= Constant-flux tracer inlet solution
{title2=$UC-\mathcal D C_x=j_0\text{ at }x=0$}
For a <passive scalar> on a half-line, impose constant total solute flux at its inlet, with initially zero <concentration> and constant positive <advection> speed $U$ and <mass diffusivity> $\mathcal D$. A <Laplace transform> in time gives
$$
\widetilde C(x,p)=\frac{2j_0}{p(U+\sqrt{U^2+4\mathcal D p})}\exp\left[\frac{U-\sqrt{U^2+4\mathcal D p}}{2\mathcal D}x\right].
$$
With $\xi=Ux/\mathcal D$, $\vartheta=U^2t/\mathcal D$ and $z_\pm=(\xi\pm\vartheta)/(2\sqrt\vartheta)$, its inverse is
$$
\frac{C(x,t)}{j_0/U}=\frac12\operatorname{erfc}(z_-)+\sqrt{\vartheta/\pi}e^{-z_-^2}-\frac12(1+\xi+\vartheta)e^\xi\operatorname{erfc}(z_+).
$$
The <complementary error function> describes the smeared front. This is a flux boundary condition, not the <constant-concentration inlet solution>. At large axial <Péclet number>, both have the leading advancing front $\tfrac12\operatorname{erfc}(z_-)$.
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