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Express the 2-dimensional continuity equation in cylindrical coordinates.(a) \(\frac{\partial(\rho v_r)}{\partial r}+\frac{1}{r}\frac{\partial(\rho v_\theta)}{\partial\theta}+\frac{\rhov_r}{r}=0\)(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»

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


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