Consider fully developed axisymmetric Poiseuille flow—flow in a round pipe of radius R(diameter D = 2 R), with a forced pressure gradient dP/dxdriving the flow as illustrated in Fig. P11–53. (dP/dxis constant and negative.) The flow is steady, incompressible, and axisymmetric about the x-axis. 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 circumferential (θ) direction and discuss the sign of the rotation.
First determine if the flow is rotational or not, if the flow is rotational then find out the component of the vorticity.
To determine if the flow is rotational or not calculate the vorticity of the flow, using the formula
If the vorticity is nonzero then the flow is rotational and if it is zero then the flow is rotational.
The given velocity components are
Substitute this components in the equation (1), the first component of the equation (1) will be zero because . And the second term si
Since the vorticity of the flow is non-zero, the flow is rotational.