Question

Express the 2-dimensional continuity equation in cylindrical coordinates.

(a) \(\frac{\partial(\rho v_r)}{\partial r}+\frac{1}{r}\frac{\partial(\rho v_\theta)}{\partial\theta}+\frac{\rho  v_r}{r}=0\)

(b) \(\frac{\partial(\rho v_r)}{\partial r}+\frac{1}{r}\frac{\partial(\rho v_\theta)}{\partial\theta}+\rho  \frac{v_r}{r}+\frac{\partial\rho}{\partial t}=0 \)

(c) \(\frac{\partial(\rho v_r)}{\partial r}+\frac{1}{r}\frac{\partial(\rho v_\theta)}{\partial\theta}+\frac{\partial\rho}{\partial t}=0\)

(d) \(\frac{\partial(\rho v_r)}{\partial r}+\frac{1}{r}\frac{\partial(\rho v_\theta)}{\partial\theta}+\rho \frac{v_r}{r}+\frac{\partial\rho}{\partial t}=0\)

This question was addressed to me in an interview for internship.

Question is taken from Numerical Methods in section Numerical Methods of Computational Fluid Dynamics

Answer 1

The correct option is (b) \(\frac{\partial(\rho v_r)}{\partial r}+\frac{1}{r}\frac{\partial(\rho v_\theta)}{\partial\theta}+\rho  \frac{v_r}{r}+\frac{\partial\rho}{\partial t}=0 \)

To explain I would say: In Cartesian coordinates, radial and angular velocities replace the x and y velocity components. Similarly, (r, θ) is the coordinate system used here. The continuity equation in this system can be given by \(\frac{\partial(\rho v_r)}{\partial r}+\frac{1}{r}\frac{\partial(\rho v_\theta)}{\partial\theta}+\rho  \frac{v_r}{r}+\frac{\partial\rho}{\partial t}=0 \).