User:Lothar.brendel/Gravity driven flow into a perfect spherical sink

From Arbeitsgruppe Kuiper
Jump to navigation Jump to search

Isothermal Bondi flow

\(\newcommand{\dr}{\partial_r}\)

The stationary continuity equation for spherical symmetry and \(\vec u=-u\vec e_r\) reads \begin{align} \label{eq:cont} 0&=\nabla\cdot(\rho\vec u)=-\dr(r^2\rho u)/r^2\\ \Leftrightarrow\quad \frac{\dot M}{4\pi}&=r^2\rho u=\text{const.}, \end{align} \(\dot M\) being the "anihilation rate" of the spherical sink. It complements the stationary Euler equation (which is invariant wrt \(u\to-u\)) \begin{align} \label{eq:eul} u\dr u+\frac{GM}{r^2}&=-\frac{\dr P}{\rho}=-\frac{\partial P}{\partial\rho}\frac{\dr\rho}{\rho}\\ &=-c^2\;r^2 u\,\dr(r^2u)^{-1} = c^2\frac{2ru+r^2\dr u}{r^2u}\\ &=c^2\left(\frac{2}{r}+\frac{\dr u}{u}\right)\\ \Leftrightarrow\quad \frac{u}{c}\frac{\dr u}{c}-\frac{\dr u}{u}&=\frac{2}{r}-\frac{2R_\text{Bo}}{r^2}~, \end{align} which suggests \(R_\text{Bo}=GM/(2c^2)\) as natural length unit and \(R_\text{Bo}/c\) as natural time unit, i.e. we continue with \begin{align} (u-1/u)\dr u &= \frac{2}{r}-\frac{2}{r^2}\\ \Leftrightarrow\quad \label{eq:du} \dr u &= \frac{2u(r-1)}{r^2(u^2-1)}\xrightarrow[r\to\infty]{u\to 0}-\frac{2u}{r}~,\qquad(1) \end{align} where in the last step we have put in the "natural" boundary condition of \(u\) vanishing for \(r\to\infty\).

The subsonic solution

The asymptotic solution of (1) is \begin{equation} \label{eq:asymp} u_\infty(r)=\frac{u_\wedge}{r^2}~, \end{equation} from which the real solution deviates strongly for \(r\not\gg 1\). As long as \(u<1\), \(\dr u\) will be negative for \(r>1\), i.e. \(u\) will increase for decreasing \(r\), but slower than \(u_\infty(r)\). We assume \(u<1\) for \(r\ge 1\), which implies \(\dr u>0\) for \(r<1\), i.e. \(u\) decreases again for decreasing \(r\), making \(u(1)=u_\text{max}<1\) a local maximum. For \(u_\text{max}\to 1\), the slope switching around \(r=1\) gets more localised, rendering the maximum more and more cusp-like.

For \(u\to 0\), (1) gets linear again: \begin{align} \dr u &= \frac{2u(r-1)}{r^2} \end{align} It yields a solution vanishing at zero together with all its derivatives: \begin{align} u_0(r) &= \frac{u_\vee}{r^2}\exp(-2/r) \end{align} The amplitudes \(u_\vee\) and \(u_\wedge\) are non-linear functions of \(u_\text{max}\), varying linearly with \(u_\text{max}\ll 1\) and saturating both at \(u_\vee\approx u_\wedge\approx 4.48\).

Larger amplitudes would lead to \(u\) reaching unity already away from \(r=1\), which corresponds to a diverging slope \(\dr u\). Hence, faster stationary, subsonic solutions with spherical symmetry do not exist.

Further reading:

Retrieved from "https://wiki.uni-due.de/agk/index.php?title=User:Lothar.brendel/Gravity_driven_flow_into_a_perfect_spherical_sink&oldid=529"