Combine your results from Probs. 4‒65 and 4‒66 to form the two-dimensional strain rate tensor Ɛij in the xy-plane,

Under what conditions would the x- and y-axes be principal axes?

PROBLEM 4 -65: For the velocity field of Prob. 4‒63, calculate the linear strain rates in the x- and y-directions.

PROBLEM 4-66: For the velocity field of Prob. 4‒63, calculate the shear strain rate in the xy-plane.

PROBLEM 4-63: A general equation for a steady, two-dimensional velocity field that is linear in both spatial directions (x and y) is

Where U and V and the coefficients are constants. Their dimensions are assumed to be appropriately defined. Calculate the x- and y-components of the acceleration field.

Solution to 63P

Step 1</p>

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

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>

Linear strain rate in x direction:

Linear strain rate in y direction:

Thus the linear strain rates in x and y directions are as follows

εxx=a1

εyy=b2

Shear strain rate is given by,

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