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

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