Converging duct flow is modeled by the steady, two-dimensional velocity field of Prob. 11–15. The pressure field is given by

where P0 is the pressure at x =0. Generate an expression for the rate of change of pressure following a fluid particle.

PROBLEM: Consider steady, incompressible, two-dimensional flow through a converging duct (Fig. P11–15). 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 ax and ay and (2) as acceleration vector .

Step 1:

Consider a duct flow which is steady, converging and incompressible. The flow happens at two dimensional space.

The pressure field is given by

P = - ----(1)

Where P0 is the pressure at x = 0

Step 2:

To generate an expression for the rate of change of pressure following a fluid particle

The material derivative of pressure function as given in the velocity field

= + + + -------(2)

Step 3:

In the case of steady flow = 0

And two dimensional flow = 0

The (2) becomes

= + -----------------------(3)

Step 4:

Differentiating (1) with respect to x

=

=

= ---------------(4)