Quadratic hindered-settling flux ($j(\varphi )=-u_s\varphi (1-\varphi )$)

For downward settling speed $u_s(1-\varphi )$ in upward coordinate $z$, particle flux is $j=-u_s\varphi (1-\varphi )$. The concentration-characteristic speed is $j^{\prime }=u_s(2\varphi -1)$ and differs from particle speed. Since $j^{\prime \prime }=2u_s>0$, decreasing concentration with height creates a compressive shock, while increasing concentration creates a rarefaction wave.

If lower initial concentration $b$ is less than upper concentration $a$, the entropy solution expands into a fan with $\varphi =[1+(z-z_m)/(u_{s}t)]/2$. Its characteristic speeds range from $u_s(2b-1)$ to $u_s(2a-1)$. The fan can be clipped by a clearing or deposition shock.

For upper concentration $a$, lower $a/2$ and interface height $4H/5$, both curved fronts meet inside the fan only if $a\ge 1-\sqrt{2/3}$. At smaller $a$, the deposition front reaches the fan upper edge first, then advances through uniform concentration $a$. The clearing front remains straight and final time is $H(1-3a/5)/[u_s(1-a)]$.

For upper concentration $a$, lower concentration $b<a$ and $L=H-z_m$, define $A=aL$ and $D=(1-b)z_m$. Once inside the settling rarefaction fan, clearing and deposition shocks obey $z_u=z_m-u_{s}t+2\sqrt{u_{s}At}$ and $z_d=z_m+u_{s}t-2\sqrt{u_{s}Dt}$. These curves apply only while their neighboring concentration remains within the fan.

If both shocks meet inside a settling rarefaction fan, $t_c=(\sqrt{A}+\sqrt{D}{)}^2/u_s$ and the last characteristic has concentration ${\varphi }_{\ast }=\sqrt{A}/(\sqrt{A}+\sqrt{D})$. It runs from the initial internal interface to the final deposit point. This construction requires $b\le {\varphi }_{\ast }\le a$; otherwise a front exits the fan before final deposition.

The Rankine-Hugoniot condition for the quadratic hindered-settling flux gives $V=u_s({\varphi }_L+{\varphi }_R-1)$, with lower and upper states labelled $L,R$. It includes a downward clearing front (${\varphi }_R=0$), an upward deposition front (${\varphi }_L=1$), and a stationary final deposit/clear-fluid interface.