QUESTION:

Problem 67P

Consider the steady, incompressible, two-dimensional flow field of  Prob. 4‒69. Using the results of Prob. 4‒69(a). do the following:

(a) From the fundamental definition of   shear strain rate (half of the rate of decrease of the angle between two initially perpendicular lines that intersect at a point), calculate shear strain rate εxy  in the xy-plane.(Hint: Use the lower edge and the left edge of the fluid particle, which intersect at 90° at the lower-left corner of the particle at the initial time.)

(b) Compare your results with those obtained from the equation for Ɛxy  in Cartesian coordinates, i.e.,

PROBLEM: Consider steady, incompressible, two-dimensional shear flow for which the velocity field iswhere a and b are constants. Sketched in Fig. P4‒69 is a small rectangular fluid particle of dimensions dx and dy at time t. The fluid particle moves and deforms with the flow such that at a later time (t + dt). the particle is no longer rectangular, as also shown in the figure. The initial location of each corner of the fluid particle is labeled in Fig. P4‒69. The lower-left corner is at (x. y) at time t. where the x-component of  velocity is u = a + by. At the later time, this corner moves to (x + u dt, y), or

(a) In similar fashion, calculate the location of each of the other three corners of the fluid particle at time t ‒ dt.

(b) From the fundamental definition of   linear strain rate (the rate of increase in length per unit length). calculate linear strain rates Ɛxx and Ɛyy.

(c) Compare your results with those obtained from the equations for Ɛxx and Ɛyy in Cartesian coordinates, i.e..

FIGURE P4‒69