Fluid kinematics

The composition of the material contains two types of terms: those involving the time derivative and those involving spatial derivatives. The time derivative portion is denoted as the local derivative, and represents the effects of unsteady flow. The local derivative occurs during unsteady flow, and becomes zero for steady flow.

The portion of the material derivative represented by the spatial derivatives is called the convective derivative. It accounts for the variation in fluid property, be it velocity or temperature for example, due to the motion of a fluid particle in space where its values are different.

The acceleration of a particle is the time rate of change of its velocity. Using an Eulerian description for velocity, the velocity field ${\displaystyle V=V(x,y,z,t)}$ and employing the material derivative, we obtain the acceleration field.