Re: Is there an equation to define the parabola caused by circular motion?

Date: Wed Feb 8 10:14:18 2006
Posted By: Tetjana Ross, Faculty, Oceanography, Dalhousie University
Area of science: Physics
ID: 1138679447.Ph

Message:

Hi Nathan,

You’re right in guessing that you need calculus to derive the equations that describe the parabola on the surface of a rotating fluid from scratch. Without calculus, however, we can still write down the equation that leads to the parabola and talk about where the different terms come from.

If we look at any two points in the rotating fluid (once it’s spun up into what’s called solid body rotation) then the difference in pressure between the points 1 and 2 (i.e. p1 and p2) will be given by:

p2 – p1 = rho w^2 (r2^2 - r1^2)/2 - rho g (z2 - z1)

where rho is the density of the fluid (it’s mass per unit volume), w is the angular velocity of the fluid, g is the gravitational acceleration, r1 and r2 are the radial distances of the two points from the center of the cylindrical container, and z1 and z2 are the vertical distances of the two points from the bottom of the cylindrical container.

Since pressure is force per unit area, this is basically an (F=ma style) force equation for the rotating fluid. On the left hand side we have force per unit area (the F). On the right hand side we have the Coriolis ‘acceleration’ and the gravitational acceleration multiplied by the density (the ma’s).

To get the parabola describing the surface of the fluid, we need to remember that atmospheric pressure requires that the surface of the fluid is at constant pressure, so any two points on the surface will have p1 = p2, so that:

w^2 (r2^2 - r1^2)/2 = g (z2 - z1)

which is a parabola. To plot it out you’ll need to arbitrarily assign an origin (say the first point is the surface height in the middle of the cylinder, making z1=r1=0).

I hope this satisfies your curiosity. If you want to learn more about the calculus behind these equations, I suggest you find a good basic fluid mechanics textbook. My favourite (which I used to help me write this response) is “Fluid Mechanics” by P. K. Kundu. If you want to learn more general info about fluid rotation and vortices you could check out the Wikipedia page on the vortex .

Cheers,
Tetjana

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