In Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Stokes’ law. When a small sphere, of radius a, moves with speed U through a fluid having a viscosity η, the drag force D is 6πηaU. This result is well known, but does it apply to a gas bubble moving in water?
I was reading through Life in Moving Fluids, by Steven Vogel, when I came across the answer. Vogel considers a fluid sphere moving in a fluid medium. His Eq. 15.8 is
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| Life in Moving Fluids, by Steven Vogel. |
Here, ηext is the viscosity of the external fluid and ηint is the viscosity of the internal fluid.
Suppose you have a sphere of water in air (say, a raindrop falling from the sky toward earth). Then ηint = ηwater = 10–3 Pa s and ηext = ηair = 2 × 10–5 Pa s. Thus ηext/ηint = 0.02. For our purposes, this is nearly zero, and the drag force reduces to Stokes’ law, D = 6πηextaU.
Now, consider a sphere of air in water (say, a bubble rising toward the surface of a lake). Then ηint = ηair = 2 × 10–5 Pa s and ηext = ηwater = 10–3 Pa s. Thus ηext/ηint = 50. For our purposes, this is nearly infinity, and the drag force becomes D = 4πηextaU. Yikes! Stokes’ law does not hold for a bubble. Who knew? (Vogel knew.)
Apparently when the sphere is a fluid, internal motion occurs, as shown in Vogel’s picture below.
Suppose you have a sphere of water in air (say, a raindrop falling from the sky toward earth). Then ηint = ηwater = 10–3 Pa s and ηext = ηair = 2 × 10–5 Pa s. Thus ηext/ηint = 0.02. For our purposes, this is nearly zero, and the drag force reduces to Stokes’ law, D = 6πηextaU.
Now, consider a sphere of air in water (say, a bubble rising toward the surface of a lake). Then ηint = ηair = 2 × 10–5 Pa s and ηext = ηwater = 10–3 Pa s. Thus ηext/ηint = 50. For our purposes, this is nearly infinity, and the drag force becomes D = 4πηextaU. Yikes! Stokes’ law does not hold for a bubble. Who knew? (Vogel knew.)
Apparently when the sphere is a fluid, internal motion occurs, as shown in Vogel’s picture below.
Note that at the edge of the sphere, the internal and external flows are in the same direction. This changes the boundary condition at the surface. A rigid sphere would obey the no-slip condition, but a fluid sphere does not because the internal fluid is moving.
Although Vogel doesn’t address this, I wonder what the drag force is on a sphere of water in water? Does this even make sense? Perhaps we would be better off considering a droplet of some liquid that has the same viscosity as water moving through water (I can imaging this might happen in a microfluidics apparatus). In that case the drag force becomes D = 5πηextaU. I must confess, I’m not sure if the derivation of the general equation is valid in this case, but I don’t see why it shouldn’t be.
There are all kinds of little jewels inside Vogel’s book. I sure wish he were still around.
Yes, this is a neat topic! I was unaware of the fluid/fluid drag coefficient until about 10 years ago. My lab used it to examine lipid vesicles -- https://doi.org/10.1103/PhysRevResearch.2.013132 . I'm fond of this paper in part because it has the shortest title of anything I've published: Lipid bilayer hydrodynamic drag
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