A long circular cylinder of diameter 2a meters is set horizontally in a steady stream (perpendicular to the cylinder axis) of velocity U m/s. The cylinder is caused to rotate at ω rad/s around its axis. Obtain an expression in terms of ω and U for the ratio of the pressure difference between the top and bottom of the cylinder divided by the dynamic pressure of the stream (i. e., the pressure coefficient difference).

The ratio of the difference of the pressure at the top and bottom of the cylinder to dynamic pressure is given as

Explanation:

As the value of the diameter is given as d=2a

The velocity is given as v=U

The rotational velocity is given as ω rad/s

Point A is at the top of the cylinder and point B is at the bottom of the cylinder

Such that the point A is at the highest point on the circumference and point B is at the bottom of the cylinder

Now the velocity at point A is given as

Now the velocity at point B is given as

Considering point B as datum and applying the Bernoulli's equation between the point A and B gives

Here P_A and P_B are the local pressures at the point A and point B.

v_A and v_B are the velocities at the point A and B

z_A and z_B is the height of point A which is 2a and that of point B is 0

Now rearranging the equation of Bernoulli gives

Putting the values

Now the dynamic pressure is given as

So the ratio of the difference of the pressure at the top and bottom of the cylinder to dynamic pressure is given as

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explanation: