In fluid mechanics, velocity and acceleration are key concepts for analyzing particle motion in both steady and unsteady flow. Consider a fluid particle moving along a pathline, where its velocity depends on its position and time. The particle's acceleration is obtained by differentiating the velocity with respect to time.

The acceleration can be generalized to any point in the flow, and expressed as components along three perpendicular directions, representing changes in velocity over time. These components reflect how the particle's velocity evolves in different spatial directions.

In steady flow, the velocity at each point remains constant over time, meaning the local time derivatives of velocity, known as local derivatives, are zero. As a result, there is no time-dependent change in the particle's velocity, and the acceleration is governed only by spatial variations in velocity.

In contrast, unsteady flow involves changes in velocity, temperature, and density over time at any given location. In this case, the local time derivatives are nonzero, contributing to the particle's acceleration. Thus, in unsteady flow, acceleration is given by the partial derivative of velocity with respect to time, highlighting the time-dependent nature of the flow.