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Place equal numbers of radii $r_1,r_2$ on alternating sites of a square nearest-neighbour lattice of spacing $d$. Nonoverlap requires $d\ge \max (r_1+r_2,\sqrt{2}{r}_1,\sqrt{2}{r}_2)$. A two-circle cell has area $2d^2$, giving the displayed packing fraction. Its maximum within this geometry occurs at large-to-small radius ratio $1+\sqrt{2}$ and is $\pi (1-1/\sqrt{2})$; smaller interstitial circles leave unused space, while larger ones force the large-circle lattice apart.

A particle suspension contains two particle species with separate particle volume fractions. In an independent-settling model, each species obeys its own flux conservation equation, while a shared stationary deposit can couple the jump conditions through its maximum total packing fraction. Such independence is an imposed model; general hindered settling need not decouple species.

Arrangement of circles in a plane without interior overlap. The covered area fraction is its packing fraction. Equal-radius square packing has fraction $\pi /4$.

Placement of geometric objects without interior overlap. A packing fraction measures how much area or volume they occupy. Circle packing and sphere packing are examples.

A square circle packing of equal radii has one circle per cell of side $2r_1$. Its packing fraction is thereforeFor the alternating-radius square circle packing shown in the source figure, let $d$ be the nearest unlike-centre separation. A two-circle repeat cell has area $2d^2$ and solid area $\pi (r_1^2+r_2^2)$. Unlike circles require $d\ge r_1+r_2$; like circles on the two sublattices require $\sqrt{2}d\ge 2r_1$ and $\sqrt{2}d\ge 2r_2$. ThusAll three nonoverlap constraints matter when varying the radius ratio.

Take $r_1\ge r_2$ as in the illustrated arrangement and set $s=r_2/r_1$. If $0\le s\le \sqrt{2}-1$, the large circles determine the spacing and $\eta =(\pi /4)(1+s^2)$ increases with $s$. If $\sqrt{2}-1\le s\le 1$, unlike contacts determine it and $\eta =(\pi /2)(1+s^2)/(1+s{)}^2$, whose derivative is proportional to $s-1$ and is nonpositive. The maximum is therefore at the joining point:The large circles then touch their large neighbours, while each small circle exactly fills a square interstice. Interchanging the species gives the reciprocal radius ratio. This is the highest packing fraction within the specified alternating square arrangement; it is not a claim about every possible unequal-circle packing.

Use downward distance $y=H-z$, with $z$ the height above the bottom, and write $W=W_s(a)>0$. This convention will also reproduce the printed negative deposit-front velocity in part (d). The ideal simple-cubic sphere packing has packing fraction $\pi /6$; leaving ${\varphi }_{\max }$ symbolic keeps the jump calculation independent of its value. In suspension, the kinematic sedimentation equation isIntegrate this conservation law across a moving discontinuity to obtain the Rankine-Hugoniot conditionThe brackets denote values on the increasing-$y$ side minus those on the other side. A sedimentation shock is a concentration jump moving at this secant slope of the particle flux. Deposited material is stationary and has zero flux.

For the upper clearing front, the states are $0$ and ${\varphi }_{\max }/8$, giving $U_1=7W/8$ downward. For the lower deposition front, the states are ${\varphi }_{\max }/8$ and ${\varphi }_{\max }$, giving $U_2=-W/8$, upward. ThusIn terms of height above the bottom, the shock paths areThey meet when $H=Wt$. Therefore complete settling occurs atThe final height also follows directly from particle volume fraction conservation, ${\varphi }_{\max }{h}_c=({\varphi }_{\max }/8)H$. The stationary deposit after the meeting carries no particle flux.

The diagram plots time vertically against height, as requested. Its shock slopes have the opposite spatial sign to $U_1,U_2$, because those velocities were defined in the downward coordinate.

Equal spheres of radius $r$ occupy the sites of a cubic lattice of spacing $2r$. A cell contains one sphere, so its packing fraction is $(4\pi r^3/3)/(2r{)}^3=\pi /6$. This is an ideal arranged deposit rather than a claim about all settling suspensions.

Arrangement of spheres without interior overlap. The occupied volume fraction is its packing fraction; it depends on the spatial arrangement and on the radii.