The ultimate purpose of this analysis is to explain the wind velocities in tropiical cyclones (hurricanes, typhoons, etc.) and tornadoes. It is however convenient for the analysis to start with water draining down drains.

Energy Changes in Drainage

When water flows from a vertical level of z₁ to vertical level of z₂ it loses potential energy per unit mass equal to gΔz where Δz equals (z₁−z_2]) and g is the acceleration due to gravity. That loss of potential energy must go into kinetic energy. The gain in kinetic energy per unit mass in the fluid is Δ(½v(z)²) where v(z) is fluid velocity at vertical level z. Consider the changes over a time interval Δt and the limits as that interval goes to zero.

Thus a balance of energy requires that

g(dz/dt) = d(½v(z)²)/dt
and hence
gv(z) = v(z)(dv(z)/dt)
and further
(dv(z)/dt) = g

This just says that acceleration is equal to g.

But this can be expressed as

d(v(z)/dz)(dz/dt) = g
or, equivalently
v(z)d(v(z)/dz) = g
or, also
d(½v(z)²./dz) = g
which, upon integration
gives
½v(z)² = C + gz

where C is a constant.

The quantity z is just the distance which the water has fallen. If z=0 then v(z)=0 so the contant C is equal to zero. Therefore

½v(z)² = gz
v(z) = (2gz)^!/2

The Components of Fluid Velocity

The fluid velocity v(z) is made up of two components: The vertical velocity v_z and the tangential velocity v_θ, where

v(z)² = v_z² + v_θ²

Assume the drain is circular with the radius R being a decreasing function of vertical level; i.e., R(z). Assume for now that the vertical velocity is the same at throughout a vertical level. The mass per unit time M that passes through the circular area at level z is given by

M = μπR(z)²v_z

where μ is the density of water.

Thus

v_z = (M/(μπ))/R(z)²
and
v_z² = (M/(μπ))²/R(z)⁴

Therefore

2gz = (M/(μπ))²/R(z)⁴ + v_θ(z)²

But the tangential velocity is proportional to the distance r from the center of the drain; i.e.;

v_θ(z, r) = r(z)ω(z)

where ω(z) is the angular rate of twisting of the water stream at level z. Its value is give by the condition

2gz = (M/(μπ))²/R(z)⁴ + R(z)ω(z)
and hence
ω(z) = 2gz/R(z) − (M/(μπ))²/R(z)⁵

(To be continued.)

Conclusions