Converging duct flow is modeled by the steady, two- dimensional velocity field of Prob. 4‒17. A fluid particle (A) is located at x = xA and y = y A at time t = 0 (Fig. P4‒57). At some later time t. the fluid particle has moved downstream with the flow to some new location x = x A., y =y A., as show n in the figure. Generate an analytical expression for the y-location of the fluid particle at arbitrary time t in terms of its initial y-location y A and constant b. In other words, develop an expression for y A.(Hint: We know that v = dyparticle/dt following a fluid particle. Substitute the equation for v. separate variables, and integrate.)

FIGURE P4‒57

Solution

Introduction

Consider a fluid particle A which is initially located at x =xA and y=yA at some later time t the fluid particle has moved and the new location x=xA1 and y=yA1.

Step 1

The velocity field is given by v=()

v=(U0 +bx) - by …………..(1)

We know that for a fluid particle v =

Substitute the value of v from equation (1) in the above equation

= -byparticle

The subscript particle can move and the above equation can write down as

= -by