Question

Consider a water tank shown in the figure. It has one wall at \(x = L\) and can be taken to be very wide in the z direction. When filled with a liquid of surface tension \(S\) and density \( \rho \), the liquid surface makes angle \( \theta_0 \) (\( \theta_0 < < 1 \)) with the x-axis at \(x = L\). If \(y(x)\) is the height of the surface then the equation for \(y(x)\) is: (take \(g\) as the acceleration due to gravity) 

• A. \( \frac{d^2 y}{dx^2} = \frac{\rho g}{S} y \)
• B. \( \frac{d^2 y}{dx^2} = \sqrt{\frac{\rho g}{S}} y \)
• C. \( \frac{d^2 y}{dx^2} = \sqrt{\frac{S}{\rho g}} y \)
• D. \( \frac{d^2 y}{dx^2} = \frac{S}{\rho g} y \)

Hint:

The shape of the liquid meniscus near a wall is determined by the balance between surface tension forces (related to the curvature) and gravitational forces (related to hydrostatic pressure). The Young-Laplace equation provides the fundamental relationship, which can be approximated for small slopes.

Answer 1

The Correct Option is A

To derive the differential equation for the height \(y(x)\) of the liquid surface, we need to consider the forces involved due to surface tension and gravity. The surface tension causes a force that depends on the curvature of the surface, while gravity causes a downward force proportional to the height of the liquid column.

The Young-Laplace equation relates the curvature of a surface to the pressure difference across the surface due to surface tension. For a surface described by a function \(y(x)\), the curvature \(\kappa\) can be approximated by the second derivative \(\kappa \approx \frac{d^2y}{dx^2}\) when \(\theta_0 \ll 1\).

The pressure difference \(\Delta P\) across the liquid surface due to surface tension is given by:

\(\Delta P = S \kappa = S \frac{d^2y}{dx^2}\)

The hydrostatic pressure due to gravity at height \(y(x)\) is:

\(P = \rho g y(x)\)

For equilibrium, set the pressure due to surface tension equal to the hydrostatic pressure:

\[ S \frac{d^2y}{dx^2} = \rho g y(x) \]

Dividing both sides by \(S\), we obtain the differential equation:

\[ \frac{d^2y}{dx^2} = \frac{\rho g}{S} y(x) \]

This is a linear second-order differential equation describing the shape of the surface of the liquid.