Drag - Dimensional Analysis / Buckingham $\pi$

Question

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?

Answer 1

In answer the specific questions that you pose:

1. Your aim is to determine the drag force on the sphere (via dimensional analysis). Have you set up a free-body diagram of the problem? This might help you think about what parameters are actually important to the problem. Hint: the list of parameters you give are sufficient for a complete description of the sedimentation of a particle under gravity, but are over sufficient for determining just the drag force. You can immediately eliminate two of these parameters to give you just 2 dimensionless (Pi) groups.

2. What is a length scale? In short, it's a characteristic distance over which forces act, e.g. the length scale for viscous forces acting on a sphere falling through a fluid is the size (radius or diameter) of the sphere. In saying that two forces are comparable, we can (more or less) equate them. In your example, what is essentially being asked is "how small does the nucleus have to be in order for thermal forces to be as large as the gravity/buoyancy forces?".

Answer 2

You are taking the dynamic viscosity as a variable, dimensions $[ML^{–1}T^{–1}$]. You should try using the kinematic viscosity, which has dimensions, $[L^2T^{–1}]$.

Answer 3

There are two densities present, one appearing in the body's kinetic energy, the other intuitively affecting how the fluid resists its movement. So we should expect $\rho_f,\,\eta$ to wait in one corner, while other variables get the dimensions of force.

Since $\rho R^3$ has the units of mass, $\rho R^3 V^2$ has the units of energy and $\rho R^2 V^2$ has the units of force.

The dependence on $\rho_f,\,\eta$ remains to be seen: they're absorbed into a dimensionless coefficient dependent on the also dimensionless Reynolds number $\propto\rho_f RV/\eta$. Of course, this parameter can be wrapped in an arbitrary function, so the precise form of the drag force is a little vaguer than in most textbook dimensional analysis problems.

There's no $g$-dependence at all. Nor should there be: we expect a finite but non-zero drag force in zero gravity.