(
).
()
tells the magnitude and direction of the velocity
of the fluid at each position in space, .
()
can be a function of time, t, in addition to being a function of
. To be more
complete we can write (,t).VELOCITY FIELD
In describing the flow of fluid,
it is useful to use a velocity field, ().
()
tells the magnitude and direction of the velocity
of the fluid at each position in space, .
()
can be a function of time, t, in addition to being a function of
. To be more
complete we can write (,t).
ACCELERATION
To apply Newton's laws of motion
to fluids, we take the laws for a drop of fluid, and re-write them in terms
of the fields that represent the fluid flow. Our first step in developing
a dynamic description of fluid flow is to calculate the acceleration of
the drop in terms of the velocity field, ().
A drop at a particular position
has a velocity which is identical to the velocity field, (),
at that point in space. The velocity carries the drop to new positions,
where ()
may be different. To cause this change in velocity, the drop must experience
an acceleration, and a net force. In addition, even if (,t)
is the same at each point in space, it can still change in time. If a drop
experiences this change in velocity it must experience an acceleration
and a net force.
We need to calculate the full
time derivative of the velocity of the drop, in terms of both the spatial
and temporal character of the velocity field(,t).
We will write this derivative in terms of ordinary x,y,z coordinate components.
Let us enclose the drop of water in a bag, and imagine that it travels
as an object, without mixing with neighboring water. The chain rule tells
us:
Now we recall that the velocity field at a point is defined as the velocity of a bag of fluid passing through that point. We replace the velocity of the bag in the expression above with the velocity field at the location of the bag:
Next, we note that the position
at which the velocity field is evaluated is determined by the position
that the bag occupies. Thus the "x" which appears in 
is identical to the x component of the position of the bag. We can replace
with 
(referring
both to the bag velocity and the velocity field at the location of the
bag).
This expression is complete,
but we will make several changes in notation. We move the velocity components
as shown below. This way we avoid any suggestion that (for example)
is differentiated with respect to 
:
Rewrite this by factoring out the derivative operator:
Factor out the velocity:
Next we define a vector operator:
and note that
where the 
denotes a scalar (or "dot") product of the vector 
with the vector operator 
.
Our final version of the acceleration of the bag of water in terms of the velocity field is
Go forward to force calculation
Return to index
Return to Euler survey