So, y’know that little dimple found at the plug holes of draining baths and sinks, in the top of hurricanes and in coffee if swirled in a circular function? Well I sure do. And I was wondering what form it would take. So I worked it out.
Using the Euler Equations for an ideal, incompressible flow in cylindrical polar coordinates, at position (r, θ, z), for a stationary flow which is independent of θ, we have
p denotes pressure, g is acceleration due to gravity, and ρ is the uniform constant density (as this is a simple model). We assume that each particle traces a horizontal circle whose centre is on the fixed vertical axis z (see the diagram).
The speed at a distance r from the axis is
(I looked this up in a set of fluid mechanics notes dealing with a similar problem)
We are concerned with the region 0 < r < a, i.e. the dip in the surface of our fluid.
If we look back at the first couple of equations, we can see that
where c(z) is an arbitrary function and
hence, c(z) = -ρgz + c0
where c0 is an arbitrary constant.
Hence, the pressure function can be expressed as
At the free surface of the water, the pressure is constant atmospheric pressure p0, if we substitute this into the pressure function and rearrange, we get
Hence, the depression in the free surface for r < a is a parabolic surface of revolution.
Note that the pressure is only ever globally defined up to an additive constant so we can take c0 = 0 or c0 = p0, if we like.
Much of this is based on the introductory fluid mechanics course delivered by Simon J A Malham at Heriot Watt University in the spring of 2013