Question

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 is unsteady and obeys the equations: u = 1 m/s v = 2 m/s 0 ≤ t < 2 s u = 0 m/s v = −1 m/s 2 s ≤ 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

Answer 1

Answer:

for t = 1 : ( x1 , y1 ) = (1,2)

for t = 2 : ( x2, y2 ) = (2,4)

for t = 3 : ( x3, y3 ) = ( 0,-3)

for t = 4 : ( x4, y4 ) = ( 0,-4)

Explanation:

Attached below is the detailed solution and the sketch

we have to make assumptions for the various values of t and integrate accordingly

for t = 1 : ( x1 , y1 ) = (1,2)

for t = 2 : ( x2, y2 ) = (2,4)

for t = 3 : ( x3, y3 ) = ( 0,-3)

for t = 4 : ( x4, y4 ) = ( 0,-4)

Given data :

origin ( x = 0, y = 0 )

u = 1 m/s,  v = 2 m/s,

$0 \leq t < 2s$  where u = 0 m/s   v = -1 m/s