Matters of state and motion - Physical Processes in Earth and Environmental Sciences - page 31

photograph of a continuously introduced dye will yield a
streakline , the locus of all fluid elements that pass through.
A photograph of an instantaneously introduced dye or of
reflective particles will yield a pathline . For a steady flow it is
possible to construct an overall flow map by drawing
streamlines . These are lines drawn such that the velocity of
every particle on the line is in the direction of the line at that
point. Numerous examples of flow visualization are given in
the text that follows (see in particular Figs 3.53-3.55).

A

(a)

r

s

Centre of
curvature

f

B

r

Angular speed, v = df / dt
Linear speed, u = rv

u

2.4.12 Flow without dynamics: “Ideal”
flow along streamlines

(b)

a

-c

From the definition of a streamline quoted above it is
obvious that streamlines cannot cross and that it is possible
to define a volume of fluid bounded by streamlines along
its length. Such an imaginary volume is termed a stream-
tube (Fig. 2.20). If the discharge into and out of a stream-
tube of any shape is constant, areas of streamline
convergence indicate flow acceleration and areas of diver-
gence indicate deceleration. Thus areas of close spacing
have higher velocity than areas with wide spacing. Some
progress may be made concerning the prediction of
streamline positions rather than the experimental visualiza-
tion considered previously by using concepts of ideal
( potential ) flow as applied to fluids in which the molecular
viscosity (see Section 3.9) is considered zero. Although
such frictionless fluids are far from physical reality, ideal
flow theory may be of great help in analyzing motions
distant from solid boundaries (i.e. away from boundary lay-
ers; see Section 4.3) and in flows where viscous effects are
negligible (at very high Reynolds' numbers; see Section 4.5).
As subsequent discussions will show, in the absence of
shearing stresses in an ideal fluid there can be no rotational
motion (vorticity), that is, all ideal flows are considered
irrotational .
Considering any ideal flow past a bounding (solid)
surface, it is apparent that discharge between the boundary
and a given streamline must be constant. Thus it is possi-
ble to label streamlines according to the magnitude of the
discharge that is carried past themselves and a distant
boundary. This discharge is known as the stream function ,

c

b

b

a

a x b = c b x a = -c

(c)

v

v

I v I

p

o r

Fig. 2.18 To illustrate curved motion angular speed and velocity.
(a) Angular speed, (b) angular velocity conventions, and (c) angular
velocity.

rotate; spinning eddies of fluid turbulence are readily
observed in rivers and from satellite images in ocean cur-
rents. Fluid vorticity is termed relative or shear vorticity
and is due to velocity differences, termed velocity gradi-
ents , across a fluid element (Section 1.19). It can be shown
(Section 3.8) that rigid body vorticity is twice the angular
velocity, that is, . Finally, vorticity must be

conserved according to the principle of the Conservation
of Absolute Vorticity (see Section 3.8).

2

2.4.11 Visualization of flow

is
obviously unique to any particular streamline and must be
constant along the streamline. Velocity is higher when
streamline spacing is closer and vice versa (Cookie 2.1).
Another useful method of analyzing ideal flow arises
from the concept of velocity potential lines , symbol

, of a streamline (Fig. 2.20). The magnitude of

No dynamical analysis may be confidently begun without
some idea of actual flow pattern. In everyday life the gusting
eddies of a wind are picked out by the motion of autumn
leaves or by the swirling pattern of snow or sleet across a
road or field. In the same way in the lab, flow visualization
introduces some marker into a flow which is then pho-
tographed (Fig. 2.19). Considering the Eulerian case, a

. These
imaginary lines are drawn normal to streamlines (Fig. 2.20).
They define a flow field, as defined in Section 2.4 and are best