fluid_mechanics

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266 Chapter 6 ■ Differential Analysis of Fluid Flow
O
v
u
O'
v δ t
u δt
F I G U R E 6.2
Translation of a fluid element.
The rate of volume
change per unit
volume is related
to the velocity
gradients.
because of the presence of velocity gradients, the element will generally be deformed and rotated
as it moves. For example, consider the effect of a single velocity gradient, 0u0x, on a small cube
having sides dx, dy, and dz. As is shown in Fig. 6.3a, if the x component of velocity of O and B
is u, then at nearby points A and C the x component of the velocity can be expressed as
u 10u0x2 dx. This difference in velocity causes a “stretching” of the volume element by an
amount 10u0x21dx21dt2 during the short time interval dt in which line OA stretches to OA¿ and BC
to BC¿ 1Fig. 6.3b2. The corresponding change in the original volume, V dx dy dz, would be
d
Change in dV a 0u dxb 1dy dz21dt2
0x
and the rate at which the volume dV is changing per unit volume due to the gradient 0u0x is
1 d1dV2
lim
(6.8)
dV dt
c 10u 0x2 dt
d 0u
dtS0 dt 0x
If velocity gradients 0v0y and 0w0z are also present, then using a similar analysis it follows that,
in the general case,
1 d1dV2
0u
(6.9)
dV dt 0x 0v
0y 0w
0z V
This rate of change of the volume per unit volume is called the volumetric dilatation rate. Thus, we
see that the volume of a fluid may change as the element moves from one location to another in the
flow field. However, for an incompressible fluid the volumetric dilatation rate is zero, since the element
volume cannot change without a change in fluid density 1the element mass must be conserved2.
Variations in the velocity in the direction of the velocity, as represented by the derivatives 0u0x, 0v0y,
and 0w0z, simply cause a linear deformation of the element in the sense that the shape of the element
does not change. Cross derivatives, such as 0u0y and 0v0x, will cause the element to rotate and
generally to undergo an angular deformation, which changes the shape of the element.
6.1.3 Angular Motion and Deformation
For simplicity we will consider motion in the x–y plane, but the results can be readily extended to
the more general three dimensional case. The velocity variation that causes rotation and angular
deformation is illustrated in Fig. 6.4a. In a short time interval dt the line segments OA and OB will
B
u
C
u +
__ ∂ u δx
∂ x
B
C
C'
δy
δy
O
u
δx
A
u +
__ ∂ u δx
∂ x
O
δx
A
A'
(a)
(b)
__ ∂ u
δ x
∂ x
δt
)
F I G U R E 6.3
Linear deformation of a fluid
element.