Use two methods to verify that the flow of Prob. 4–64 is incompressible: 

(a) By calculating the volume of the fluid particle at both times,

(b) By calculating the volumetric strain rate. Note that Prob. 4–64 should be completed before this problem.

Data from problem 64

Consider steady, incompressible, two-dimensional shear flow for which the velocity field is

where a and b are constants. Sketched in Fig. P4–64 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. P4–64. 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 (x + (a + by) dt, y)

Transcribed Image Text:

V = (u, v) = (a + by)i + 07