= Bidisperse gravity-current deposition
{title2=$f_1(x)$}
In a <bidisperse particle suspension> transported by a <steady depositing gravity current>, species with concentrations $c_i$ and downward <settling velocity> magnitudes $v_i$ satisfy $c_i(x)=c_{i0}e^{-v_ix/q}$. Their local <particle deposition fluxes> are $v_ic_i$. The local deposited particle-volume fraction belonging to species one is
$$
f_1(x)=\frac{v_1c_{10}e^{-v_1x/q}}{v_1c_{10}e^{-v_1x/q}+v_2c_{20}e^{-v_2x/q}}.
$$
For $v_1>v_2>0$ and both species present, $f_1$ decreases strictly downstream, since $f_1'=(v_2-v_1)f_1(1-f_1)/q$. If $g'_i=g_{0i}c_i$, use $c_i=g'_i/g_{0i}$ rather than buoyancy ratios unless $g_{01}=g_{02}$. A number fraction additionally divides each species volume flux by its single-particle volume. Integrating deposition over $[0,x]$ instead gives cumulative weights $q c_{i0}(1-e^{-v_ix/q})$.
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