The jet is oriented so that it is vertical and pointing downward. We will start with the following definitions:
If the volumetric flow rate out of the nozzle is Q, then continuity/mass conservation gives
Bernoulli's equation can be used if viscous effects are neglected. The surface of the bell is a stream line. Assuming that the surface tension is constant such that the pressure does not change along the streamline, the resulting equation for the velocity is:
where u is the velocity as a function of the arc length and is the nozzle velocity. A force balance normal to the bell surface will be used to determine the shape of the water bell. Each component of the force balance can be dealt with separately:
The normal force balance results in the following with forces that pull the surface in on the LHS of the equation and those that pull the surface out on the RHS of the equation.
or in other words
In the process of nondimensionalizing these equations, Taylor produces a length scale which becomes a repeated length scale in all of the work on thin fluid sheets. However, Taylor's discussion of the nondimensionalization leaves much to be desired and will be discussed here in more detail so it becomes more evident how the length scale is determined.
We define the length and velocity scales to be and respectively. retains its definition as the nozzle velocity. will be determined in the course of this nondimensionalization. The variables in this problem all now scale as:
Preliminary substitution of these scaling terms into equation 4 gives
The continuity equation yields
The thickness of the sheet of water can be eliminated from equation 6 using the continuity equation such that
Substituting this result into the normal force balance equation yields
From the last term of equation 9, the undetermined length scale, , can be defined such that . This allows for simplification to Taylor's equation I7
The Bernoulli's equation, nondimensionalized with the newly defined length scale becomes Taylor's equation I5