Periodic orbits of four point-particles settling under gravity in a viscous fluid

M.L. Ekiel-Jeżewska and M. Sikora

What is a point-particle?

What does it mean a viscous fluid'?


I. Initially, the particles are aligned horizontally in the following symmetric configuration:


Here L is the length unit, and C is a parameter.


II. Guess: how will the particles move?

Click on the arrows below to see how the particle motion depends on the parameter C.




(time unit τ = 8 π η L2)


1. The particles perform a periodic motion in addition to settling down.

2. Shape of the trajectory depends on a value of the parameter C.

3. After a certain number of oscillations, the system destabilizes and splits into two pairs of particles.

4. Period, amplitude and destabilization time increase when C is increased.



III. How does the motion look like in the reference frame moving with the center of mass?


The center of mass moves with velocity VCM equal to the mean velocity of all the particles.

In general, VCM is not vertical and it varies with time.


Click on the arrows below to observe the particle dynamics in the reference frame moving with the center of mass.





1. Motion of the particles in the laboratory frame is the superposition of the center-of-mass motion and the relative particle motion observed from the center-of-mass reference frame.

2. The relative motion of the particles is periodic, and the particle trajectories are closed.

3. Shape and size of the trajectories depend on C.

4. The orbits seem to be unstable - owing to a small inaccuracy of the computations, after a certain time they are destroyed.



IV. How does the trajectory shape depend on the initial configuration?

Move the arrow along the bar to choose value of the parameter C. Remember you are in the frame of reference moving with the center of mass.

Keep in mind that the vertical and horizontal scales on the plot are different!

V. Remarks

The class of the periodic motions shown above was found and analyzed by Hocking [1], for C≤1.9. Our calculations confirm that periodic motions exist also for higher values of C, up to C=2.41. The period increases significantly with C. The question if there exists a limiting value of C, above which there is no periodicity, remains open.

[1] L. M. Hocking, J. Fluid Mech. 20, 129 (1964).