Question

A horizontal tube AB of length L, open at A and closed at B, is filled with an
ideal fluid. The end B has a small orifice. The tube is set in rotation in the horizontal plane with angular velocity ω about a vertical axis passing through A.
Show that the efflux velocity of the fluid is given by v = ωlroot(2L/l− 1)where l is the length of the fluid.

Answer 1

Answer:

Proof is given below.

Explanation:

Given,

A horizontal tube AB of length L, open at A and closed at B, is filled with an

ideal fluid.

The end B has a small orifice. The tube is set in rotation in the horizontal plane with angular velocity ω about a vertical axis passing through A.

To prove: the efflux velocity of the fluid is given by v = ωlroot(2L/l− 1)where l is the length of the fluid.

Solution:

Let the mass element be dm from at x from y-axis.

Now, we are required to find centrifugal force on mass dm.

dF = dmω^2x = dm dv/dt = dm. v dv/dx

or, v dv = ω²x dx

Integrating the equation above,

$\displaystyle \int v \ dv = \omega^2 \int x \ dx\\\\\dfrac{v^2}{2} = \dfrac{\omega^2}{2} x^2 |_{L-l}^L\\\\v = \omega l\sqrt{\dfrac{2L}{l}-1}$