1. While one airstream moves up the tube and the other down it, both rotate in the same direction at the same angular velocity. That is, a particle in the inner stream completes one rotation in the same amount of time as a particle in the outer stream. However, because of the principle of conservation of angular momentum (tendency of a rotating object to keep rotating at the same speed about the same axis of rotation), the rotational speed of the smaller vortex might be expected to increase. (The conservation principle is demonstrated by spinning skaters who can slow or speed up their spin by extending or drawing in their arms.) But in the vortex tube, the speed of the inner vortex remains the same. Angular momentum has been lost from the inner vortex. The energy that is lost shows up as heat in the outer vortex. Thus the outer vortex becomes warm, and the inner vortex is cooled.
2. Temperature is an average of how fast the particles are bumping into each other. Because it's an average, it means that air is composed of fast and slow (hot and cold) particles. A vortex tube merely separates the hot which goes to the peripheral of the spinning vortex and exhausts out the hot side and the cold air is diverted back through the center to exit the cold end.
3. Turbulent eddies (whirlpools) expanding and contracting are causing the heat transfer.
4. Adiabatic compression and adiabatic expansion is causing all the heat separation.
5. Acoustic streaming theory states there is a relationship between the acoustic resonance
frequencies and the forced vortex motion frequency. The energy separation inside the VT is due to the damping of the acoustic streaming along the axis of the tube towards the hot exhaust.
6. Secondary circulation states that some of the separation of heat is caused by a closed flow of medium inside the VT because studies show cumulative mass flow over the cross section of the vortex tube in the cold end direction, is larger than the cold exhaust flow.
All these theories focus on thermodynamics (compression and expansion), turbulent flow, viscous friction, internal heat transfer, acoustics and flow pattern.