Tension in a chain fountain

I was reading the following paper: http://arxiv.org/abs/1310.4056

There were a few things I couldn't follow:

1) Equation (0.1)

$T_C/r=\lambda v^2/r$

I understand this takes the form of Newton's 2nd law, where $T_C$ is the force and $v^2/r$ is the acceleration. I don't understand why the tension is divided by $r$. I'm guessing it has to do with the dimensions of $\lambda$ (mass/length), but why is the radius of curvature the correct length to divide by? Is the sum of the tensions across the string a meaningful quantity?

2) Equations (0.2) and (0.3), and Figure 1

Do the tensions $T_T,T_C,T_F$ describe the tension at any point in their respective regions, or just the points at which they are shown in the figure? I.e., Is the tension uniform in those 3 respective regions or is it constantly varying along the chain? What I'm trying to understand is, could any point in (for example) the curved region be chosen to represent $T_C$? If not, why aren't the chosen points arbitrary?

Sorry for the abundance of questions. I'm guessing there's something fundamentally wrong with my understanding of tension, and that an answer correcting my misconception could potentially answer most or all of the questions simultaneously.

-
–  Qmechanic Feb 3 at 21:54