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Fluid Kinematics

Kinematics is the study of the velocity and acceleration of the fluid motion. There are two methods to study the motion of fluid. One way can focus on fluid particles by tracking their position vector, i.e. and determining their associated properties as a function of time, i.e. velocity vector . This is the Lagrangian description of fluid motion. The other way of describing fluid motion is the Eulerian description of fluid motion, which focuses on the fluid flows in and out of a control volume in terms of field variables. Instead of following individual fluid particles, the description describes the variables value of  the location only and does not care whichever fluid particle occupying that location inside the control volume  as a function of space and time, i.e.  pressure field  , velocity field to define the flow field.

Velocity Field

Fluid property can be represented as a function of spatial coordinates in term of a field representation. For a fluid flow, it is also a function of time.

When the velocity value of a location equals to zero, i.e. , it is called a stagnation point.

Acceleration Field

In a fluid flow, the velocity of a particle is a function of its location and time. Imply . By definition, the acceleration of a particle is the time rate of change of its velocity, . Using chain rule of differentiation, and because

imply .

Since the equation can be used to describe any particle, the acceleration field can be obtained from the velocity field as
,or
;
where is named local acceleration or unsteady acceleration, which equals to zero for steady flow and is named advective acceleration or convective acceleration, which equals to non-zero even for steady flow.

And the total acceleration can be expressed as

or

Since the total derivative operator is derived from a fluid particle, it is called material derivative, . i.e.

The total acceleration is therefore also called material acceleration.