Solution Found!
Combine your results from Probs. 4?65 and 4?66 to form the
Chapter 4, Problem 63P(choose chapter or problem)
Problem 63P
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.
Questions & Answers
QUESTION:
Problem 63P
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.
ANSWER:
Solution to 63P
Step 1
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