Rim-Lamella Model Dynamics

• The Rim-Lamella Model is a framework where a radially spreading thin liquid sheet (lamella) supplies mass to an outer, thicker rim, influencing its stability and fragmentation.
• Key dynamics are driven by the balance of inertia and surface tension, with the lamella’s flux determining rim thickness, ligament formation, and the regime of breakup.
• The model applies to droplet impacts and laser-driven experiments, and extends to variational and morphological analyses in systems like block copolymer bicelles.

Searching arXiv for the cited Rim-Lamella papers to ground the article in current preprints. The Rim-Lamella Model denotes, in its most explicit current usage, a class of descriptions for systems in which a thin expanding lamella is bounded by a thicker rim, and the rim thickness, stability, ligament formation, and breakup are governed by its coupling to the sheet that feeds it. In fluid mechanics this framework is used for droplet impact, laser-driven metal-sheet expansion, and related splashing problems; its recent extension isolates the rim from the lamella by severing the liquid connection and thereby distinguishes fluid-fed rim dynamics from the capillary breakup of a freely expanding toroidal rim [(Kharbedia et al., 2 Aug 2025); (Amirfazli et al., 2023); (Juarez et al., 2011)].

1. Definition and conceptual structure

In the fluid-mechanical setting, the model resolves a two-part morphology. A thin lamella spreads radially, while liquid accumulates at its edge to form a thicker outer rim. The lamella provides mass influx, and that influx is not ancillary: it helps set the rim thickness, controls ligament growth, and affects fragmentation. In the laser-driven tin experiments, a laser pulse turns a tin microdroplet into a radially expanding thin sheet with a thick, unstable bounding rim; a second, weak laser pulse then vaporizes the thin neck connecting the rim to the sheet, producing a freely expanding toroidal rim with no further liquid supply from the sheet (Kharbedia et al., 2 Aug 2025).

A related but distinct formulation appears in the inviscid droplet-impact literature. There, stage two of impact is treated as an axisymmetric structure consisting of a thin lamella and a thicker rim at the lamella edge. Roisman et al.’s model, as analyzed rigorously in later work, represents the lamella by depth-averaged velocity and thickness fields and the rim by its volume, radial position, and velocity (Amirfazli et al., 2023).

This usage is not limited to one forcing scenario. In drop impact on geometric targets, the same physical decomposition appears: impact redirects momentum into a thin, expanding lamella, fluid accumulates at the sheet edge to form a thick outer rim, and the subsequent splash pattern is selected by competition between imposed azimuthal perturbations and the rim’s natural Plateau–Rayleigh-type instability (Juarez et al., 2011). This suggests that “Rim-Lamella Model” is best understood not as a single universal equation set, but as a family of coupled rim-sheet descriptions whose details depend on forcing, geometry, and the role of influx.

2. Fluid-fed rim dynamics and thickness selection

A central principle of the connected model is that the attached rim is continuously fed by the lamella. In the severance study, prior work on attached rims is summarized by the local instantaneous Bond number condition

$${\rm Bo}=\frac{\rho(-\ddot r_s)b^2}{\sigma}\approx 1,$$

which implies a dynamically selected rim thickness

$b \sim \sqrt{\frac{\sigma}{\rho(-\ddot r_s)}}.$

Within this picture, rim thickness is not arbitrary; it is selected by the balance of inertia and surface tension while the rim remains connected to the sheet (Kharbedia et al., 2 Aug 2025).

For the laser-driven tin case, the sheet expansion law is taken from Wang & Bourouiba and adapted as

$$2R_s \equiv 2\frac{r_s}{d_0} = {\rm We}_d^{1/2}\left[b_3(T-T_m)^3+b_2(T-T_m)^2+b_0\right],\qquad T=\frac{t}{\tau_c}.$$

The measured coefficients are approximately

$$T_m \approx 0.38,\qquad b_0 \approx 0.14,\qquad b_2 \approx 0.58,\qquad b_3 \approx 0.43.$$

In the authors’ interpretation, $b_0$ encodes the maximum normalized radius, $b_2$ the rim deceleration, and $b_3$ the initial expansion rate (Kharbedia et al., 2 Aug 2025).

