Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 314 / 4 / c

Solution

Along the open line,

B_{p} \sim F / r^{2}

and the mass-loading relation with

u_{p} \to u_{\infty}

gives

ρ \sim \frac{k F}{u _{\infty} r ^{2}}, M_{A}^{2} \sim \frac{μ _{0} k u _{\infty}}{F} r^{2} ⟶ \infty.

The solutions of part b consequently have

u_{ϕ} \sim \frac{ℓ}{r}, B_{ϕ} \sim - \frac{ω F}{u _{\infty} r} .

The azimuthal Alfvén speed therefore approaches the nonzero constant

∣ v_{A ϕ} ∣ = \frac{∣ B _{ϕ} ∣}{μ _{0} ρ} ⟶ ∣ ω ∣ \frac{F}{μ _{0} k u _{\infty}} .

For an unconfined outflow it is natural to take

Φ \to 0

and

h \to 0

at infinity; also

u_{ϕ} \to 0

- \frac{r ω B _{ϕ}}{μ _{0} k} ⟶ \frac{ω ^{2} F}{μ _{0} k u _{\infty}} .

ϵ = \frac{1}{2} u_{\infty}^{2} + \frac{ω ^{2} F}{μ _{0} k u _{\infty}} .

There are no discussions about this article yet.

There are currently no matching articles.