how the choice of boundary condition fundamentally dictates the long-term equilibrium and the resulting flow (flux) across the boundary of a material

The comparison between the two boundary conditions for the convection-diffusion system reveals how a boundary constraint fundamentally dictates the flow state. The Robin boundary condition (zero total flux, $-D \frac{d u}{d x}+v_0 u=0$ ) forces the concentration $u(L)$ to self-adjust to a non-zero value, establishing a dynamic equilibrium where the outward convective flow is precisely balanced by the inward diffusive flow, resulting in zero net material transfer. Conversely, the Dirichlet boundary condition (fixed concentration, $u(L)=0$ ) acts as a strong external sink, creating an unbalanced system where the concentration gradient is steepened dramatically. This results in a large, non-zero outward flux (dominated by diffusion), as material flows freely out of the domain to perpetually maintain the imposed low concentration at the edge.

🎬Narrated Video

https://youtu.be/GR4SV7exgr8

🏗️Structural clarification of Condensed Notes

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