In the previous lectures, we took-up couple of examples on the application of the NS-equation, in particular, the flow in a circular tube. You must have realized that the NS–equation is non–linear on velocity, and the velocity and pressure–fields are coupled. In general, it is difficult to solve the NS equation, and more often numerical techniques are used to solve the equation. In this context. approximations are made to simplify the equation for solution. In this lecture, we will make use of the dimensional analysis to non–dimensionalize the NS equation, and then, explore the possibility of approximation.

x-momentum

Non–dimensionalize the variables as follows:

 ,

The characteristic variables are generally chosen to make the dimensionless quantities vary between 0-1. Therefore, L may be the length of a tube or channel.