In the inviscid axisymmetric impact model, the lamella and rim are coupled through mass and momentum balances. The lamella obeys

$$u_t + u\,u_r = 0,\qquad (h)_t + \frac{1}{r}(urh)_r = 0,$$

with

$u(r,t)=\frac{r}{t+t_0}.$

The rim satisfies

$\frac{dV}{dt}=2\pi R\,(u_0-U)\,h(R,t),$

$b \sim \sqrt{\frac{\sigma}{\rho(-\ddot r_s)}}.$0

$b \sim \sqrt{\frac{\sigma}{\rho(-\ddot r_s)}}.$1

A rigorous analysis of these ODEs derives upper and lower bounds for the maximum spreading radius, both with $b \sim \sqrt{\frac{\sigma}{\rho(-\ddot r_s)}}.$2 scaling, and proves that once a rim forms, its height will invariably exceed that of the lamella (Amirfazli et al., 2023).

3. Severance, ballistic toroidal rims, and rim re-formation

The severance experiment isolates the rim from the lamella by removing the fluid reservoir. After detachment, the rim follows ballistic motion with essentially the same radial velocity it had at the instant of severance:

$b \sim \sqrt{\frac{\sigma}{\rho(-\ddot r_s)}}.$3

The measured rim trajectories are linear in time, and the vaporization pulse does not measurably propel the rim (Kharbedia et al., 2 Aug 2025).

Once severed, several changes occur simultaneously. The rim can no longer receive liquid from the sheet; inherited corrugations do not get replenished; the rim thins further because it continues expanding; and breakup occurs faster than for a smooth, fed rim. The severed rim breaks in two main places: at the base of inherited ligaments and at the junctions between neighboring corrugations. These breakup times are not strongly separated, because the local curvatures are similar (Kharbedia et al., 2 Aug 2025).

Regime	Liquid supply	Reported behavior
Attached rim	Fed by the sheet	Thickness self-adjusts; corrugations select ligaments
Severed rim	No further liquid supply	Ballistic expansion; ligaments stop growing and pinch off
Re-forming rim	Continued outward flow in the sheet	A new rim reforms on the remaining sheet

The characteristic breakup scale after severance is related to a Rayleigh-Plateau capillary time for a cylinder of diameter $b \sim \sqrt{\frac{\sigma}{\rho(-\ddot r_s)}}.$4,

$b \sim \sqrt{\frac{\sigma}{\rho(-\ddot r_s)}}.$5

The measured first breakup events occur earlier than the full-fragmentation prediction because the rim is already corrugated before severance, those thickness variations seed earlier local pinch-off, and expansion after severance further thins the rim (Kharbedia et al., 2 Aug 2025).

The remaining sheet does not stay rimless. A new rim reforms due to continued outward flow in the sheet. The proposed re-formation timescale is

$b \sim \sqrt{\frac{\sigma}{\rho(-\ddot r_s)}}.$6

with filling time

$b \sim \sqrt{\frac{\sigma}{\rho(-\ddot r_s)}}.$7

The reported timescale is about $b \sim \sqrt{\frac{\sigma}{\rho(-\ddot r_s)}}.$8 early in the expansion and a few $b \sim \sqrt{\frac{\sigma}{\rho(-\ddot r_s)}}.$9 later in the expansion (Kharbedia et al., 2 Aug 2025).

4. Corrugation selection, ligaments, and fragment counts

For severed expanding rims, the corrugation number is taken from prior theory on unsteady expanding rims as

$$2R_s \equiv 2\frac{r_s}{d_0} = {\rm We}_d^{1/2}\left[b_3(T-T_m)^3+b_2(T-T_m)^2+b_0\right],\qquad T=\frac{t}{\tau_c}.$$0

with $$2R_s \equiv 2\frac{r_s}{d_0} = {\rm We}_d^{1/2}\left[b_3(T-T_m)^3+b_2(T-T_m)^2+b_0\right],\qquad T=\frac{t}{\tau_c}.$$1. The ligament count is predicted by

$$2R_s \equiv 2\frac{r_s}{d_0} = {\rm We}_d^{1/2}\left[b_3(T-T_m)^3+b_2(T-T_m)^2+b_0\right],\qquad T=\frac{t}{\tau_c}.$$2

