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Gaetano Napoli

Publications and source records attributed to Gaetano Napoli.

At least 19 recordsLinked to original sources
arXiv2026

Nonholonomic collective flows: velocity--orientation locking in a continuum with microstructure

We develop a continuum theory for a fluid of elongated particles that advance along their own axes. The same kinematics is ideally shared by flocks of sheep, self-propelled rods, vehicular traffic, and turning flocks of birds. Within the framework of continua with vectorial microstructure, in which each point also carries an orientation, we impose the no-side-slip (skate) condition $\vv=u\,\nn$ as an ideal, non-integrable internal constraint. We derive the pure equations of motion and the equation governing the constraint reaction from the principle of virtual power. Using a constitutive closure provided by the Ericksen-Leslie theory of nematic liquid crystals, we show how the constraint also shapes collective effects. It turns parabolic orientational diffusion into hyperbolic orientation-density waves and forbids any steady simple shear. It fixes the empirical Toner-Tu convective coefficient to the flow-alignment ratio measured in colloidal rollers, thereby providing a mechanical foundation for angular sound in micropolar active hydrodynamics. It also recovers the inertial spin model of bird flocks, with the addition of a banking force and a turn-density coupling. Finally, it recovers the classical macroscopic models of one-dimensional traffic flow, while describing steering, lateral tyre forces, and road geometry in two dimensions.

cond-mat.soft↗
arXiv2026

Boundary-layer analysis of the partial engulfment of a small particle by a lipid membrane

We study the axisymmetric partial engulfment of a small rigid sphere by a fluid Helfrich membrane. When the size ratio $\eps$ between the particle and the membrane is small, the neck that joins the wrapped cap to the surrounding membrane is an elastic boundary layer, and we analyse it by matched asymptotic expansions. The leading inner surface is a catenoid, a minimal surface that stores no Helfrich energy, so that the energy of partial engulfment is carried by the first correction and appears only at order $\eps^{2}\ln(1/\eps)$. We obtain it by solving the inhomogeneous Jacobi equation of the catenoid, forced by the spontaneous curvature and by the ambient mean curvature that the neck must match, and we give its coefficient in closed form. The boundary layer can then be integrated out, and the neck replaced by a scalar self-energy carried at the pole, so that the outer field can be closed independently of the inner one. For a membrane coupled to a tension reservoir we derive the binding threshold, which turns out to be independent of both the tension and the spontaneous curvature, the complete-wrapping threshold, and the hysteresis of the envelopment transition. The neck self-energy law is confirmed against the full nonlinear shape equations.

cond-mat.soft↗
arXiv2026

Odd elasticity in a three-link microswimmer: feedback equivalence, global controllability, and the cost of non-reciprocity

We study a Purcell three-link microswimmer whose joints are \emph{odd-elastic}: the torsional stiffness is non-Hermitian, its antisymmetric part $k_o$ injecting mechanical work so that the internal elastic drift is non-conservative. Our main finding is a sharp separation between geometry and cost --- odd elasticity is invisible to the control geometry of the swimmer and visible only in the energy of a manoeuvre. The mechanism is a single algebraic fact: the drift lies in the span of the two control vector fields, so the system is feedback-equivalent to a driftless one and the odd modulus enters the bracket structure only through the scalar $\det\textbf{K}=k^2+k_o^2$. From this we deduce that the abnormal extremals of the energy problem are unchanged by $k_o$, that the swimmer is globally controllable for every non-reciprocity with no threshold, and that its nilpotent model is the Cartan $(2,3,5)$ sub-Riemannian structure, deformed only by the metric scaling $g_\chi=(1+\chi^2)g_0$. The odd modulus acts solely on the cost: casting the optimal-steering problem in sub-Finsler (Randers) form, we prove that a prescribed reorientation is strictly cheaper for either sign of $k_o$. Full Resistive-Force-Theory simulations confirm the analysis and reveal that at isotropic drag the swimmer becomes a pure rotator, turning without translating.

