differential
equation (ODE), which is what we will be mainly working in the following example. Not
even all ODE systems can be derived analytically, so whenever you deal with such
systems you would usually then go to modeling softwares like COMSOL, ANSYS, or
FEniCS, or computational fluid dynamics (CFD) softwares. As mentioned earlier,
systems like pipes and reactors tend to be done in cylindrical coordinates (z, r, θ).
Systems like flow between two plates, or most microfluidic systems would more often
use Cartesian coordinates (x, y, z).
Here’s a fairly simple example: steady-state fluid flow through a pipe with radius, R, and
length, L (sorry about the poorly drawn axes).

CAN SOMEONE SHOW ME ONE EXAMPLE OFF THIS!!!! PLS