From 91485b606969d2f8f63695118a111c970ba4069a Mon Sep 17 00:00:00 2001 From: Walter Dal'Maz Silva Date: Mon, 15 Apr 2024 10:48:06 +0200 Subject: [PATCH] Added note --- .../Analysis of Transport Phenomena.md | 23 +++++++++++++++++++ 1 file changed, 23 insertions(+) diff --git a/docs/src/Teaching/Transport Phenomena/Analysis of Transport Phenomena.md b/docs/src/Teaching/Transport Phenomena/Analysis of Transport Phenomena.md index 1d503b4bc..d079a6c96 100644 --- a/docs/src/Teaching/Transport Phenomena/Analysis of Transport Phenomena.md +++ b/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