simulation of the time intervals between every two drops as a function of the drop interval
finding strange attractors by creating Poincare maps, and trying different parameters like drop mass
critical speed and flow rate.
I will write a Runge-Kutta algorithm in order to solve the motion equations involved. the algorithm with fortran and run it on phelafel.
I will present all graphics using matlab.

Overview

I will use a model suggested by A.D'Innocenzo and L.Renna, Dripping faucet, 'International Journal Of Theoretical Physics' Vol. 35 (1996) page 941.

The model starts with a standard mass-on-spring setup: mass M which grows linearly with time, pulling on a spring with stretch constant k. the spring produces a linear restoring force of -kx, where x is the position of the forming drops center of mass (CM) and the spring constant k represents the surface tension. The damping of the residual oscillations is included in the 'friction' force -bv, where v=dx/dt is the velocity of the center of mass (CM).
We can describe this model by the equation:

The mass of the drop grows in a flow rate R:

Where a is a proportionality parameter to be suitably adjusted.

With regard to the initial condition of the residue mass mr=Mc-m When a drop falls, we can consider two different model systems of the breaking drop at the critical point xc:
  a) a spherical drop and a residue point mass (one sphere).
  b) a spherical drop falling off and another one forming a residue of successive drop (two spheres).

The left side of this figure shows the system of point mass and sphere. the system CM is at xc. the initial position for the residue mass is given by simple calculations to be:

Whereand Ro is the liquid density.

Where the two radii can be found from the same equation as the previous model.

NOTE: the two sphere model does not work in the simulation yet.

In their article D'Innocenzo and Renna wrote that they were still working on it.
If you'd like, you can investigate the problem using this simulation, and perhaps YOU can find the proper modulations needed to be put in the model so it'll work.
 

Now that we have all the equations of motion and initial conditions we can obtain the numerical solutions by means of standard forth order Runge-Kutta methods.
 

Running the simulation

First  download  all the files from my  ftp  site to one directory.
then open an x-terminal and type f77 chaos.f -o chaos.ex on the relevant directory (the one you downloaded all the files to) in order to compile chaos.f .
Now type chaos.ex to run the program.
The program will offer a few menus in which you can change the simulation parameters.
If you don't change anything, the simulation will run on it's default parameters.
Run the simulation from the main menu and wait until it is done.
For  help  with chaos.ex press  here .

Then run matlab on the relevant directory and type help chaos for instructions or chaos to run the program .
in order to get help on the matlab program just enter the number 0 while the program is running.
now matlab will read data from a file called chaos.dat which was produced by the simulation, and  plot the time interval between every two following drops as a function of the drop index (or drop number) such as the following:

Then you will be asked if you would like to create Poincare maps .
If you answer 'yes' you will be asked to choose a data range from the graph.
For example: these are poincare maps which were created from drop indices 6000 to 6500
of the above figure:

Then matlab will plot two diagrams one is a 2-dimentional Poincare map and the other is a 3-dimentional Poincare map.
 

Experimental results

Here you can see a few experimental results which I collected on the winter semester of 1999.

Time intervals plots	Poincare maps