math-ph↗
arXiv2026

Nematic Bubbles and the Breaking of Spherical Symmetry

The emergence of nematic order on deformable closed surfaces plays a pivotal role in the morphogenesis of active biological matter, such as the regeneration of Hydra. In this work, we present a continuum model that couples the two-dimensional Landau-de Gennes order tensor, describing in-plane nematic ordering, with the mechanics of a mass-conserving, deformable spherical shell. By investigating the isotropic-to-nematic phase transition driven by a reduction in temperature, mimicking the natural induction of nematic order in actomyosin fibres, we perform both linear and weakly non-linear bifurcation analyses. The onset of nematic ordering spontaneously breaks spherical symmetry, yielding distinct equilibrium morphologies governed by the shell's deformability. Axisymmetric configurations, featuring two +1 defects at the poles, emerge via a discontinuous bifurcation, resulting in a globally stable prolate shape, alongside a metastable oblate shape. Non-axisymmetric configurations, featuring four +1/2 defects arranged in a square, arise via a continuous bifurcation. Shell softness drives the first-order character of the transition, while in the limit of infinite stiffness all bifurcations become continuous. Integer defects strongly couple with local mass redistribution, manifesting as shell thinning or thickening, whilst half-integer defects induce no such local deformation. These findings provide a purely mechanical framework for understanding body-axis formation and defect-mediated morphogenesis in biological vesicles.

cond-mat.soft↗
arXiv2025

Controllability and Displacement Analysis of a Three-Link Elastic Microswimmer: A Geometric Control Approach

This study investigates the dynamics and controllability of a Purcell three-link microswimmer equipped with passive elastic torsional coils at its joints. By controlling the spontaneous curvature, we analyse the swimmers motion using both linear and weakly nonlinear approaches. Linear analysis reveals steady harmonic solutions for small-amplitude controls but does not predict any net displacement, whereas weakly nonlinear analysis predicts translation along the orientation of the central link. Using geometric control theory, we prove that the system is small time locally controllable near equilibrium and derive displacement estimates for periodic piecewise constant controls, which are validated through numerical simulations. These findings indicate that oscillatory controls can enable motion in all directions near equilibrium. This work offers foundational insights into the controllability of elastic microswimmers, paving the way for advanced motion planning and control strategies.

math-ph↗
arXiv2022

Circumferential buckling of a hydrogel tube emptying upon dehydration

A cylindrical hydrogel tube, completely submerged in water, hydrates by swelling and filling its internal cavity. When it comes back into contact with air, it dehydrates: the tube thus expels the solvent through the walls, shrinking. This dehydration process causes a depression in the tube cavity, which can lead to circumferential buckling. Here we study the occurrence of such buckling using a continuous model that combines non-linear elasticity with Flory-Rehner theory, to take into account both the large deformations and the active behavior of the hydrogel. In quasi-static approximation, we use the incremental deformation formalism, extended to the chemo-mechanical equations, to determine the threshold value of the enclosed volume at which buckling is triggered. This critical value is found to depend on the shell thickness, chemical potential and constitutive features. The results obtained are in good agreement with the results of the finite element simulations of the complete dynamic problem.

cond-mat.soft↗
arXiv2022

Growth of a flexible fibre in a deformable ring

We study the equilibrium configurations related to the growth of an elastic fibre in a confining flexible ring. This system represents a paradigm for a variety of biological, medical, and engineering problems. We consider a simplified geometry in which initially the container is a circular ring of radius $R$. Quasi-static growth is then studied by solving the equilibrium equations as the fibre length $l$ increases, starting from $l = 2R$. Considering both the fibre and the ring as inextensible and unshearable, we find that beyond a critical length, which depends on the relative bending stiffness, the fibre buckles. Furthermore, as the fibre grows further it folds, distorting the ring until it induces a break in mirror symmetry at $l>2 πR$. We get that the equilibrium shapes depend only on two dimensionless parameters: the length ratio $μ= l/R$ and the bending stiffnesses ratio $κ$. These findings are also supported by finite element simulation. Finally we experimentally validate the theoretical results showing a very good quantitative prediction of the observed buckling and folding regimes at variable geometrical parameters.

