The correct choice is (b) 2ρVr + ρVθcot⁡θ + ρ\(\frac {∂(V_θ)}{∂θ}\) + Vθ\(\frac {∂(ρ)}{∂θ}\) = 0

To elaborate: The general continuity EQUATION is given by

\(\frac {∂}{∂t}\) + ∇.(ρV) = 0.

Since the flow is assumbed to be steady, \(\frac {∂}{∂t}\) = 0.

For SPHERICAL coordinated of a cone, the del operator is expanded as

∇.(ρV) = \(\frac {1}{r{^2}} \frac {∂}{∂r}\)(r^2ρVr) +\(\frac {1}{r sinθ} \frac {∂}{∂θ}\) (ρVθsin⁡θ) + \(\frac {1}{r sinθ} \frac {∂(ρV_ϕ)}{∂ϕ}\) = 0

Solving the partial derivatives, we GET

\(\frac {1}{r{^2}}\bigg [ \)r^2\(\frac {∂}{∂r}\)(ρVr) + ρVr\(\frac {∂(r^2)}{∂r} \bigg ] + \frac {1}{r sinθ} \bigg [ \)ρVθ\(\frac {∂}{∂θ}\)(sin⁡θ ) + sin⁡θ\(\frac {∂(ρV_θ)}{∂θ} \bigg ] + \frac {1}{r sinθ} \frac {∂(ρV_ϕ)}{∂ϕ}\) = 0

This is equal to

\(\frac {1}{r{^2}}\bigg [ \)r^2\(\frac {∂}{∂r}\)(ρVr) + ρVr(2r)\( \bigg ] + \frac {1}{r sinθ} \bigg [ \)ρVθ(cos⁡θ) + sin⁡θ\(\frac {∂(ρV_θ)}{∂θ} \bigg ] + \frac {1}{r sinθ} \frac {∂(ρV_ϕ)}{∂ϕ}\) = 0

Since the flow PROPERTIES are constant along a ray, \(\frac {∂}{∂r}\)(ρVr) = 0 and \( \frac {∂(ρV_ϕ)}{∂ϕ}\) = 0

The equations becomes \(\frac {1}{r{^2}}\)[ρVr(2r)] + \(\frac {1}{r sinθ} \bigg [ \)ρVθ(cos⁡θ) + sin⁡θ \(\frac {∂(ρV_θ)}{∂θ} \bigg ] \) + 0

Multiplying the final equation with r: 2ρVr + ρVθcot⁡θ + ρ\(\frac {∂(ρV_θ)}{∂θ}\) + Vθ\(\frac {∂(ρ)}{∂θ}\) = 0