Solution Found!
Answer: Converging duct flow is modeled by the steady,
Chapter 4, Problem 56P(choose chapter or problem)
Problem 56P
Converging duct flow is modeled by the steady, two- dimensional velocity field of Prob. 4‒17. As vertical line segment AB moves downstream it shrinks from length η to length η+ ∆η as sketched in Fig. P4‒58. Generate an analytical expression for the change in length of the line segment, ∆η. Note that the change in length, ∆η, is negative. (Hint: Use the result of Prob. 4‒57.)
FIGURE P4‒58
Questions & Answers
QUESTION:
Problem 56P
Converging duct flow is modeled by the steady, two- dimensional velocity field of Prob. 4‒17. As vertical line segment AB moves downstream it shrinks from length η to length η+ ∆η as sketched in Fig. P4‒58. Generate an analytical expression for the change in length of the line segment, ∆η. Note that the change in length, ∆η, is negative. (Hint: Use the result of Prob. 4‒57.)
FIGURE P4‒58
ANSWER:
Solution
Introduction
The vertical line segment AB moves downstream it shrinks its length from η to η+is shown in the figure below
Step 1
For the particle A at time t , its location is yA1 = yA. E-bt
Similarly the location for the particle B at a time is yB1 =yB. E-bt
Initial length η is nothing but the separation between the two particles initial position
η= yB - yA
The length η+ is the separation between the two particles at the secondary position
η+ = yB1 - yA1