The Kelvin-Helmotz instability apear when there is two fluid in contact, flowing out with non equal velocities. the interface between the two fluid is is subject of deformation. These deformations can get smaller and smaller or it can be amplified. in tne case of it's mplification we are in the presence of the Kelvin-Helmotz instability.
The stabilisator factor in this process is the superficiel tension.
to study this instability we translate the velocity field as it shwon in the figure.
we express the two velocity from each side of the flow.
we then express the velocity y-component with the deformation amplitude.
the stability condition is the folloween one