V is the unit tangent vector of congruence of geodesics, it is used to know the behavior of the congruence which are seen as the flow lines.
In 
,the acceleration of the flow line, one can recognize the non-linear term 
which show that the field of motion transports itself.
the vorticity tensor . It can be seen as the component of the curl of 
.
the expansion tensor is called 
in the Fluids Dynamics literature.
The trace of that tensor is 
, the volume expansion . but 
can be also seen as the divergence of the motion 
.
They are used to define the shear tensor 
. It has the same name and symbol in Fluids Dynamics.