Question

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).

Answer 1

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

`(4a (U\omega- g))/(U^2)`

Step-by-step 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

`v_A=U-(d)/(2)\omega\\v_A=U-(2a)/(2)\omega\\v_A=U-a\omega\\`

Now the velocity at point B is given as

`v_B=U+(d)/(2)\omega\\v_B=U+(2a)/(2)\omega\\v_B=U+a\omega\\`

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

`(P_A)/(\rho g)+(v_A^2)/(2 g)+z_A=(P_B)/(\rho g)+(v_B^2)/(2 g)+z_B`

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

`(P_A-P_B)/(\rho g)=(v_B^2-v_A^2)/(2 g)+z_B-z_A`

Putting the values

`(P_A-P_B)/(\rho g)=(v_B^2-v_A^2)/(2 g)+z_B-z_A\\(P_A-P_B)/(\rho g)=((U+a\omega)^2-(U-a\omega)^2)/(2 g)+0-2a\\(P_A-P_B)/(\rho g)=((U^2+a^2\omega^2+2Ua\omega)-(U^2+a^2\omega^2-2Ua\omega))/(2g)-2a\\(P_A-P_B)/(\rho g)=(U^2+a^2\omega^2+2Ua\omega-U^2-a^2\omega^2+2Ua\omega))/(2g)-2a\\(P_A-P_B)/(\rho g)=(4Ua\omega)/(2g)-2a\\(P_A-P_B)/(\rho g)=(2Ua\omega)/(g)-2a\\P_A-P_B=(2Ua\omega)/(g)*\rho g-2a*\rho g\\`

`P_A-P_B=2Ua\omega\rho-2a\rho g`

Now the dynamic pressure is given as

`P_D=(1)/(2)\rho U^2`

`(P_A-P_B)/(P_D)=(2Ua\omega\rho-2a\rho g)/(1/2 \rho U^2)\\(P_A-P_B)/(P_D)=(2a\rho (U\omega- g))/(1/2 \rho U^2)\\(P_A-P_B)/(P_D)=(4a\rho (U\omega- g))/(\rho U^2)\\(P_A-P_B)/(P_D)=(4a (U\omega- g))/(U^2)`

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

`(4a (U\omega- g))/(U^2)`