fluid dynamics - bubble/drop Reynolds number

The bubble/drop Reynolds number makes me confused and I hope someone can help me on this please!

Normally (as I read in every books and papers) that when a bubble or drop rises in a fluid, the bubble/drop Reynolds number is calculated by:

Re = ρUD/μ

where U is particle velocity, D can be particle diameter, and ρ and μ are density and viscosity of continuous fluid

my question is why don't use ρ and μ of bubble/drop? why use values of surrounding fluid? what is the physical meaning of this Re?

Thanks in advance.

Answer

Actually all of them have to be considered. The flow depends on $\rho,\mu,D,U,\rho_{bubble},\mu_{bubble}$. Any quantity of interest would be a function of dimensionless parameters that are formed from these variables. You may make use of Buckingham's Pi theorem to determine what they are. In this case mere inspection tells you that there are three dimensionless numbers: \begin{align} Re\equiv \frac{\rho U d}{\mu},\quad \frac{\rho_{bubble}}{\rho}, \quad \frac{\mu_{bubble}}{\mu} \end{align} However dependence on three parameters is complicated, so we seek simplification. Usually if we may adopt the approximation that one of the parameters tends to $0$ or $\infty$, then we may drop that parameter from the list. Another case in which you may drop parameters is if, in the particular case you are studying, those parameters are held constant.

For example if you are studying air bubble in liquid, as an approximation you may analyse governing equations in the limit, $\frac{\rho_{bubble}}{\rho}\to 0, \frac{\mu_{bubble}}{\mu}\to 0$, in which case solutions obtained will be a function of $Re$ alone. If you are an experimenter studying motion of air bubbles of different size $d$ in water, then $\frac{\rho_{bubble}}{\rho}, \frac{\mu_{bubble}}{\mu}$ are constant (for this experiment) and results depend on $Re$ alone. But if you change your experiment to studying motion of bubbles in water of identical size but varying gas inside the bubble, then $Re$ remains constant and results depend on $\frac{\rho_{bubble}}{\rho}$ and $\frac{\mu_{bubble}}{\mu}$. In doing experiments, you exploit the freedom to form dimensionless numbers any way you want in such a way that, as far as possible, resulting dimensionless numbers can be varied independently of each other in an experiment. My point is that there is nothing sacred about adopting $Re$ as defined above, but the choice depends on context and the simplification that results.

Suppose instead of the definition above for dimensionless parameters, you had the following altered definitions \begin{align} Re'\equiv \frac{\rho_{bubble} U d}{\mu_{bubble}},\quad \frac{\rho_{bubble}}{\rho}, \quad \frac{\mu_{bubble}}{\mu} \end{align} In principle there is nothing wrong with this new set of definitions. Now suppose you conduct the very last experiment of changing gas inside the bubble. Then whenever you change the gas inside the bubble, all three parameters above would change, while with the first set of definitions only two parameters would change. So if you wanted to plot some dependent quantity, say drag on the bubble (suitably non-dimensionalised), then with the first choice you have two independent variables, but with second choice you have three independent variables. However both graphs are equivalent in the sense that they are interconvertible, i.e. they contain the same information. So the first choice in this case would make your life easy.