I'm working on dimensional analysis and I'm having trouble. Here's a problem from my book I'm working on. I'm supposed to consider a small sphere experiencing acceleration due to gravity $g$. The sphere is of radius $R$ and density $\rho$ and surrounded by a fluid of density $\rho_{f}$ and viscosity $\eta$.
I am supposed to determine the drag force on the sphere by dimensional analysis. But I really don't understand. I'd appreciate someone walking me through this.
Parameters:
- Force Drag (F) - $ML / T^2$
- Velocity (V) - $L / T$
- Radius of Sphere (R) - $L$
- Density of Sphere ($\rho$) - $M / L^3$
- Density of Liquid ($\rho_{f}$) - $M / L^3$
- Visocity ($\eta$) - $M / LT$
- Effect of Gravity ($g$) - $L/T^2$
First, are these the right parameters?
Now I have $7 - 3 = 4$ $\pi$ groups. I can figure out the exponents and what not - but I'm confused how I deal with multiple $\pi$ groups once I set up the dimensional analysis and get the exponents. Note, the end goal is to solve for a terminal velocity so I need an equation - setting up the $\pi$ groups isn't enough.
The problem also suggests me think of the sphere as a nucleus inside a cell and then to determine at what length scale that thermal forces, give by $kT$ (the Boltzmann constant times the temperature), are comparable to gravity and buoyant forces. What is meant by length scale and how do I apply dimensional analysis to get these quantities?