Question

A cylindrical jet of water is unstable many a times. In this problem we will try to find the situation when the cylinder is stable under the forces of surface tension. You might have seen that a stream of water coming from a tap breaks into droplets before it drops to ground (figure-2). Consider a cylinder radius a of water which is disturbed by a small disturbance.

r = a + b cos($\frac{2\pi x}{\lambda}$) (figure-1).

here r is the radius of cylinder at position x. At cross-section A, the radius of cylinder is maximum and at cross section B, the radius is minimum. The surface tension of water is $\sigma$. The jet is stable if excess pressure at B is less than excess pressure at A. Then water will move from A to B to bring it to stable situation. Otherwise water will move from B to A till it splits into drops as shown in figure-2. Assume b << a and b << $\lambda$.

What is excess pressure at C.

• A. (A) $\frac{\sigma}{a}$
• B. (B) $\frac{2\sigma}{a}$
• C. (C) $\frac{\sigma}{a-b}$
• D. (D) $\frac{\sigma}{a+b}$

Answer 1

Answer: (A)

(A) $\frac{\sigma}{a}$

Answer 2

The excess pressure inside a curved liquid surface is given by the Young-Laplace equation. For a cylindrical surface of radius R, the excess pressure is $\Delta P = \frac{\sigma}{R}$. At point C, the radius of the cylinder is 'a'. Therefore, the excess pressure at C is $\frac{\sigma}{a}$.