A two-dimensional fluid element of dimensions dx and dy translates and distorts as shown in Fig. P4‒74 during the infinitesimal time period dt = t 2‒t 1· The velocity components at point P at the initial time are u and v in the x- and y-directions, respectively. Show that the magnitude of the rate of rotation (angular velocity) about point P in the xy-plane is

FIGURE P4‒74

Problem 70P

Solution 70P

Step 1 of 6 :

In this question,we have two dimensional fluid, we need to show the magnitude of rate of rotation at point P as

Considering the given figure

Step 2 of 6:

From the definition of angular velocity we have, the rate of rotation at a point is the average rotation rate of two initially perpendicular lines intersect at a point P

Here Line a rotates along an angle and Line b rotates along an angle

Thus we have average angle of rotation as

Step 3 of 6 :

During time increment

The point P moves distance along right and upwards

The point A moves distance along right and upwards

The point B moves distance along right and upwards