fluid_mechanics

6.5 Some Basic, Plane Potential Flows 287
or
v r 1 0c
r 0u
v u 0c
0r
(6.75)
where the stream function was previously defined in Eqs. 6.37 and 6.42. We know that by defining
the velocities in terms of the stream function, conservation of mass is identically satisfied. If we
now impose the condition of irrotationality, it follows from Eq. 6.59 that
0u
0y 0v
0x
and in terms of the stream function
0
0y a 0c
0y b 0
0x a0c 0x b
or
0 2 c
0x 2 02 c
0y 2 0
Thus, for a plane irrotational flow we can use either the velocity potential or the stream function—
both must satisfy Laplace’s equation in two dimensions. It is apparent from these results that the velocity
potential and the stream function are somehow related. We have previously shown that lines of constant
c are streamlines; that is,
dy
dx `
along cconstant
v u
(6.76)
y
b
a
ψ
a_
b
a
b
b_
a
× (
– )
= –1
x
The change in f as we move from one point 1x , y2 to a nearby point dx, y dy2 is given by
the relationship
df 0f
0x
Along a line of constant f we have df 0 so that
1x
0f
dx dy u dx v dy
0y
dy
dx `
along fconstant
u v
(6.77)
A comparison of Eqs. 6.76 and 6.77 shows that lines of constant f 1called equipotential lines2
are orthogonal to lines of constant c 1streamlines2 at all points where they intersect. 1Recall that
two lines are orthogonal if the product of their slopes is 1, as illustrated by the figure in the
margin.2 For any potential flow field a “flow net ” can be drawn that consists of a family of
streamlines and equipotential lines. The flow net is useful in visualizing flow patterns and can
be used to obtain graphical solutions by sketching in streamlines and equipotential lines and
adjusting the lines until the lines are approximately orthogonal at all points where they intersect.
An example of a flow net is shown in Fig. 6.15. Velocities can be estimated from the flow net,
since the velocity is inversely proportional to the streamline spacing, as shown by the figure in
the margin. Thus, for example, from Fig. 6.15 we can see that the velocity near the inside corner
will be higher than the velocity along the outer part of the bend. (See the photographs at the
beginning of Chapters 3 and 6.)
Streamwise acceleration
Streamwise deceleration
6.5.1 Uniform Flow
The simplest plane flow is one for which the streamlines are all straight and parallel, and the
magnitude of the velocity is constant. This type of flow is called a uniform flow. For example,
consider a uniform flow in the positive x direction as is illustrated in Fig. 6.16a. In this instance,
u U and v 0, and in terms of the velocity potential
0f
0x U 0f
0y 0