fluid_mechanics

322 Chapter 6 ■ Differential Analysis of Fluid Flow
6.15 For each of the following stream functions, with units of
m 2 /s, determine the magnitude and the angle the velocity vector
makes with the x axis at x 1 m, y 2 m. Locate any stagnation
points in the flow field.
(a) c xy
(b) c 2x 2 y
6.16 The stream function for an incompressible, two-dimensional
flow field is
c ay by 3
where a and b are constants. Is this an irrotational flow? Explain.
6.17 The stream function for an incompressible, two-dimensional
flow field is
c ay 2 bx
where a and b are constants. Is this an irrotational flow? Explain.
6.18 The velocity components for an incompressible, plane flow are
v r Ar 1 Br 2 cos u
v u Br 2 sin u
where A and B are constants. Determine the corresponding stream
function.
6.19 For a certain two-dimensional flow field
u 0
v V
(a) What are the corresponding radial and tangential velocity components?
(b) Determine the corresponding stream function expressed in
Cartesian coordinates and in cylindrical polar coordinates.
6.20 Make use of the control volume shown in Fig. P6.20 to derive
the continuity equation in cylindrical coordinates 1Eq. 6.33 in text2.
y, ft
1.0
O
1.0
A
x, ft
6.24 The radial velocity component in an incompressible, twodimensional
flow field 1v z 02 is
v r 2r 3r 2 sin u
F I G U R E P6.23
Determine the corresponding tangential velocity component, v u ,
required to satisfy conservation of mass.
6.25 The stream function for an incompressible flow field is given
by the equation
c 3x 2 y y 3
where the stream function has the units of m 2 s with x and y in meters.
(a) Sketch the streamline1s2 passing through the origin. (b) Determine
the rate of flow across the straight path AB shown in Fig. P6.25.
y, m
1.0 B A
z
y
r
dθ
F I G U R E P6.20
θ
dr
Volume element
has thickness d z
6.21 A two-dimensional, incompressible flow is given by u y
and v x. Show that the streamline passing through the point
x 10 and y 0 is a circle centered at the origin.
6.22 In a certain steady, two-dimensional flow field the fluid density
varies linearly with respect to the coordinate x; that is, r Ax
where A is a constant. If the x component of velocity u is given by
the equation u y, determine an expression for v.
6.23 In a two-dimensional, incompressible flow field, the x component
of velocity is given by the equation u 2x. (a) Determine
the corresponding equation for the y component of velocity if
v 0 along the x axis. (b) For this flow field, what is the magnitude
of the average velocity of the fluid crossing the surface OA of
Fig. P6.23? Assume that the velocities are in feet per second when
x and y are in feet.
x
y
0 1.0 x, m F I G U R E P6.25
6.26 The streamlines in a certain incompressible, two-dimensional
flow field are all concentric circles so that v r 0. Determine the
stream function for (a) v and for (b) v u Ar 1 u Ar
, where A is a
constant.
*6.27 The stream function for an incompressible, twodimensional
flow field is
c 3x 2 y y
For this flow field, plot several streamlines.
6.28 Consider the incompressible, two-dimensional flow of a nonviscous
fluid between the boundaries shown in Fig. P6.28. The velocity
potential for this flow field is
f x 2 y 2
q
A
B
(x i , y i )
F I G U R E P6.28
q ψ = 0
x