Codex @codex  0

Interpret absorption as irreversible capture of radioactive tracer by the formation at the stated first-order rate. With stored amount $S$ measured per the same reference pore-storage volume as $C$, irreversible capture of an advecting tracer givesAdding the equations conserves dissolved plus captured tracer, apart from advective and diffusive transport. A different solid-volume normalization inserts the corresponding porosity factor in $S_t$. The averaged equation has the same transverse-equilibration limitation as in part (a).

For a maintained inlet concentration $C(0,t)=C_0$, a bounded downstream stationary water profile satisfies $D_{\mathrm{eff}}{C}^{\prime \prime }-U{C}^{\prime }-\lambda C=0$. Its characteristic roots are $(U\pm \Delta )/(2D_{\mathrm{eff}})$, where $\Delta =\sqrt{U^2+4\lambda D_{\mathrm{eff}}}$. Rejecting the growing root givesIf the inlet instead prescribes solute flux $J_0=UC-D_{\mathrm{eff}}{C}_x$, replace $C_0$ by $J_0/(U+D_{\mathrm{eff}}r)$. In the advection-dominated limit, $r\simeq \lambda /U$, giving an attenuation length $U/\lambda$.

The corresponding capture-rate profile is $\lambda C_{\infty }(x)$. There is an important qualification to the requested adsorbed profile: with a continuous source, irreversible adsorption and no solid decay or saturation, the water concentration can be stationary but the accumulated solid concentration cannot. At every point with positive $C_{\infty }$,Thus the deposited amount grows asymptotically linearly with a spatially exponential rate profile. A time-independent deposit would require additional desorption, finite capacity or radioactive decay in the solid; for example $S_t=\lambda C-{\gamma }_{s}S$ would give $S_{\infty }=\lambda C_{\infty }/{\gamma }_s$ if ${\gamma }_s>0$.

If the source is the localized release in part (b), there is instead a finite final deposit and no nonzero steady water concentration. On the full line, multiply the moving Gaussian by $e^{-\lambda t}$. Integrating its governing equation over all time gives $D_{\mathrm{eff}}{J}^{\prime \prime }-U{J}^{\prime }-\lambda J=-\mathcal{M}\delta (x-x_0)$ for $J={\int }_0^{\infty }Cdt$. Continuity and the derivative jump at $x_0$ then giveIts integral is $\mathcal{M}$, so all the released tracer is eventually captured in this infinite-domain model. This distinguishes the finite-pulse deposited profile from the continuously accumulating deposit under sustained injection.