Consider fully developed two-dimensional Poiseuille flow—flow between two infinite parallel plates separated by distance h,with both the top plate and bottom plate stationary, and a forced pressure gradient dP/dxdriving the flow as illustrated in Fig. P11–48. (dP/dxis constant and negative.)

FIGURE P11-48

The flow is steady, incompressible, and two-dimensional in the xy-plane. The velocity components are given by

where μ is the fluid’s viscosity. Is this flow rotational or irrotational? If it is rotational, calculate the vorticity component in the z-direction. Do fluid particles in this flow rotate clockwise or counterclockwise?

Step 1 of 4:

Consider the steady flow in two-dimensional xy plane. The flow is also considered as Poiseuille flow. It happens between two infinitely long parallel plates. The distance between the two plates is h. And the flow is stationary at the plates. The velocity field components are given by

= ----(1)

Where is the fluid’s viscosity

is the pressure gradient.

= 0 ------(2)

To find whether the flow is rotational or irrotational.

Step 2 of 4:

To find whether the flow is rotational or irrotational.

This is done by calculating the vorticity vector of z component. Vorticity is the local rotation or the spin of the flow. In the case of two-dimensional flow,

= -----(3)

From (2), differentiating with respect to x

= 0 -----(4)

From (1), differentiating with respect to y

=

= ------(5)