Velocity and Acceleration in Steady and Unsteady Flow

Consider a fluid particle moving along a pathline in a flow. The particle's velocity is denoted by the function of its location and time.

Now, the acceleration of the particle can be determined by differentiating the expression of velocity with respect to time.

If this velocity is generalized to any point, then the acceleration can also be expressed generally.

Further, the components of acceleration can be indicated in three perpendicular directions, representing how the velocity of an object varies over time.

Finally, the acceleration can be expressed as the derivative of velocity with respect to time.

In these equations, time derivatives are denoted as local derivatives and are equal to zero in a steady flow, and the local effect vanishes in this case.

In unsteady flow, parameters such as velocity, temperature, and density can vary over time at any given location.

That is, the spatial derivatives will reduce to zero, and the acceleration will reduce to a partial derivative of the velocity with respect to time.