The preceding article concerned the acceleration of a particular particle of fluid. The definition of the acceleration in this Lagrangian approach is clear cut. The meaning of the Eulerian approach is perhaps somewhat elusive, but can always be made more concrete by the connecting relations of the two systems. The advantage of the Eulerian system should now be obvious in that the representation of trajectories is avoided altogether, or at least postponed. Instead, we inquire into the nature of the fields of velocity and of the acceleration of the fluid.
Consider that the field of velocity 
is known.
This implies the existence of the three functions
representing the velocity components in the x, y, z directions,
respectively. The field of acceleration 
can
be represented by the material derivative of 
or, specifically
where the operator D/Dt is the same as implied in Eq. (24), i.e.
or, in abbreviated notation when applid to Eq. (85)
However, the operation when performed on the vector is not
readily comprehensible, although one can always resort to the component
equations implied by (87) which are, when written out in full
The advantage of the abbreviated from (87) should be obvious.
The connecting equations between the Lagrangian and Eulerian representations of acceleration are straightforward. It is evident that the latter representation must lead to the result expressed by the Lagrangian system when applied to a particular particle. Thus
or, alternately,
Eq. (89) or (90) represent in effect three scalar equations which, as a unit, express the compatibility of the two systems of representation of acceleration. If we expand Eq. (90) in full we obtain
As a simple example, consider the one-dimensional velocity field of case A, Art. 1.03). The selected field was
for which the path histories of particles were found to be
and 
and 
are constant. The acceleration is therefore
This result could likewise be obtained as follows:
and
thus leading to the identical result obtained by the Lagrangian approach.
As an illustration for the two dimensional case, consider the conditions stipulated in case C, Art. 1.03, namely:
The trajectory of that particle starting from the position
(0, 0) at 
was found to be (Eq. 56):
The acceleration is therefore given by the components
the magnitude of which is (since
),
and therefore constant. The direction of 
is however not
constant.
The same result can be obtained with the Eulerian system. Applying Eq. (88) we obtain
Thus
Q.E.D.
Returning now to Eq. (87), it may be stated in behalf of the
interpretation thereof that the acceleration of that particle
which happens to occupy the point x, y, z at the instant t is
equivalent to the time rate of change of the velocity if we follow
along with the fluid motion. The total acceleration is the sum
of two effects: local time rate of change of the velocity
of the fluid at the point in question, and the field accelerations
which are related to the gradients of the velocity components.
In reconciling these effects with the Lagrangian description, it may
be noted that the field accelerations represent, in effect, the
accelerations, normal and tangential, associated with the streamline
pattern of the flow existing at any instant; the ``local''
acceleration term represented by 
on the
other hand accounts for the movement and/or intensification of the
flow pattern and amounts to a correction term which adjusts the
acceleration associated with the instantaneous streamlines so as to
give the true value associated with the trajectories.
The above statement may be clarified by actually giving an alternate
form of Eq. (87) which indicates the role of the streamlines and
local changes more explicitly than Eq. (87). Let 
represent
the unit vector tangential to the streamline which passes through
the point x, y, z at time t, and let 
be the unit vector
normal to the streamline at this same point and time. The sense of
is taken towards the centr of curvature of the streamline.
The unit vectors and are also tangential and
normal, respectively, to the path of that particle which happens to
occupy the point x, y, z at the instant t. This must be the
case since both streamline and particle path at a given point at
a particular instant are both oriented tangential to the velocity
vector through that point at the given instant. The important thing
is that the curvature of the streamline differs from that of the
path, so that if we follow along with the particle then at a later
instant we will no longer be on the same streamline. Furthermore,
the variation of the magnitude V along the streamline is
different from the variation along the path. However, it can be
readily shown that the acceleration of that particle occupying
the position x, y, z at instant t can be expressed in the form
where 
represents the angle between the velocity
vector and some fixed reference line. The choice of
sign on the second term depends upon the sense of turning
relative to the streamline pattern. The term in brackets
represents the term 
of Eq. (87) and
is to be evaluated along the instantaneous streamline through
the point in question. The first two terms represent the
local acceleration of Eq. (87).
