In continuum mechanics
a material line element
is a small displacement vector
between two particle
s located at x
is "built into" the medium. As the medium
ed the displacement between the particles changes, so dl
depends on time
The main reason why we are interested in material line elements is that we for many purposes (especially integration) like to think of curves as being approximable by a chain of small displacement vectors. If we know what happens to the material line elements we therefore also know something about the behaviour of curves that are fixed in the medium.
For this reason we wish to find the precise form of the time-dependence of dl. Consider the change in dl during a short time dt. In this time the position of the particles will have changed to x + u(x)dt, x+dl + u(x+dl)dt, respectively, where u is the velocity field of the medium. Therefore to linear order
d(dl) = ((x+dl + u(x+dl)dt) - (x + u(x)dt) - dl) = u(x+dl)dt - u(x)dt = (dl.∇)udt
Using the substantial derivative operator D/Dt to indicate that we move with the medium while we measure the change in dl we can write this as
Ddl/Dt = (dl.∇)u
The interpretation of this is that if u increases in the direction of dl then dl will be stretched out by the motion of the medium.