Converging duct flow is modeled by the steady, two dimensional velocity field of Prob. 4–17. A fluid particle (A) is located on the x-axis at x = x_A at time t = 0 (Fig. P4–51).

At some later time t, the fluid particle has moved downstream with the flow to some new location x = x_A´, as shown in the figure. Since the flow is symmetric about the x-axis, the fluid particle remains on the x-axis at all times. Generate an analytical expression for the x-location of the fluid particle at some arbitrary time t in terms of its initial location x_A and constants U₀ and b. In other words, develop an expression for x_A´.

Data from Problem 17.

Consider steady, incompressible, two-dimensional flow through a converging duct (Fig. P4–17). A simple approximate velocity field for this flow is

where U₀ is the horizontal speed at x = 0. Note that this equation ignores viscous effects along the walls but is a reasonable approximation throughout the majority of the flow field. Calculate the material acceleration for fluid particles passing through this duct. Give your answer in two ways:

(1) As acceleration components a_x and a_y 

(2) As acceleration vector a(vector)

Fluid particle at some later time t Fluid particle at time t = 0