3.2 Motion of a Continuum
57
can record the velocity and time, but they do not know the displacement for each
passing particle, i.e., they know v(x, t) but not v(X, t). How can the acceleration
field be determined from these data alone?
First, we note that the velocity field represents the velocity of individual particles,
whether we record the velocity of each particle as it moves through space (material
description) or record the velocities of different particles as they pass each point in
space (spatial description). Regardless of how the field is described, x represents
the current position of a given particle and is therefore a function of time for each
particle. Thus, Eqs. (3.6) and the chain rule give
a =
d
dt
v(x, t) =
∂v
∂x
∂x
∂t
+
∂v
∂t
=
∂v
∂x
v +
∂v
∂t
.
(3.9)
For the present example, Eq. (3.8) 1 yields the partial derivatives
∂v
∂x
=
2αt
1 + αt 2
∂v
∂t
=
2αx(1 − αt 2 )
1 + αt 2
2 .
Substituting these expressions, along with v from Eq. (3.8) 1 , into (3.9) and combining terms give
a =
2αx
1 + αt 2 ,
which agrees with Eq. (3.8) 2 .
Example 3.1 Determine x(X, t) from the spatial form of the velocity given by
Eq. (3.8) 1 .
Solution
Since v = dx/dt, we have
dx
dt
=
2αxt
1 + αt 2
or
dx
x
=
2αt
1 + αt 2 dt.
Integrating both sides yields
ln x = ln(1 + αt
2 ) + C,