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Acceleration of a fluid particle is the rate of change of the velocity of that fluid particle. As acceleration is a function of time, the components its position vector can be written as-

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

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

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

Where,

u, v and w are the velocities in the direction x, y and z respectively.

Thus,

Acceleration in x-direction can be given by,

ax = du/dt =∂u/∂x.dx/dt + ∂u/∂y.dy/dt+ ∂u/∂z.dz/dt + ∂u/∂t

Or,

ax = u ∂u/∂x +v∂u/∂y+ w ∂u/∂z + ∂u/∂t

Here,

u = dx/dt

v = dy/dt

w = dz/dt

Similarly,

Acceleration in y-direction is given by,

ay = u ∂v/∂x + v∂v/∂y+ w ∂v/∂z+ ∂v/∂t

And,

Acceleration in y-direction is given by,

az = u ∂w/∂x + v∂w/∂y+ w ∂w/∂z + ∂w/∂t

Note:

ax , ay and az are material or substantial accelerations

∂u/∂t ,∂v/∂tand∂w/∂tare temporal or local accelerations

u ∂u/∂x + v∂u/∂y + w ∂u/∂z , u ∂v/∂x + v∂v/∂y + w ∂v/∂zand u ∂w/∂x + v∂w/∂y + w ∂w/∂zare convective accelerations