In a diverging pipe section like the following,
the pipe of radius $r$ splits into two pipes of radius $r/2$.
Consider a solute transported by convection from node 1. $$\frac{\partial C}{\partial t} = - v\frac{\partial C}{\partial x}$$
According to Gauss divergence theorem,
"The flux out of each region is equal to the sum of the flux through its two faces"
Therefore, would it be right to interpret $$\frac{\partial C}{\partial t}_\text{at point 2} = - (1/delx)[v_{12}C_{12}- v_{24}C_{24} - v_{23}C_{23}] \hspace{0.5cm} (2)$$
where $v$ and $c$ are the velocity and average concentration in the respective pipe section, respectively.
I am confused because, if the fluid is incompressible $$\nabla .v = 0; $$
This implies incoming flow is equal to the outgoing flow.
The continuity equation is in terms of flow and in the solute transport equation the products, $vC$, give flux in and flux out (in terms of moles/area/s).
In some references that I looked up, the fluxes in the solute transport equation are written in terms of moles/s. So, I am confused whether flux should be expressed in unit area i.e moles/area/time or moles/time while modeling flows through varying volumes.
Should the transport equation itself be expressed in terms of moles/time instead of moles/time/volume?
EDIT:
The condition for continuity is the following, $ A_{12}v_{12}C_{12} = A_{24}v_{24}C_{24} + A_{23}v_{23}C_{23} $
Could someone explain how the above condition can be incorporated in the discretized version of the solute transport equation?
Multiplying equation (2) by area changes the units of RHS to moles/length/time, whereas the units of LHS is in moles/volume/time. I am not sure how this has to be handled so that the units are the same on both sides.
Also, I couldn't completely understand the implication of Gauss Divergence theorem.In the sense, from what I understand the gauss-divergence talks about flux (per unit moles/area/time) that is entering and leaving a control volume element. However, here we are looking at the flux ( in terms of moles/time) entering and leaving the control volume (node 2).