The severance experiments find that the number of ligaments is largely inherited from the pre-severance sheet evolution, and the reported scalings are

$$2R_s \equiv 2\frac{r_s}{d_0} = {\rm We}_d^{1/2}\left[b_3(T-T_m)^3+b_2(T-T_m)^2+b_0\right],\qquad T=\frac{t}{\tau_c}.$$3

with final fragment number $$2R_s \equiv 2\frac{r_s}{d_0} = {\rm We}_d^{1/2}\left[b_3(T-T_m)^3+b_2(T-T_m)^2+b_0\right],\qquad T=\frac{t}{\tau_c}.$$4 essentially identified with the inherited corrugation number,

$$2R_s \equiv 2\frac{r_s}{d_0} = {\rm We}_d^{1/2}\left[b_3(T-T_m)^3+b_2(T-T_m)^2+b_0\right],\qquad T=\frac{t}{\tau_c}.$$5

The rationale is that after severance there is little time for corrugations to merge or be scavenged (Kharbedia et al., 2 Aug 2025).

In geometric-target drop impact, the same instability-selection problem is posed differently. The target supplies a finite-amplitude azimuthal perturbation

$$2R_s \equiv 2\frac{r_s}{d_0} = {\rm We}_d^{1/2}\left[b_3(T-T_m)^3+b_2(T-T_m)^2+b_0\right],\qquad T=\frac{t}{\tau_c}.$$6

while the rim’s intrinsic Plateau–Rayleigh mode is estimated by

$$2R_s \equiv 2\frac{r_s}{d_0} = {\rm We}_d^{1/2}\left[b_3(T-T_m)^3+b_2(T-T_m)^2+b_0\right],\qquad T=\frac{t}{\tau_c}.$$7

If

$$2R_s \equiv 2\frac{r_s}{d_0} = {\rm We}_d^{1/2}\left[b_3(T-T_m)^3+b_2(T-T_m)^2+b_0\right],\qquad T=\frac{t}{\tau_c}.$$8

the imposed geometry dominates and the splash is regular; if

$$2R_s \equiv 2\frac{r_s}{d_0} = {\rm We}_d^{1/2}\left[b_3(T-T_m)^3+b_2(T-T_m)^2+b_0\right],\qquad T=\frac{t}{\tau_c}.$$9

the rim is controlled by the intrinsic instability and the splash becomes irregular. For $$T_m \approx 0.38,\qquad b_0 \approx 0.14,\qquad b_2 \approx 0.58,\qquad b_3 \approx 0.43.$$0, breakup into exactly $$T_m \approx 0.38,\qquad b_0 \approx 0.14,\qquad b_2 \approx 0.58,\qquad b_3 \approx 0.43.$$1 filaments is reported; for the cylinder and for polygons with $$T_m \approx 0.38,\qquad b_0 \approx 0.14,\qquad b_2 \approx 0.58,\qquad b_3 \approx 0.43.$$2, the number and placement of filaments become independent of target geometry (Juarez et al., 2011).

Direct numerical simulations of colliding liquid rims further emphasize that ligaments can originate from the initial corrugated geometry of the perturbed rim surface rather than from a purely late-time linear instability. In that setting the lamella bulk follows a quasi-inertial similarity solution $$T_m \approx 0.38,\qquad b_0 \approx 0.14,\qquad b_2 \approx 0.58,\qquad b_3 \approx 0.43.$$3, while the rim is capillary-decelerated and the ligament population evolves through merging. The proposed decay law is

$$T_m \approx 0.38,\qquad b_0 \approx 0.14,\qquad b_2 \approx 0.58,\qquad b_3 \approx 0.43.$$4

which is then connected to fragment statistics (Tang et al., 2023). Taken together, these results delimit several selection mechanisms: inherited corrugations, imposed azimuthal forcing, capillary deceleration, and geometric merging.

