fluid_mechanics

164 Chapter 4 ■ Fluid Kinematics
The orientation of
the unit vector
along the streamline
changes with
distance along the
streamline.
are curved, both the speed of the particle and its direction of flow may change from one point to
another. In general, for steady flow both the speed and the flow direction are a function of location—
V V1s, n2 and ŝ ŝ1s, n2. For a given particle, the value of s changes with time, but the value
of n remains fixed because the particle flows along a streamline defined by n constant. 1Recall
that streamlines and pathlines coincide in steady flow.2 Thus, application of the chain rule gives
or
D1V ŝ2
a
Dt
DV
Dt ŝ V Dŝ
Dt
a a 0V
0t 0V ds
0s dt 0V dn
0n dt b ŝ V a 0ŝ
This can be simplified by using the fact that for steady flow nothing changes with time at a given
point so that both 0V0t and 0ŝ0t are zero. Also, the velocity along the streamline is V dsdt and
the particle remains on its streamline 1n constant2 so that dndt 0. Hence,
a aV 0V 0ŝ
b ŝ V aV
0s 0s b
The quantity 0ŝ0s represents the limit as ds S 0 of the change in the unit vector along the
streamline, dŝ, per change in distance along the streamline, ds. The magnitude of ŝ is constant
1 0ŝ0 1; it is a unit vector2, but its direction is variable if the streamlines are curved. From Fig. 4.9
it is seen that the magnitude of 0ŝ0s is equal to the inverse of the radius of curvature of the
streamline, r, at the point in question. This follows because the two triangles shown 1AOB and
A¿O¿B¿ 2 are similar triangles so that dsr 0d ŝ 0 0ŝ 0 0dŝ 0 , or 0dŝds 0 1r. Similarly, in the
limit ds S 0, the direction of dŝds is seen to be normal to the streamline. That is,
0ŝ
0s lim dŝ
dsS0 ds nˆr
0t 0ŝ
0s
ds
dt 0ŝ dn
0n dt b
Hence, the acceleration for steady, two-dimensional flow can be written in terms of its streamwise
and normal components in the form
a V 0V
0s ŝ V 2
r nˆ or a s V 0V
0s , a n V 2
r
The first term, a s V 0V0s, represents the convective acceleration along the streamline and the
second term, a n V 2 r, represents centrifugal acceleration 1one type of convective acceleration2
normal to the fluid motion. These components can be noted in Fig. E4.5 by resolving the
acceleration vector into its components along and normal to the velocity vector. Note that the unit
vector nˆ is directed from the streamline toward the center of curvature. These forms of the
acceleration were used in Chapter 3 and are probably familiar from previous dynamics or physics
considerations.
(4.7)
O
O
δθ
δθ
^
n
B
δs
δs
A
B'
B
^
δ s
s
δθ A'
A
O'
F I G U R E 4.9 Relationship between the unit vector along the
streamline, ŝ , and the radius of curvature of the streamline, r.
^
s(s)
s(s + s)
^ δ
s(s + s)
^ δ
^
s(s)