The velocity field for a plane source located distance h = 1m above an infinite wall aligned along the x axis is given by V → = q 2 π [ x 2 + ( y − h ) 2 ] [ x i ^ + ( y − h ) j ^ ] + q 2 π [ x 2 + ( y + h ) 2 ] [ x i ^ + ( y + h ) j ^ ] where q = 2 m 3 /s/m. The fluid density is 1000 kg/m 3 and body forces are negligible. Derive expressions for the velocity and acceleration of a fluid particle that moves along the wall, and plot from x = 0 to x = + 10 h . Verify that the velocity and acceleration normal to the wall are zero. Plot the pressure gradient ∂ p /∂ x along the wall. Is the pressure gradient along the wall adverse (does it oppose fluid motion) or not? P6.6
The velocity field for a plane source located distance h = 1m above an infinite wall aligned along the x axis is given by V → = q 2 π [ x 2 + ( y − h ) 2 ] [ x i ^ + ( y − h ) j ^ ] + q 2 π [ x 2 + ( y + h ) 2 ] [ x i ^ + ( y + h ) j ^ ] where q = 2 m 3 /s/m. The fluid density is 1000 kg/m 3 and body forces are negligible. Derive expressions for the velocity and acceleration of a fluid particle that moves along the wall, and plot from x = 0 to x = + 10 h . Verify that the velocity and acceleration normal to the wall are zero. Plot the pressure gradient ∂ p /∂ x along the wall. Is the pressure gradient along the wall adverse (does it oppose fluid motion) or not? P6.6
Solution Summary: The author calculates the expression for velocity and acceleration of fluid particle, and plots pressure gradient along the wall.
The velocity field for a plane source located distance h = 1m above an infinite wall aligned along the x axis is given by
V
→
=
q
2
π
[
x
2
+
(
y
−
h
)
2
]
[
x
i
^
+
(
y
−
h
)
j
^
]
+
q
2
π
[
x
2
+
(
y
+
h
)
2
]
[
x
i
^
+
(
y
+
h
)
j
^
]
where q = 2 m3/s/m. The fluid density is 1000 kg/m3 and body forces are negligible. Derive expressions for the velocity and acceleration of a fluid particle that moves along the wall, and plot from x = 0 to x = + 10h. Verify that the velocity and acceleration normal to the wall are zero. Plot the pressure gradient ∂p/∂x along the wall. Is the pressure gradient along the wall adverse (does it oppose fluid motion) or not?
Home Work (steady continuity equation at a point for incompressible
fluid flow:
1- The x component of velocity in a steady, incompressible flow field in the
xy plane is u= (A /x), where A-2m s, and x is measured in meters. Find
the simplest y component of velocity for this flow field.
2- The velocity components for an incompressible steady flow field are u= (A
x* +z) and v=B (xy + yz). Determine the z component of velocity for
steady flow.
3- The x component of velocity for a flow field is given as u = Ax²y2 where
A = 0.3 ms and x and y are in meters. Determine the y component of
velocity for a steady incompressible flow. Assume incompressible steady
two dimension flow
4. The velocity vectors of three flow fileds are given as V, = axĩ + bx(1+1)}+ tk ,
V, = axyi + bx(1+t)j , and V3 = axyi – bzy(1+t)k where coefficients a and b have
constant values. Is it correct to say that flow field 1 is one-, flow filed 2 is two-, and
flow filed 3 is three-dimensional? Are these flow fields steady or unsteady?
1.6 An incompressible Newtonian fluid flows in the z-direction in space between two par-
allel plates that are separated by a distance 2B as shown in Figure 1.3(a). The length and
the width of each plate are L and W, respectively. The velocity distribution under steady
conditions is given by
JAP|B²
Vz =
2µL
B
a) For the coordinate system shown in Figure 1.3(b), show that the velocity distribution
takes the form
JAP|B?
v, =
2μL
Problems
11
- 2B --– €.
(a)
2B
(b)
Figure 1.3. Flow between parallel plates.
b) Calculate the volumetric flow rate by using the velocity distributions given above. What
is your conclusion?
2|A P|B³W
Answer: b) For both cases Q =
3µL
Chapter 6 Solutions
Fox and McDonald's Introduction to Fluid Mechanics
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