Last weekend, I was making another batch of my friend Jean's
dosa recipe, and I started thinking about the whirlpool in my blender:
 |
| Water, not batter, so it's easier to see |
I was curious if I could figure out the shape of the well, and how it depended on the speed and ingredients.
The basic idea is that we have water molecules with some angular momentum, rising to some height
h based on the distance from the center,
r. If we ignore drag/viscosity, we can just set the rotational energy equal to the gravitational energy:
where
ω is the angular velocity, and
m is the mass. Solving for
h gives
This has a parabolic cross-section, which is characteristic of a rotational vortex, according to
Wikipedia.
Really though, there's some drag on the liquid: It's moving quickly in the center, where the blades are, but at the walls it has to slow down. I looked at the models for
viscosity of liquids, and I was hopeful I could find a way to apply it here:
F is the force between one sheet of liquid and the next, 𝜇 is a constant that depends on the liquid,
A is the area of the liquid, and
v is its rotational speed. In our case we can write
I was hoping I could solve this, with an input velocity at the center, and an output velocity at the outside, but I'm not sure what to do with the time derivative. Looking at the photo up there, I'm inclined to say a parabola is good enough, unless one of you readers can suggest a way to solve this!
I think you want the steady state solution, when presumably partial v/partial t is zero. But I think since you got there from F, you want the full time derivative of v, which should just be r*omega^2.
ReplyDeleteIf that's the right idea, then you still need the detailed shape of partial v/partial r to get the answer. I guess you can plug in the parabola for h and a linear function for v and iterate until it converges. Or something.