cond-mat.soft↗
arXiv2021

Nematic versus ferromagnetic shells: new insights in curvature-induced effects

Within the framework of continuum theory, we draw a parallel between ferromagnetic materials and nematic liquid crystals confined on curved surfaces, which are both characterized by local interaction and anchoring potentials. We show that the extrinsic curvature of the shell combined with the out-of-plane component of the director field gives rise to chirality effects. This interplay produces an effective energy term reminiscent of the chiral term in cholesteric liquid crystals, with the curvature tensor acting as a sort of anisotropic helicity. We discuss also how the different nature of the order parameter, a vector in ferromagnets and a tensor in nematics, yields different textures on surfaces with the same topology as the sphere. In particular, we show that the extrinsic curvature governs the ground state configuration on a nematic spherical shell, favouring two antipodal disclinations of charge +1 on small particles and four $+1/2$ disclinations of charge located at the vertices of a square inscribed in a great circle on larger particles.

cond-mat.soft↗
arXiv2021

Cooling a spherical nematic shell

Within the framework of Landau-de Gennes theory for nematic liquid crystals, we study the temperature-induced isotropic-nematic phase transition on a spherical shell. Below a critical temperature, a thin layer of nematic coating a microscopic spherical particle exhibits non-uniform textures due to the geometrical frustration. We find the exact value of critical threshold for the temperature and determine exactly the nematic textures at the transition by means of a weakly nonlinear analysis. The critical temperature is affected by the extrinsic curvature of the sphere, and the nematic alignment is consistent with the Poincaré-Hopf index theorem and experimental observations. The stability analysis of the bifurcate textures at the isotropic-nematic transition highlight that only the tetrahedral configuration is stable.

cond-mat.soft↗
arXiv2020

Dehydration induced mechanical instabilities in active elastic spherical shells

Active-elastic instabilities are common phenomena in the natural world which have the aspect of sudden mechanical morphings. Frequently, the driving force of the instability mechanisms has a chemo-mechanical nature which makes these kind of instabilities very different from standard elastic instabilities. In this paper, we describe and study the active-elastic instability occurring in a swollen spherical closed shell, bounding a water filled cavity, during a de-hydration process. The description is given through the outcomes of a few numerical experiments based on a stress-diffusion model which allows to glance at the phenomenon. The study is carried on from a chemo-mechanical perspective through a few simplifying assumptions which allow to derive a semi-analytical model which takes into account both the stress state and the water concentration into the walls of the shell at the onset of the instability. Moreover, also the invariance of the cavity volume at the onset of instability, which is due to the impossibility to instantaneously change the cavity volume filled with water, is considered. It is shown as the semi-analytic model matches very well the outcome of the numerical experiments. It is also shown as a wider range of mechanical instabilities can be produced when inhomogeneous shells are considered, even when the inhomogeneities can be described by a small number of parameters.

cond-mat.soft↗
arXiv2018

Influence of the Extrinsic Curvature on 2D Nematic Films

Nematic interfaces are thin fluid films, ideally two-dimensional, endowed with an in-plane degenerate nematic order. In this letter we examine a generalisation of the classical Plateau problem to an axisymmetric nematic interface bounded by two coaxial parallel rings. The equilibrium interface shape results from the competition between surface tension, which favours the minimization of the interface area, and the nematic elasticity which instead promotes the alignment of the molecules along a common direction. We find two classes of equilibrium solutions with intrinsically uniform alignments: one in which the molecules are aligned along the meridians, the other along parallels. Depending on two parameters, one geometric and the other constitutive, the Gaussian curvature of the equilibrium interface may be negative, vanishing or positive. The stability of these equilibrium configurations is investigated.

cond-mat.soft↗
arXiv2016

The delamination of a growing elastic sheet with adhesion

We study the onset of delamination blisters in a growing elastic sheet adhered to a flat stiff substrate. When the ends of the sheet are kept fixed, its growth arouses residual stresses that lead to delamination. This instability can be viewed as a discontinuous buckling between the complete adhered solution and the buckled solution. We provide an analytic expression for the critical deformation at which the instability occurs. We show that the critical threshold scales with a single dimensionless parameter that comprises information from the geometry of the sheet, the mechanical parameters of material and the adhesive features of the substrate.

