Yes, and you will, you will keep on seeing that stuff many times. That is the essence of fluid mechanics. You better understand it. Even though some people here and there claim to be engineers, I have never seen such a fantastic maneuver of practical engineering such as that proposed by former scientists on the field to non dimensionalize the equations and split up the physical phenomena into modes. The symbol O is what is called a big O of Landau, and it is read as "of the same order-magnitude than..". For instance, consider the dimensional momentum equation:
\rho\overline{u}\cdot \nabla\overline{u}=-\nabla P+\mu\Delta\overline{u}
and assume that your problem has a characteristic velocity U and length a, such as those stemming from the motion of a body in a fluid. The non dimensionalization of the velocity and spatial coordinates is clear: \overline{u}=\overline{u}/U and \overline{x}=\overline{x}/a. The non dimensionalization of the pressure IS NOT trivial at all. Two characteristic pressures emerge from the body motion: \rho U^2 and \mu U/a. At first sight one may think it does not make sense to non dimensionalize pressure with a measure proportional to the kinetic energy in slow flows, such as in hydrodynamic lubrication, where the flow is slow but the overpressures are large. Similarly, it wouldn't make sense to non dimensionalize pressure with the viscous stress for a slightly viscous fluid. Both non dimensionalizations have a famous name:
1)P=P/\rho U^2 is called the Reynolds Scaling. The non dimensional momentum equation then reads:
\overline{u}\cdot \nabla\overline{u}=-\nabla P+\frac{1}{Re}\Delta\overline{u}
so that \nabla P\sim O(\overline{u}\cdot \nabla\overline{u}) if the Reynolds number Re\gg 1. If you calculate the Reynolds number of your flow and turns out to satisfy this condition, the above scaling is well posed. Moreover, the resultant equation is the Euler equation for inviscid flow:
\overline{u}\cdot \nabla\overline{u}=-\nabla P
2)P=P/\mu U/a is called the Stokes Scaling. The non dimensional momentum equation then reads:
Re\overline{u}\cdot \nabla\overline{u}=-\nabla P+\Delta\overline{u}
so that \nabla P\sim O(\Delta\overline{u}) if Re\ll 1. If you calculate the Reynolds number of your flow and turns out to satisfy this condition, this scaling is well posed. Moreover, the resultant equation is the Stokes equation for viscous flow:
\nabla P=\Delta\overline{u}
I don't know what is a rotlet but I can help you with the Stokeslet. This stuff can be explained by using Green's functions of the stokes equation, but I'm avoiding such a mess and put an easy example. Imagine a sphere translating through a quiescent viscous fluid. The non dimensional stream function can be written as:
\psi=\frac{1}{4}\left(3r-\frac{1}{r}\right)sin^2\theta
The first and second terms are called Stokeslet and Doublet or Dipole respectively. Surprisingly, the Stokeslet is the only term that is rotational. Look at distances very far away from the sphere. The stream function can be written at first approximation as \psi\sim 3rsin^2\theta/4, so that the only term that matters in that region is the Stokeslet. That term represents the flow field caused by a point force acting in a viscous fluid. Therefore, the effect of the moving sphere can be approximated far away as the effect of a point force acting at the center of the sphere (it sounds like the definition of a Green's function, isn't it?).
Hanson, all the above analysis is INTIMATELY-CLOSELY related to ASYMPTOTIC ANALYSIS. In both Stokes and Reynolds Scalings, the Reynolds number or its inverse appears as a small parameter, multiplying the neglected terms. But nothing is absolutely negligible in nature. Asymptotic analysis takes care of those SMALL REMAINDERS carried when you neglect some effect. Turns out to be that those apparently small effects neglected DO HAVE sometimes an extraordinary effect on the fluid motion. It is because of that reason that the Reynolds number in the Reynolds and Stokes scalings is referred as a SINGULAR PERTURBATION PARAMETER. That is for instance, what happens with the Euler equation and the Stokes equation in a Boundary Layer flow and the Viscous flow around a sphere respectively, where to leading order in 1/Re and Re the flow is described by the mentioned Euler and Stokes equations (respectively), leading to classical paradoxes in fluid mechanics (the boundary layer one solved by Prandtl and the sphere motion- The Whitehead's paradox- solved by O'seen).
I don't know if you get the picture, but to my understanding, a well posed non dimensionalization is the first step on the way of "expanding in Asymptotic Series" the equations of motion around the flow regime (Re) in study. That's what we do in physics of fluids, we work out the scales of the problem, we localize small parameters, and we expand the equations around those small parameters. Man, that's all the artifact needed to formulate any problem in this field, but unfortunately understanding that artifact and locating the point of expansion is something that is only in the range of people with a lot of experience on the field. If you plan to get deeper in Fluid Mechanics as a graduate student, maybe you luckily happen not to have a problem statement for your thesis but you have to set it up by yourself (like me). In case of that being an analytical problem, you'll be surely enforced to use these powerful analytic methods.
Hope this helped.
