Imagine that we mark one of the drops of fluid with a colored speck so we can watch it. In a small interval of time , this drop will move from to a different location We can write all the 's in terms of From the definition of a derivative the acceleration is and so the acceleration of a fluid element is This says that there can be non-zero acceleration even when the velocity at a given point is not changing, i.e., when . Streamlines in a steady fluid flow track the path of fluid elements. The term represents the change of in the direction of , and vanishes only when there is no variation of in the direction of . If all the fluid elements follow straight streamlines then this acceleration term is zero.
Acceleration occurs where streamlines change direction.
As an example, water flowing through a converging nozzle can be in steady state so that at any location the velocity remains constant, however the water accelerates as the cross-section of the nozzle decreases and the streamlines converge. Of course, a fluid element can also accelerate by changing its speed along a straight streamline. Now we have all we need for the equation of motion, and the rest is just math, or where viscosity has been omitted. Giving the final form for the equation of motion which is Euler equation. The vector field is called the vorticity .