I'm trying to develop a very basic scaling law/unit analysis for viscous droplet formation, and I'd like to get some rough numerical values of the Reynolds number to play with. To be specific, I'm looking at the behavior of the smaller of the two droplets shown in the picture below (experimental setup shown sideways):
The darker fluid is a glycerol/water mixture and the lighter fluid is mineral oil - so it's a viscous fluid dropping into another (dissimilar) viscous fluid. I'm trying to understand how the viscosity of the dark fluid effects the size of the small droplet - the large droplet pretty much remains the same size, but the small one gets smaller for lower viscosities.
Since I'm trying to work only with dimensionless parameters, I'd like to work with something like Reynolds number instead of viscosity. I know that $\text{Re}=\frac{\rho \mathbf{v} L}{\mu}$ is the "standard" formula, but I want to make sure this applies here, and if so, that I'm using the correct values for the parameters. I can calculate $\bf{v}$ from my high speed video, and I can calculate $\rho$ and $\mu$ for both fluids using a formula, but I'm wondering:
- Do I need to use a ratio of the two densities and viscosities, or do I just use one? Would I use the viscosity of the stationary fluid or the moving fluid?
- Does the width of the channel make sense for the characteristic length $L$, or should I go with something more related to the droplet?
Feel free to suggest a book or online resource if this doesn't have a simple answer. I have to admit I'm very hazy on the "physicist's" viewpoint on fluids (I'm a math grad student). Thanks in advance!