Thin-film mass flux

The thin-film mass flux is the mass transported along a thin layer per unit width and per unit time. For constant density $\rho$, thickness $h$ and depth-averaged streamwise velocity $\overline{u}$,The quantity $q=h\overline{u}$ is instead the volume flux per unit width. Their units differ by density: $j_m$ has units $\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$, whereas $q$ has units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$.

For a constant-density incompressible film with no sources, mass conservation becomes $h_t+q_x=0$. In a steady region, $q$ is constant; in a profile moving at speed $c$, the conserved relative flux is $q-ch$. Thus a slowly varying or moving transition does not automatically have constant laboratory-frame flux. Keeping the density factor and the reference frame explicit prevents confusing a geometric thickness balance with the transported mass.