Dimensional or dimensionless constant While deriving new equations , how do theoretical physicists know whether the proportionality constant in their equation will be dimensional or dimensionless?
I mean, say for example, we consider Stokes' Force in hydro-statics.
There the experimenter saw that $F$ varies as $\eta$,$r$ and $v$ where the symbols have the usual meanings. Hence he got the equation $F = k\eta^{x} r^{y} v^{z}$ where $k$ is the proportionality constant and it is strictly dimensionless. So he found $F = 6\pi r\eta v$ where $k=6\pi$ by statistical analysis. (He could have gone for other powers of the variables so that $k$ has a unit.) Same is for the expression for Reynold's Number.
Again if we consider Newton's Law of Gravitation, there the experimenter saw that $F$ varies as $m_1$, $m_2$ and $\frac{1}{r^2}$ and concluded the relation $F = \frac{Gm_1m_2}{r^2}$ where $G$ is the proportionality constant and it is strictly dimensional with proper units. (He could have gone for other powers of the variables so that $G$ has no unit.) Same is for the expression for Resistivity.
I don't know if I have been able to express my question through the examples. But if I have , please help.
 A: There is no answer to this. When you are taught to use dimensional analysis at school the teacher invariably selects an easy example (it's almost always the pendulum) to keep things simple. In the real world there is no guarantee that you have a dimensionless constant.
It's actually quite rare to use dimensional analysis to derive equations in the real world. The sorts of simple systems that are amenable to dimensional analysis are usually already well known. However it's very, very useful to use dimensional analysis to check that an equation you derive is dimensionally consistent. 
For example suppose you're working through a differential equation for some quantity, and after covering many sheets of paper with scribbles you end up with a final equation. It's very easy to make a minor mistake along the way, so the first thing you check is that your final equation is dimensionally consistent, i.e. the dimensions of the left and right sides are the same. If they aren't that means you've made a mistake somewhere. I routinely do this in my answers to questions on this site!
A: 
He could have gone for other powers of the variables so that $G$ has no unit.

No he couldn't have!  Then the equation wouldn't have accurately described the force of gravitational attraction between two objects.  Scientists can't just "choose" the equation to look like whatever they want; it has to accurately describe experiment.
Dimensionless constants don't need units in order to get units to play nice; constants with dimensions do.
