Problem 57P

Using the results of Prob. 4‒58 and the fundamental definition of linear strain rate (the rate of increase in length per unit length), develop an expression for the linear strain rate in the y-direction (εyy) of fluid particles moving down the channel. Compare your result to the general expression for εyy,in terms of the velocity field, i.e., εyy = ∂vl∂y. (Hint: Take the limit as time t → 0. You may need to apply a truncated series expansion for e ‒bt.)

Solution

Introduction

The strain rate at some points within the material measures the rate at which the distances of adjacent parts of the material with time in the neighborhood of that point.

Step 1

The linear strain rate is the rate of increase in length of a line segment per unit length of the line segment

The linear strain rate in y direction is

ε yy =

ε yy= ………….(1)