Combine your results from Problem given below to form the two-dimensional strain rate tensor in the ɛij in the xy-plane,

Are the x- and y-axes principal axes?

PROBLEM: 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 4 - 100: 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 98P

Step 1</p>

We need to find the two dimensional strain rate tensor for a two-dimensional Poiseuille flow with the velocity field equation as given below,

Where, 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.

The strain rate tensor εij in the x and y plane is given by,

Where, εxx and εyy are the linear strain rates in x and y directions respectively and εxy and εyx are the shear strain rates in x and y directions respectively.

By symmetry,

εxy=εyx

Step 2</p>

εxx=εyy=0 since,

And,

Now,

u and v are velocity components in x and y directions.

since v=0