next up previous
Next: Solution Up: The dynamics of thin Previous: Introduction

Governing Equations

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

  equation186

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 tex2html_wrap_inline326 is constant such that the pressure does not change along the streamline, the resulting equation for the velocity is:

  equation188

where u is the velocity as a function of the arc length and tex2html_wrap_inline330 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 tex2html_wrap_inline332 . The direction that the force acts will be determined by the sign of tex2html_wrap_inline334 . In the normal/tangential coordinate system, while tex2html_wrap_inline316 is less than 90 tex2html_wrap_inline338 , the force pulls the surface inwards. iSURFACE TENSION FORCES: The surface tension force also acts to pull the surface inwards. It is given as

equation191

the meridian radius of curvature is defined as tex2html_wrap_inline340 and the axisymmetric radius of curvature is defined as tex2html_wrap_inline342 iCENTRIFUGAL FORCE: The centrifugal force acts to pull the surface outwards and is tex2html_wrap_inline344 iPRESSURE FORCE: A positive net pressure balance tex2html_wrap_inline346 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.

eqnarray194

or in other words

  equation196

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 tex2html_wrap_inline348 and tex2html_wrap_inline330 respectively. tex2html_wrap_inline330 retains its definition as the nozzle velocity. tex2html_wrap_inline348 will be determined in the course of this nondimensionalization. The variables in this problem all now scale as:

eqnarray198

Preliminary substitution of these scaling terms into equation 4 gives

  equation200

The continuity equation yields

  equation202

The thickness of the sheet of water can be eliminated from equation 6 using the continuity equation such that

equation204

Substituting this result into the normal force balance equation yields

  equation206

From the last term of equation 9, the undetermined length scale, tex2html_wrap_inline348 , can be defined such that tex2html_wrap_inline358 . This allows for simplification to Taylor's equation I7

  equation208

The Bernoulli's equation, nondimensionalized with the newly defined length scale becomes Taylor's equation I5

  equation210


next up previous
Next: Solution Up: The dynamics of thin Previous: Introduction

brenner@math.mit.edu