A solute is carried down a straight circular tube of radius a=0.001 m by a fully…

A solute is carried down a straight circular tube of radius $a=0.001\,\mathrm{m}$ by a fully developed Poiseuille flow of cross-sectional mean speed $U=0.01\,\mathrm{m}/\mathrm{s}$. The molecular diffusivity is $D=1\times 10^{-6}\,\mathrm{m}^2/\mathrm{s}$. After a long time the cross-sectionally averaged concentration obeys a one-dimensional advection–diffusion equation with the Taylor–Aris dispersivity \[ D_{\mathrm{eff}}=D+\frac{U^2 a^2}{48 D}. \] 1. Prove $D_{\mathrm{eff}}=3.083333333\times 10^{-6}\,\mathrm{m}^2/\mathrm{s}$. The radial diffusion time is $a^2/D$. Prove that this time is $1\,\mathrm{s}$ and that the observation instant $t=100\,\mathrm{s}$ is therefore a long-time station. 2. An areal pulse of strength $\Gamma=1$ (concentration times length) is released at $z=0$ when $t=0$. Prove that the mean field is \[ \bar c(z,t) =\frac{\Gamma}{\sqrt{4\pi D_{\mathrm{eff}} t}} \exp\Bigl(-\frac{(z-Ut)^2}{4 D_{\mathrm{eff}} t}\Bigr). \] At $t=100\,\mathrm{s}$ prove that the centroid is at $z=1\,\mathrm{m}$, that the axial variance is $2 D_{\mathrm{eff}} t=6.166666667\times 10^{-4}\,\mathrm{m}^2$, that $\sigma=0.024832774\,\mathrm{m}$, and that the peak value is $16.06515163$ (same units as $\bar c$). 3. Audit both sentences: (i) “axial spreading uses the molecular $D$ only, so $D_{\mathrm{eff}}=D$ and the variance is $2Dt$”; (ii) “the Gaussian variance is $D_{\mathrm{eff}} t$ rather than $2 D_{\mathrm{eff}} t$.” Also refuse to treat the mean field as a bare Continuum/PDE Laplacian that ignores the Taylor shear contribution. Do not replace the long-time pulse by a Graetz developing-thermal scaling or by a ternary Maxwell–Stefan inversion.

$U^2 a^2=10^{-10}$ and $48 D=4.8\times 10^{-5}$, so the shear piece is $2.083333333\times 10^{-6}$ and $D_{\mathrm{eff}}=3.083333333\times 10^{-6}$.