Tiny hydrogen bubbles are being used as tracers to visualize a flow. All the bubbles are generated at the origin ( x = 0, y = 0). The velocity field is unsteady and obeys the equations: u = 1 m / s υ = 2 m / s 0 ≤ t < 2 s u = 0 υ = − 1 m / s 0 ≤ t ≤ 4 s Plot the pathlines of bubbles that leave the origin at t = 0, 1, 2, 3, and 4 s. Mark the locations of these five bubbles at t = 4 s. Use a dashed line to indicate the position of a streakline at t = 4 s.
Tiny hydrogen bubbles are being used as tracers to visualize a flow. All the bubbles are generated at the origin ( x = 0, y = 0). The velocity field is unsteady and obeys the equations: u = 1 m / s υ = 2 m / s 0 ≤ t < 2 s u = 0 υ = − 1 m / s 0 ≤ t ≤ 4 s Plot the pathlines of bubbles that leave the origin at t = 0, 1, 2, 3, and 4 s. Mark the locations of these five bubbles at t = 4 s. Use a dashed line to indicate the position of a streakline at t = 4 s.
Solution Summary: The following table shows the pathlines and the streaklines of the bubbles.
Tiny hydrogen bubbles are being used as tracers to visualize a flow. All the bubbles are generated at the origin (x = 0, y = 0). The velocity field is unsteady and obeys the equations:
u
=
1
m
/
s
υ
=
2
m
/
s
0
≤
t
<
2
s
u
=
0
υ
=
−
1
m
/
s
0
≤
t
≤
4
s
Plot the pathlines of bubbles that leave the origin at t = 0, 1, 2, 3, and 4 s. Mark the locations of these five bubbles at t = 4 s. Use a dashed line to indicate the position of a streakline at t = 4 s.
Consider the velocity field given by u = y/(x2 + y2) and v = −x/(x2 + y2). For the velocity field given , calculate the circulation around a circular path of radius 5 m. Assume that u and v given are in units of meters per second.
Find the stream lines and path lines of the particles of the two dimensional velocity field u=\frac{x|1+t}, v=y, w=0.
Consider the flow field shown. Coordinates are measured in meters. For the particle that passes through the point ðx, yÞ = ð1, 2Þ at the instant t = 0, plot the pathline during the time interval from t = 0 to 3 s. Compare this pathline with the streakline through the same point at the instant t=3s.
Chapter 2 Solutions
Fox and McDonald's Introduction to Fluid Mechanics
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