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