A variable force solenoid (VFS) is an electro-hydraulic device that controls pressure proportionally or inversely proportionally to a signal (voltage or current) obtained from the on-board controller of a powertrain.
A low flow VFS is used as a signal level devices for transmission line pressure control or application of clutches.
Out of necessity there is a circular air gap that can only provide radial force (also known as side loading).
The rate of change of the working air gap stored energy provides the axial magnetic force.
The images below show flux lines and field density for an example solenoid design [2]