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Solution

The solution for this equation can be achieved analytically if both the pressure effects, , and the gravitational effects, , are small. In this event, equation 11 reveals that the velocity U is 1 and the force balance equation (10 simplifies to

The solution presented here will now take a different course than the one that Taylor did and follow the work of Boussinesq to find the radius of the bell as a function of the distance away from the jet Z.

Define the radius of the bell as a function of the distance away from the jet, . Some geometric facts that are very useful:

Beginning with equation 15, taking the derivative with respect to Z yields:

Substituting equation 12 and using equations 13 and 14 results in a familiar final ordinary differential equation for R(Z)

with the boundary conditions that the initial slope be given by and the initial radius be zero. The solution of the ODE is a catenary of the form

Applying the boundary condition regarding the initial slope yields that . Applying the zero initial radius condition requires that such that

The form of this equation is quite straight forward and easily verifiable through experiments. Taylor used a horizontal jet to verify his equation, which after inspection, is essentially the same as equation 19. The choice of a horizontal jet is quite interesting because it allows him to say that the gravity term is not that important because of the symmetry he obtains in his experiments. His selection of the orifice size made that quite true (proof by selection, so to speak).

brenner@math.mit.edu