Page 65 - Mechanical Engineers' Handbook (Volume 4)

54 Fluid Mechanics

The equations of streamlines are described by stream functions , from which the ve-
locity components in two-dimensional flow are u / y and v / x. Streamlines
are lines of constant stream function. In polar coordinates
1
v and v
r
r
r
Some streamline patterns are shown in Figs. 11, 12, and 13. The lines at right angles
to the streamlines are potential lines.

4.3 Deformation of a Fluid Element
Four types of deformation or movement may occur as a result of spatial variations of velocity:
translation, linear deformation, angular deformation, and rotation. These may occur singly
or in combination. Motion of the face (in the x-y plane) of an elemental cube of sides x,
y, and z in a time dt is shown in Fig. 14. Both translation and rotation involve motion or
deformation without a change in shape of the fluid element. Linear and angular deformations,
however, do involve a change in shape of the fluid element. Only through these linear and
angular deformations are heat generated and mechanical energy dissipated as a result of
viscous action in a fluid.
For linear deformation the relative change in volume is at a rate of
u v w
—
( — V dt — V )/ V div V
0
0
x y z
which is zero for an incompressible fluid, and thus is an expression for the continuity equa-
tion. Rotation of the face of the cube shown in Fig. 14d is the average of the rotations of
the bottom and left edges, which is
dt
1 v u
2 x y
The rate of rotation is the angular velocity and is
v about the z axis in the x-y plane
1
u
z
y
2
x
1
w
v
x
2 y z about the x axis in the y-z plane

Figure 11 Flow around a corner in a duct. Figure 12 Flow around a corner into a duct.