5. Mathematical extensions and related rim-lamella morphologies

Outside free-surface impact, related rim-lamella pictures appear as morphological or variational analogues rather than as the same hydrodynamic model. In the anisotropic Ohta–Kawasaki problem on the flat torus, the object of study is a horizontal lamella

$$T_m \approx 0.38,\qquad b_0 \approx 0.14,\qquad b_2 \approx 0.58,\qquad b_3 \approx 0.43.$$5

minimizing

$$T_m \approx 0.38,\qquad b_0 \approx 0.14,\qquad b_2 \approx 0.58,\qquad b_3 \approx 0.43.$$6

For uniformly elliptic anisotropy, strict stability yields strict $$T_m \approx 0.38,\qquad b_0 \approx 0.14,\qquad b_2 \approx 0.58,\qquad b_3 \approx 0.43.$$7-local minimality with a quadratic lower bound; for horizontally flat anisotropy, the lamella is an isolated $$T_m \approx 0.38,\qquad b_0 \approx 0.14,\qquad b_2 \approx 0.58,\qquad b_3 \approx 0.43.$$8-local minimizer for every $$T_m \approx 0.38,\qquad b_0 \approx 0.14,\qquad b_2 \approx 0.58,\qquad b_3 \approx 0.43.$$9; and in the planar case the horizontal lamella is the unique global minimizer for sufficiently small $b_0$0 under the stated anisotropic condition (Fiorini, 15 Apr 2026). This is a variational stability theory for lamellar morphology, not a sheet-and-rim flow model, but it is explicitly motivated by lamellar or rim-lamella-type morphologies.

A more literal structural rim-lamella picture is reported for disk-shaped bicelles in block copolymer/homopolymer blends. Self-consistent field theory finds a flat central region that is lamella-like and a thicker rim enriched in micelle-forming diblocks. The segregation is not perfect: the concentration of micelle formers is of the order of $b_0$1 on the flat central surface of the bicelle. A finite disk radius is stable only when both micelle-forming and lamella-forming diblocks are present, because then the free-energy density has a minimum as a function of radius; with only lamella former present, the free-energy density decays monotonically, indicating instability with respect to further aggregation (Greenall, 2016).

These related usages support a broader morphological interpretation. A plausible implication is that “rim-lamella” can denote a composite architecture in which a relatively flat lamellar interior coexists with a distinct boundary region whose curvature, composition, or stability properties differ from those of the interior. The governing mechanisms, however, remain domain-specific.

6. Interpretation, scope, and common misconceptions

A recurring misconception is to treat the rim as merely the geometric edge of the lamella. The severance study rejects that simplification: the sheet is the active fluid reservoir that stabilizes and tunes the rim. While attached, the rim thickness self-adjusts and ligament growth is sustained; after severance, thickness can no longer remain slaved to the sheet-fed balance, ligaments stop growing, and fragmentation is accelerated at inherited weak points (Kharbedia et al., 2 Aug 2025).

A second misconception is that rim breakup is governed by a single universal instability. The available formulations instead show multiple regimes. In geometric-target impacts, breakup can be locked to an imposed azimuthal perturbation or governed by the fastest-growing Plateau–Rayleigh mode depending on $b_0$2 (Juarez et al., 2011). In severed tin rims, the final fragment number is largely inherited from pre-severance corrugations (Kharbedia et al., 2 Aug 2025). In colliding liquid rims, ligament formation is tied to a corrugated initial surface and later merging dynamics rather than to a single wavelength-selection argument (Tang et al., 2023).

A third misconception is that the standard rim-lamella ODE model is purely heuristic. In the inviscid spreading problem, later analysis proves exact mass conservation for the chosen thickness profile, derives upper and lower bounds with $b_0$3 scaling for the maximum spreading radius, and demonstrates self-consistency of the decomposition by showing that once a rim forms, its height will always exceed that of the lamella (Amirfazli et al., 2023). At the same time, that analysis also specifies the regime of validity: the model agrees closely with experiments and DNS approximating inviscid or free-slip conditions, whereas more viscous experiments are overpredicted, as expected.

The model’s present significance lies in this separation of roles. The lamella supplies unsteady influx and sets the conditions for rim self-adjustment; the rim concentrates mass, capillary deceleration, corrugation growth, ligament formation, and fragment emission. Recent severance experiments sharpen that distinction by showing that once the supply is removed, the rim behaves as a freely expanding toroidal structure with little opportunity for reorganization, while the remaining sheet rapidly regenerates a new rim and resumes the instability cycle (Kharbedia et al., 2 Aug 2025).