In previous articles on this website focused on finite control volumes. Normally, this is a very practical. This is because detailed knowledge of pressure and velocity variations inside the control volume do not need be considered. Instead, only knowledge of what is occurring on the control surface is needed. However, what if you need to have detailed knowledge the flow field? If this is the case than the finite control volume method will no longer work. In stead you will have to use a differential analysis.

Some examples of when you may need to use a differential analysis are the following. One example would be if you want to know how the fluid velocity varies as it passes over a pipe. Another example would be would be determining how pressure and shear stress varies along the wing of an airplane. Whatever the case,, you will need to take an approach that involves an infinitesimal control volume, which will allow you to perform a differential analysis. As a result, you will need to know how to apply and solve differential equations.

Now I will warn you, since differential equations will need to be used, some of the equations in this section will become quite complicated. There will also be cases where there will be non-linear differential equations that can’t be solved accept for a few cases. In other words most of what you see in this section will be more practical to run on a computer. However, if you ever start using computational fluid dynamic (CFD) software, what you will see in this section will provide you with an idea of what is occurring behind the scenes when the solver runs.