For the axisymmetric Poiseuille flow of Prob. 4-108. calculate the linear strain rates in the x- and r-directions, and calculate the shear strain rate ɛxr. The strain rate tensor in cylindrical coordinates (r, θ. x) and (ur,uθ. ux). is

PROBLEM: Consider fully developed axisymmetric Poiseuille flow—flow in a round pipe of radius R(diameter D = 2 R), with a forced pressure gradient dP/dx driving the flow as illustrated in Fig. P11–53. (dP/dx is 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.

FIGURE P11-53

,

and

.

So the linear strain along the direction is

Since the component of the velocity has no dependence.