IV - 1
Ch. IV
Differential Relations for a Fluid Particle
This chapter presents the development and application of the basic differential
equations of fluid motion.
Simplifications in the general equations and common
boundary conditions are presented that allow exact solutions to be obtained.
Two
of the most common simplifications are 1). steady flow and 2). incompressible
flow.
The Acceleration Field of a Fluid
A general expression of the flow field velocity vector is given by:
V
(
r
,
t
)
=
°
i u
x
,
y
,
z
,
t
(
)
+
°
j
v
x
,
y
,
z
,
t
(
)
+
°
k w
x
,
y
,
z
,
t
(
)
One of two reference frames can be used to specify the flow field characteristics:
eulerian ± the coordinates are fixed and we observe the flow field
characteristics as it passes by the fixed coordinates.
lagrangian
- the coordinates move through the flow field following individual
particles in the flow.
Since the primary equation used in specifying the flow field velocity is based on
Newton²s second law, the acceleration vector is an important solution parameter.
In cartesian coordinates, this is expressed as
a
=
d
V
d t
=
∂
V
∂
t
+
u
∂
V
∂
x
+
v
∂
V
∂
y
+
w
∂
V
∂
z
=
∂
V
∂
t
+
V
⋅
∇
(
)
V
total
local
convective
The acceleration vector is expressed in terms of three types of derivatives:
Total acceleration =
total derivative of velocity vector
= local derivative
+
convective derivative
of velocity vector

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