• Continuity Equation in Cartesian Coordinates:
• General Form (Mass Conservation):

• Universality of the Equation:
• Steady and Unsteady flow: Directly accounts for temporal density fluctuations ($\frac{\partial \rho}{\partial t} = 0$ for steady flow).
• Uniform and Non-uniform flow: Spatial derivative terms handle velocity and area variations along coordinate axes.
• Compressible and Incompressible fluids: Retains fluid density $\rho$ inside spatial gradients for gas dynamics while simplifying to $\nabla \cdot \vec{v} = 0$ when $\rho = \text{constant}$.
• Conclusion: (A), (B) and (C) (Option 3).

Correct option: (3) (A), (B) and (C).

• Capillary Rise & Minimum Piezometer Tube Diameter:
• Capillary Rise Equation:

For clean water in a glass tube, contact angle $\theta = 0^\circ \implies \cos\theta = 1$.

• Calculation:
• Surface tension $\sigma = 0.073575\text{ N/m}$
• Density $\rho = 1000\text{ kg/m}^3$, $g = 9.81\text{ m/s}^2$
• Maximum allowable capillary rise $h = 3\text{ mm} = 0.003\text{ m}$

• Conclusion: Minimum diameter of glass tube is 1 cm (Option 4).

Correct option: (4) 1cm