Added note

docs/src/Teaching/Transport Phenomena/Analysis of Transport Phenomena.md

@@ -40,6 +40,29 @@ $$
 \frac{\partial{}b}{\partial{}t} + \nabla\cdotp{\vec{F}} - B_{v} = 0
 $$
 
+So far we have a PDE describing the conservation *inside* the RVE of the system being described. To be able to solve such a system we also need to provide a description of its interactions with its surroundings through the specification of *boundary conditions*.
+
+Assume a volume of thickness $l$ enclosing a boundary $S$ splitting domains $V_{1}$ and $V_{2}$, where discontinuities in properties and all quantities describing the system may be present. We can write the continuity equation for this *thick* boundary as
+
+$$
+\frac{d}{dt}\left(\int_{V}bdV\right) = -\int_{\Omega}\vec{n}\cdotp\vec{F}dA+\int_{V}B_{v}dV+\int_{\Omega}B_{s}dA
+$$
+
+where the last term has been added to describe the net surface production rate of $b$. Collapsing the volume over the boundary $S$ by taking the limit where $l\to{}0$, it should be evident that all volume integrals in the above vanish. The integral form of boundary condition then simplifies to
+
+$$
+\int_{\Omega}\vec{n}\cdotp\vec{F}dA=\int_{\Omega}B_{s}dA
+$$
+
+Defining as positive the normal of $S$ pointing outwards $V_{1}$ the integrant on right-hand side can be written as $\vec{n}\left(\vec{F}_{2}-\vec{F}_{1}\right)$. Applying mean value theorem as if all terms where in the same side of the equation (that vanishes overall) leads to the differential form of boundary condition
+
+$$
+\vec{n}\left(\vec{F}_{2}-\vec{F}_{1}\right)=B_{s}
+$$
+
+This expression has a very straightforward interpretation. If there is no creation rate $B_{s}$ at the interface, flux is continuous across interface; otherwise some arbitrary form of discontinuity should arise, whose form would depend on the volume governing equations at each side of $S$.
+
+
 ---
 ## Mass and energy diffusion equations