## 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 asor in component form as We integrate these to obtain the parametric equationsIt may be possible to solve these to obtain a single solution relatingand(orand). 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 functionwith

The equations are(3) and(4).

From (3)and from (4)

Hence