Question

Express the 2-dimensional continuity equation in cylindrical coordinates.

(a) (frac{partial( ho v_r)}{partial r}+frac{1}{r}frac{partial( ho v_ heta)}{partial heta}+frac{ ho  v_r}{r}=0)

(b) (frac{partial( ho v_r)}{partial r}+frac{1}{r}frac{partial( ho v_ heta)}{partial heta}+ ho  frac{v_r}{r}+frac{partial ho}{partial t}=0 )

(c) (frac{partial( ho v_r)}{partial r}+frac{1}{r}frac{partial( ho v_ heta)}{partial heta}+frac{partial ho}{partial t}=0)

(d) (frac{partial( ho v_r)}{partial r}+frac{1}{r}frac{partial( ho v_ heta)}{partial heta}+ ho frac{v_r}{r}+frac{partial ho}{partial t}=0)

Answer 1

The correct option is (b) (frac{partial( ho v_r)}{partial r}+frac{1}{r}frac{partial( ho v_ heta)}{partial heta}+ ho  frac{v_r}{r}+frac{partial ho}{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( ho v_r)}{partial r}+frac{1}{r}frac{partial( ho v_ heta)}{partial heta}+ ho  frac{v_r}{r}+frac{partial ho}{partial t}=0 ).