User:HarryJA/Surface vibrations of a droplet

Surface Vibrations of a Droplet can be observed at the surface of liquid droplets and, in principal, any other type of particle assuming that the driving forces is large enough. Such vibrations can occur naturally; for example surface oscillations are seen in falling raindrops. The same mathematics applies to bubbles and other analogous systems. Certain properties of the fluid can be deduced from the nature of these vibrations. The mathematics of this system was first developed by the astrophysicist Subrahmanyan Chandrasekhar in his paper 'Oscillations of a Viscous Liquid Globe'.

Equilibrium Shape
In the absence of external forces pressure difference will be uniform across the droplet surface. The Young-Laplace equation shows a proportional relationship between the pressure difference and curvature. Therefore the curvature must be equal across the surface and the equilibrium shape must be a perfect sphere. 1 Hydrostatic pressure Weber number

Frequency and Surface Tension Relationship
A series of vibrational modes can be described using n = 0, 1, 2, 3.... The n = 0 mode requires that the entire surface moves uniformly, it is therefore a volume dilation. n=1 refers to a translation. Consequently these two modes are not relevant to the problem in hand and n = 2 shall be considered to be the primary mode from hereon. 1

Attenuation and Viscosity Relationship
Attenuation arises from viscous damping of the shape oscillation. Viscosity measurements on both levitated droplets and droplets residing on a surface have been attempted using this method.