A more convenient form of Eq. (92) is
where the subscript s on the first terms in each bracketed expression indicates that the quantity is to be evaluated along the streamline. The terms
and
therefore represent the magnitude of the tangential and normal components of the acceleration, respectively, as determined from the instantaneous streamline and isovel pattern. The terms
and
represent, in effect, ``correction terms'' which take into
account the local rate of intensification of the velocity field
and the movement of the streamline pattern. The
term 
is the local rate of change of the
magnitude of the velocity, and the term
is the local rate of turning
of the velocity vector.
The complete first term in brackets in Eq. (93), i.e.
represents the tangential component of acceleration associated with the path of that particle which occupies the position x, y, z at the instant t. The complete second term in brackets, i.e.
represents the normal component of the acceleration associated with the particle path. The curvature K of the particle path in terms of the streamline curvature and the local rate of turning of the streamlines is, consequently
Similarly, the rate of change of V along the path
is
The choice of sign of the term 
in Eqs. (93) and (94) must be determined by inspection of a series
of synoptic flow patterns. It can be determined from such a series
whether the local turning term tends to give a greater or smaller
curvature to the particle path through the point in question.
As an example, consider the moving circular streamline pattern
represented in Figure 1.04-7. At point (1) the angle
is increasing locally with time due to the movement of the
streamline pattern. The effect of this is that the curvature
of the path at point (1) is less than that of the streamlines,
hence we must choose the negative sign in Eq. (94). At point
(2), the magnitude of is decreasing with time but it can
be seen that the path curvature must still be less than the streamline
curvature, so the negative sign in Eq. (94) must again be chosen.
By inspection of the situations at points (3) and (4) it will be
found that the particle paths at both points have a greater
curvature than than of the streamline and hence the positive
sign is selected in Eq. (94). In fact, as a rule the path
curvature is less than the streamline curvature on the side of
a moving vortex system where the fluid velocity has a component
in the direction of the storm; on the other hand the path
curvature is greater than the streamline curvature on the side of
the moving vortex system where the fluid velocity has a component
opposite to the direction of movement of the storm. It will be
noted the the rule is consistent with the flow conditions
illustrated in Fig. 1.03-5. The actual magnitude of the difference
in the path and streamline curvatures must be ascertained from the
second term in Eq. (94).
If the pattern of flow is changing intensity as it moves along, but is maintaining its form then the above rule still applies. However, if the streamlines are additionally changing shape then the local turning of the velocity vectors depends on this effect as well as the effect of the movement of the pattern of flow, and the above rule may no longer apply. Nevertheless, it is always possible to ascertain the appropriate sign in Eq. (94) by careful inspection of the changing situation of flow.
The special case of a moving vortex system with circular
streamlines is important enough to warrant further deductions from
Eq. (94). It will be assumed that the system undergoes no change in
shape as it moves. The distribution of V will be considered
arbitrary and may even be changing with time. The change in
intensity of the system, without change in the form of the
streamlines is conceivable. The effect of the changing intensity
is to modify the tangential acceleration and is included in the
term 
of Eq. (95). However, there is
no effect on the normal acceleration component.
Referring to Figure 1.04-8, it can be seen that the angular
change of the velocity vector in time 
is
where 
is the radius vector from the center of the streamline
pattern to point P, and is the angle measured counterclockwise
from the streamline propagational velocity 
to the
radius vector . Thus the magnitude of the local
turning is
Making use of this
result and noting that 
we can write
Eq. (94) in the form
where the negative sign is chosen so as to be consistent with
the rule given earlier. It will be noted that for a point on
the lower side of the vortex pattern is greater than
180 degress and hence 
is negative. Thus the sign
of the second term in Eq. (98) is automatically taken into
account, and is consistent with the aforementioned rule for
all values of .
The normal acceleration for particles governed by the moving vortex system is, consequently
It is evidence that if the vortex is stationary then U vanishes,
, and 
. This is the special
case where the streamlines and particle paths are identical.
It must be emphasized, however, that the results (98) and (99)
apply only to a vortex system with circular streamlines.
In more general situations Eq. (93b) must be employed in order
to evaluate the normal component of acceleration.
In passing it should be remarked that two systems have been
employed for representing components of acceleration. Equations
(88) give the components relative to standard rectilinear
coordinates while Eq. (93a,b) give the components tangential
and normal to the streamlines of flow. The two systems are merely
two ways of representing the same thing. In the latter system
each of the vectors 
and 
may be
considered as composed of three rectilinear components. If we
add the corresponding components of each we should obtain the
components 
. The choice of the system
of representation is arbitrary.