Fluid Kinematics And Dynamics: Velocity, Acceleration

Velocity

Velocity of a fluid along any direction can be defined as the rate of change of displacement of the fluid along that direction.

u = dx / dt

Where dx is the distance traveled by the fluid in time dt.

Velocity of a fluid element is a vector, which is a function of space and time.

Let V be the resultant velocity of a fluid along any direction and u, v and w be the velocity components in x, y and z directions respectively.

Mathematically the velocity components can be written as

u = f ( x, y, z, t )

v = f ( x, y, z, t )

w = f ( x, y, z, t )

Acceleration

Acceleration of a fluid element along any direction can be defined as the rate of change of velocity of the fluid along that direction.

If a_x , a_y and a_z are the components of acceleration along x, y and z directions respectively, they can be mathematically written as

a_x  = du/ dt.

But u = f (x, y, z, t) and hence by chain rule, we can write,

_{ax =} ?u dx ₊ ?u dy ₊ ?u dz ₊ ?u ?x dt ?y dt ?z dt ?t

Similarly