cond-mat.soft↗
arXiv2016

Hydrodynamic theory for nematic shells: the interplay among curvature, flow and alignment

We derive the hydrodynamic equations for nematic liquid crystals lying on curved substrates. We invoke the Lagrange-Rayleigh variational principle to adapt the Ericksen-Leslie theory to two-dimensional nematics in which a degenerate anchoring of the molecules on the substrate is enforced. The only constitutive assumptions in this scheme concern the free-energy density, given by the two-dimensional Frank potential, and the density of dissipation which is required to satisfy appropriate invariance requirements. The resulting equations of motion couple the velocity field, the director alignment and the curvature of the shell. To illustrate our findings, we consider the effect of a simple shear flow on the alignment of a nematic lying on a cylindrical shell.

cond-mat.soft↗
arXiv2015

Snap buckling of a confined thin elastic sheet

A growing or compressed thin elastic sheet adhered to a rigid substrate can exhibit a buckling instability, forming an inward hump. Our study shows that the strip morphology depends on the delicate balance between the compression energy and the bending energy. We find that this instability is a first order phase transition between the adhered solution and the buckled solution whose main control parameter is related to the sheet stretchability. In the nearly- unstretchable regime we provide an analytic expression for the critical threshold. Compressibility is the key assumption which allows us to resolve the apparent paradox of an unbounded pressure exerted on the external wall by a confined flexible loop.

cond-mat.soft↗
arXiv2013

Growth-induced blisters in a circular tube

The growth of an elastic film adhered to a confining substrate might lead to the formation of delimitation blisters. Many results have been derived when the substrate is flat. The equilibrium shapes, beyond small deformations, are determined by the interplay between the sheet elastic energy and the adhesive potential due to capillarity. Here, we study a non-trivial generalization to this problem and consider the adhesion of a growing elastic loop to a confining \emph{circular} substrate. The fundamental equations, i.e., the Euler Elastica equation, the boundary conditions and the transversality condition, are derived from a variational procedure. In contrast to the planar case, the curvature of the delimiting wall appears in the transversality condition, thus acting as a further source of adhesion. We provide the analytic solution to the problem under study in terms of elliptic integrals and perform the numerical and the asymptotic analysis of the characteristic lengths of the blister. Finally, and in contrast to previous studies, we also discuss the mechanics and the internal stresses in the case of vanishing adhesion. Specifically, we give a theoretical explanation to the observed divergence of the mean pressure exerted by the strip on the container in the limit of small excess-length.

cond-mat.soft↗
arXiv2011

Surface free energies for nematic shells

We propose a continuum model to describe the molecular alignment in thin nematic shells. By contrast with previous accounts, the two-dimensional free energy, aimed at describing the physics of thin films of nematics deposited on curved substrates, is not postulated but it is deduced from the conventional three-dimensional theories of nematic liquid crystals. Both the director and the order-tensor theories are taken into account. The so-obtained surface energies exhibit extra terms compared to earlier models. These terms reflect the coupling of the geometry of the shell with the nematic order parameters. As expected, the shape of the shell plays a key role in the equilibrium configurations of nematics coating it.

cond-mat.soft↗
arXiv2006

Bulk and surface biaxiality in nematic liquid crystals

Nematic liquid crystals possess three different phases: isotropic, uniaxial, and biaxial. The ground state of most nematics is either isotropic or uniaxial, depending on the external temperature. Nevertheless, biaxial domains have been frequently identified, especially close to defects or external surfaces. In this paper we show that any spatially-varying director pattern may be a source of biaxiality. We prove that biaxiality arises naturally whenever the symmetric tensor $\Sb=(\grad \nn)(\grad \nn)^T$ possesses two distinct nonzero eigenvalues. The eigenvalue difference may be used as a measure of the expected biaxiality. Furthermore, the corresponding eigenvectors indicate the directions in which the order tensor \QQ is induced to break the uniaxial symmetry about the director \nn. We apply our general considerations to some examples. In particular we show that, when we enforce homeotropic anchoring on a curved surface, the order tensor become biaxial along the principal directions of the surface. The effect is triggered by the difference in surface principal curvatures.

cond-mat.soft↗