For the two-dimensional Poiseuille flow of Prob. 4-100, calculate the linear strain rates in the x- and y-directions, and calculate the shear strain rate ɛxy.

PROBLEM: 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?

Solution to 97P

Step 1</p>

We need to find the linear strain rates in x and y directions and the shear strain rate εxy for the velocity field equation given below.

u is the component of velocity in x direction, v is the component of velocity in y direction, is the pressure gradient, μ is the fluid viscosity, h is the height.

The fluid is assumed to be incompressible, steady and two dimensional.

Step 2</p>

Linear strain rates:

We know that the linear strain rates are given by the following equations,

Where is the linear strain rate for a given coordinates, u and v are velocity components in the given direction.

Linear strain rate in x direction:

Linear strain rate in y direction:

Thus linear strain rates in both the directions is zero.