Question

Which of these is the continuity equation for an axisymmetric flow?

(a) ρVθcot⁡θ + ρ$\frac {∂(V_θ)}{∂θ}$ + Vθ$\frac {∂(ρ)}{∂θ}$ = 0

(b) 2ρVr + ρVθcot⁡θ + ρ$\frac {∂(V_θ)}{∂θ}$ + Vθ$\frac {∂(ρ)}{∂θ}$ = 0

(c) 2ρVr + ρVθcot⁡θ = 0

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

I have been asked this question in examination.

Question is from Quantitative Formulation topic in portion Linearized and Conical Flows of Aerodynamics

Answer 1

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