fluid_mechanics

156 Chapter 4 ■ Fluid Kinematics
y
y
t = 0
v 0 /u 0
Pathlines of
particles at origin
at time t
t = π/2ω Pathline
u 0
Streaklines
through origin
at time t
v 0
–1 0 1
x
0
x
(c)
(d)
F I G U R E E4.3(c), (d)
4.2 The Acceleration Field
V4.11 Pathlines
Acceleration is the
time rate of change
of velocity for a
given particle.
As indicated in the previous section, we can describe fluid motion by either 112 following individual
particles 1Lagrangian description2 or 122 remaining fixed in space and observing different particles
as they pass by 1Eulerian description2. In either case, to apply Newton’s second law 1F ma2 we
must be able to describe the particle acceleration in an appropriate fashion. For the infrequently
used Lagrangian method, we describe the fluid acceleration just as is done in solid body dynamics—
a a 1t2 for each particle. For the Eulerian description we describe the acceleration field as a
function of position and time without actually following any particular particle. This is analogous
to describing the flow in terms of the velocity field, V V 1x, y, z, t2, rather than the velocity for
particular particles. In this section we will discuss how to obtain the acceleration field if the velocity
field is known.
The acceleration of a particle is the time rate of change of its velocity. For unsteady flows
the velocity at a given point in space 1occupied by different particles2 may vary with time, giving
rise to a portion of the fluid acceleration. In addition, a fluid particle may experience an acceleration
because its velocity changes as it flows from one point to another in space. For example, water
flowing through a garden hose nozzle under steady conditions 1constant number of gallons per
minute from the hose2 will experience an acceleration as it changes from its relatively low velocity
in the hose to its relatively high velocity at the tip of the nozzle.
4.2.1 The Material Derivative
Consider a fluid particle moving along its pathline as is shown in Fig. 4.4. In general, the particle’s
velocity, denoted V A for particle A, is a function of its location and the time. That is,
V A V A 1r A , t2 V A 3x A 1t2, y A 1t2, z A 1t2, t4
z
Particle path
V A (r A , t)
Particle A at
time t
v A (r A , t)
r A
w A (r A , t)
u A (r A , t)
x
y A (t)
z A (t)
x A (t)
y
F I G U R E 4.4
Velocity and position of particle A
at time t.