Before proceeding with a discussion of acceleration it may be well to restate the definition of velocity as given essentially by Eq. (30) in a modified form.
Consider again a particular particle of fluid A whose trajectory is
known. The position of the particle A at any instant is
. In effect, the position at time t is
a vector (from the origin to the point in question), which will
be referred to as the radius vector of the particle and will
be denoted by 
. Thus, the function
describes the trajectory. This is illustrated
for the two-dimensional case in Figure 1.04-1.
In this figure the radius vectors for the instnats 
and
are indicated. The vector difference
also indicated in the figure, represents the net incremental displacement of the particle A in the elapsed time
It will be noted that (73a) can be rearranged so as to state
that 
is the increment vector diplacement
that must be added to 
to get the radius
vector 
on the trajectory after an elapsed
time 
.
If the time interval is reduced, the vector
is accordingly reduced, and its orientation
becomes more and more tangent to the trajectory curve.
In addition, the magnitude ratio 
approaches a limit. This limit is the magnitude of the
velocity and its direction is tangent to the trajectory.
Thus, the velocity 
at the instant is
which, when applied to the radius vector at
any instant t, an be written as
This relation is actually nothing more than a concise statement of Eqs. (30), representing in effect three scalar equations. The definition (75) holds equally well for three dimensions, but the geometrical interpretation is slightly more complex in respect to the concept of tangency to a skew curve in space.
We may now proeceed with the definition and interpretation of acceleration
in an analogous manner.
The trajectory of a particular particle A is again depicted
in two dimensions in Figure 1.04-2a. The reference axes are
considered to be fixed in space. The velocity vectors at different
points along the trajectory are indicated by the solid arrows.
If the vectors are all referred to a common origin (as indicated
in the adjoining Figure 1.04-2b) then the end points describe a
curve which will be referred to as the hodograph of the
velocity vectors. This curve is accurately defined by all the
vectors 
applicable to the motion of
particle A. The coordinates of the hodograph are nominally
labeled x and y but should more accurately be labeled
and 
, i.e. the components of the velocity
in the x and y directions, respectively.
The vector difference
represents the net change in velocity for the elapsed time
defined by Eq. (73b). As can be seen by comparison
of Figures 1.04-2a and b, the direction of
is not necessarily oriented along the
trajectory. This would be true only if the trajectory is
a straight line.
The limiting direction of , as is
reduced to zero, is along the tangent to the hodograph curve
at the instant . The limit of the ratio
, if it exists, is the acceleration
of the particle A at the instant . Thus, if we let
represent the acceleration of the particle
at any instant then
The acceleration can also be expressed directly in terms of the radius vector function which describes the trajectory of A, since it follows from Eq. (75) that
These definitions again hold equally well for three-dimensional vectors.
There are, categorically speaking, essentially two modes of acceleration which a particle can possess. These are, specifically: (1) acceleration along the trajectory, and (2) acceleration normal to the trajectory. The first merely represents the rate of change of the magnitude of the velocity. The second is associated with the rate of change of the direction of the velocity vector.
Let s be the path distance along the trajectory of
particle A and ds a differential displacement between
points (1) and (2) on the path. The velocities of
particle A at points (1) and (2) are
and 
(see Fig. 1.04-3).
The differential change of the magnitude of
, denoted by 
, is indicated in the
inset of Fig. 1.04-3. It actually represents the projection
of the differential vector change 
along
the path. Thus, the acceleration along the path can be
expressed by the derivative 
. It is positive
if the magnitude of 
is increasing as the
particle proceeds along its path and negative if the particle
is slowing down. The component acceleration
along the path can be expressed alternately by the
forms
However, ds/dt is merely the magnitude of , thus
We can also refer to this acceleration as the tangential
acceleration since it has a direction tangential to the path
of the particle. It should be remarked that
is not the same as 
.
The latter quantity is the magnitude of the total acceleration
while the tangential acceleration is merely a component of
the acceleration representing the rate of change of the magnitude
of the velocity.
An expression for the second mode of acceleration can be
derived by again referring to Fig. 1.04-3. The second
component of the differential vector change
is 
and is directed normal to the velocity
vector and hence normal to the path of the particle at the
point in question. The differential angle 
represents
the change in direction of the velocity in the distance ds
along the trajectory. The magnitude of the normal component
of acceleration 
of particle A is therefore
This can be expressed in the form
where 
is the curvature of the path
of the particle. In Figure 1.04-3 the path is shown in two
dimensions only. This may be regarded as the projection of the
three-dimensional trajectory in the x-y plane. The projections
in the x-z and y-z planes are, in general, curved paths as
well. We may regard the trajectory as merely the common intersection
of the three cylindrical surfaces formed by the extensions of the
projected paths on the three reference planes (see Figure 1.04-4).
In general therefore the curvature K is to be interpreted as the
rate of change of 
with s along the three-dimensional
trajectory. It is possible to interpret the curvature as a vector
normal to the path. This is discussed by
Holmboe et al. (1945) (Arts. 7.04-7.09).
In terms of the radius of curvature of the path, the normal acceleration of the particle becomes
where
For the special case where the path of A is a plane circle then
R is constant. If the speed 
is also constant then the
acceleration is everywhere directed towards the center of the
circular path and is of constant magnitude 
.
The acceleration associated with this special type of motion
is commonly referred to as centripetal
acceleration. The latter term might also be used to apply
to the normal acceleration in the general case, where it is to be
kept in mind that R is not necessarily constant for the
particular path being considered.
The normal acceleration occuring at different
points along an example trajectory is shown by the arrows in
Figure 1.04-5. The total
acceleration at any point is given by the vector sum of
and 
as illustrated in
Figure 1.04-6.
By way of emphasis, it should again be remarked that it is the curvature of the trajectories which determines the normal acceleration component of a particular particle. This must be kept in mind in later developments where streamline patterns are frequently the object of discussion. It is a rather common mistake to use the curvature of streamlines in order to ascertain the normal acceleration. This procedure is correct only if the pattern of flow within the fluid is steady.