Question

The incompressible continuity equation in polar coordinates is written as

• A. \(\dfrac{\partial \rho}{\partial r} + \dfrac{1}{r} \dfrac{\partial \rho}{\partial \theta} = 0\)
• B. \(\dfrac{\partial v_r}{\partial r} + \dfrac{\partial v_\theta}{\partial \theta} = 0\)
• C. \(\dfrac{\partial v_r}{\partial r} + \dfrac{1}{r} \dfrac{\partial v_\theta}{\partial \theta} = 0\)
• D. \(\dfrac{1}{r} \dfrac{\partial (r v_r)}{\partial r} + \dfrac{1}{r} \dfrac{\partial v_\theta}{\partial \theta} = 0\)

Hint:

Incompressibility means divergence of velocity is zero even in polar coordinates.

Answer 1

The Correct Option is D

The incompressible continuity equation in polar coordinates is fundamental in fluid mechanics and can be derived based on the conservation of mass principle, considering a control volume in polar coordinates. The general form involves velocities in the radial and angular directions, denoted as \(v_r\) and \(v_\theta\) respectively. For an incompressible fluid, where the density \(\rho\) remains constant, the continuity equation simplifies the conservation of mass.

The correct form of the incompressible continuity equation in polar coordinates is:

\[\frac{1}{r} \frac{\partial (r v_r)}{\partial r} + \frac{1}{r} \frac{\partial v_\theta}{\partial \theta} = 0\]

This equation is derived by considering the following:

• Radial Component: \(\frac{\partial}{\partial r}\) represents the change in radial direction. The term \(r v_r\) accounts for the radial velocity multiplied by the radial distance, preserving the form \(\frac{1}{r} \frac{\partial (r v_r)}{\partial r}\) to maintain dimensional consistency.
• Angular Component: \(\frac{\partial v_\theta}{\partial \theta}\) represents the change in the angular velocity, scaled by the factor \(\frac{1}{r}\) to accommodate the polar coordinate system.

Both terms work together to respect the conservation of mass in a control volume, ensuring inflow matches outflow for steady-state, incompressible fluid flow.