Pathlines
Pathlines in fluids are the paths taken by the individual particles of the fluid as they travel from point to point.
We can find the pathlines taken by the particles of a fluid if we know the velocity of the particles of the fluid. In two dimensions, we write down the differential equations for the pathlines, in vector form as
or in component form as
We integrate these to obtain the parametric equations
It may be possible to solve these to obtain a single solution relating
and
(or
and
). A particular pathline might be found using a given initial point.
Example: Find the equations of the pathlines for the velocity function
The equations are
(1) and
(2).
From (1)
a constant and from (2)
The pathlines are concentric circles around the origin. Asincreases, the speed decreases in proportion.
Example: Find the equations of the pathlines for the velocity function
with 
The equations are
(3) and
(4).
From (3)
and from (4)
Hence
