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# Governing Equations

The jet is oriented so that it is vertical and pointing downward. We will start with the following definitions:

iz is the vertical distance from the nozzle, positive pointing in the direction of gravity ir is the radius of the bell and it is a function of z i is the angle that the bell makes with the +z axis, CCW is defined positive. is is the arc length along the bell it is the thickness of the interface

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:

iGRAVITATIONAL FORCES: The normal component of the gravity force is . The direction that the force acts will be determined by the sign of . In the normal/tangential coordinate system, while is less than 90 , the force pulls the surface inwards. iSURFACE TENSION FORCES: The surface tension force also acts to pull the surface inwards. It is given as

the meridian radius of curvature is defined as and the axisymmetric radius of curvature is defined as iCENTRIFUGAL FORCE: The centrifugal force acts to pull the surface outwards and is iPRESSURE FORCE: A positive net pressure balance will push the surface outwards.

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

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