I was studying about Accretion disks and found some difficulties regarding some concepts. My questions are(adsbygoogle = window.adsbygoogle || []).push({});

1. For an accretion disk, ## v_r<<c_s ## Why?

2. The conservation of mass equation is derived as,

The mass in an annulus of radius ##\Delta r## is ##2\pi r \Delta r\Sigma ## and it is changing due to the mismatch of mass outflow and inflow which is written as,

$$ \frac{\partial}{\partial t}(2πr\Delta r\Sigma) = v_r(r,t)2πr\Sigma(r,t)−v_r(r+\Delta r,t)2π(r+\Delta r)\Sigma(r+\Delta r,t)\\ \approx-2\pi\Delta r\frac{\partial (r\Sigma v_r)}{\partial r} $$

up to this I have no problem but it further says that this equation in the ## \Delta r\rightarrow0 ## limit gives,

$$r\frac{\partial\Sigma}{\partial t}+\frac{\partial (r\Sigma v_r)}{\partial r}=0$$

and I can't able to derive how that limit gives that equation.

3. The angular momentum conservation equation is derived from the Navier-Stokes equation as,

$$ \frac{\partial \vec{v}}{\partial t}+(\vec{v}.\vec{\nabla})\vec{v}=-\frac{1}{\rho}(\vec{\nabla}P-\vec{\nabla}.\sigma)-\vec{\nabla}\Phi $$

In the axisymmetry assumption and no momentum loss or gain from the ## z ## direction one gets,

$$\Sigma\Big(\frac{\partial v_\phi}{\partial t}+\frac{v_rv_\phi}{r}+v_r\frac{\partial v_\phi}{\partial r}\Big)=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 T_{r\phi}) $$

Now my question is, shouldn't the RHS be ## \frac{1}{r}\frac{\partial}{\partial r}(r T_{r\phi}) ## instead of ## \frac{1}{r^2}\frac{\partial}{\partial r}(r^2 T_{r\phi}) ## because the ## r ## part of ## \vec{\nabla}.\vec{f}## in the cylindrical coordinate is ## \frac{1}{r}\frac{\partial}{\partial r}(r f_r) ##.

Thanks in advance..

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A Questions about Accretion disk

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**