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

how the choice of boundary condition fundamentally dictates the long-term equilibrium and the resulting flow (flux) across the boundary of a material-L.mp4

🗒️how the Robin boundary condition naturally arises in physical problems involving both diffusion and convection when the flux across a boundary must be zero

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