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Alfredo Marzocchi

Publications and source records attributed to Alfredo Marzocchi.

11 recordsLinked to original sources
arXiv2026

Existence and uniqueness of minimizers for axisymmetric nematic films

Nematic surfaces are thin liquid films endowed with in-plane orientational order. We study a variational model in which the nematic director is constrained to lie in the tangent space of an axisymmetric surface, and the associated surface energy accounts for both surface tension and elastic nematic contributions. Here we adopt the surface gradient as the differential operator on the surface, we restrict our analysis to revolution surfaces spanning two coaxial rings, and we assume that the nematic director is aligned along parallels. In this setting, the energy functional reduces to a one-dimensional variational problem. We rigorously prove the existence and uniqueness of minimizers and we provide their complete geometric characterization. Finally, we run some numerical simulations.

math.AP↗
arXiv2025

A variational analysis of nematic axisymmetric films: the covariant derivative case

Nematic surfaces are thin fluid structures, ideally two-dimensional, endowed with an in-plane nematic order. In 2012, two variational models have been introduced by Giomi [11] and by Napoli and Vergori [29,28]. Both penalize the area of the surface and the gradient of the director: in [11] the covariant derivative of the director is considered, while [28] deals with the surface gradient. In this paper, a complete variational analysis of the model proposed by Giomi is performed for revolution surfaces spanning two coaxial rings.

math.AP↗
arXiv2024

Elastic membranes spanning deformable boundaries

We perform a variational analysis of an elastic membrane spanning a closed curve which may sustain bending and torsion. First, we deal with parametrized curves and linear elastic membranes proving the existence of equilibria and finding first-order necessary conditions for minimizers computing the first variation. Second, we study a more general case, both for the boundary curve and for the membrane, using the framed curve approach. The infinite dimensional version of the Lagrange multipliers' method is applied to get the first-order necessary conditions. Finally, a numerical approach is presented and employed in several numerical test cases.

math.AP↗
arXiv2024

Thermal convection in a higher velocity gradient and higher temperature gradient fluid

We analyse a model for thermal convection in a class of generalized Navier-Stokes equations containing fourth order spatial derivatives of the velocity and of the temperature. The work generalises the isothermal model of A. Musesti. We derive critical Rayleigh and wavenumbers for the onset of convective fluid motion paying careful attention to the variation of coefficients of the highest derivatives. In addition to linear instability theory we include an analysis of fully nonlinear stability theory. The theory analysed possesses a bi-Laplacian term for the velocity field and also for the temperature field. It was pointed out by E. Fried and M. Gurtin that higher order terms represent micro-length effects and these phenomena are very important in flows in microfluidic situations. We introduce temperature into the theory via a Boussinesq approximation where the density of the body force term is allowed to depend upon temperature to account for buoyancy effects which arise due to expansion of the fluid when this is heated. We analyse a meaningful set of boundary conditions which are introduced by Fried and Gurtin as conditions of strong adherence, and these are crucial to understand the effect of the higher order derivatives upon convective motion in a microfluidic scenario where micro-length effects are paramount. The basic steady state is the one of zero velocity, but in contrast to the classical theory the temperature field is nonlinear in the vertical coordinate. This requires care especially dealing with nonlinear theory and also leads to some novel effects.

physics.flu-dyn↗
arXiv2024

Self-contact preventing energy for tubular rods

We introduce a generalization of Mobius energy for knots to an energy functional for tubular neighbourhoods of closed inextensible curves. We prove the continuity of the energy and its boundedness for physically admissible tubes without self-contact and in particular for the torus. The functional allows to distinguish isotopy classes of the centerline through its physical inspiration to a self-repulsive electrostatic energy. If the tube has zero thickness, O'Hara's functional is recovered. Finally, a discussion on the possible exponents in the functional is carried out.

math-ph↗
arXiv2023

Geometric invariants of non-smooth framed curves

We compare the Serret-Frenet frame with a {\em relatively parallel adapted frame} (RPAF) introduced by Bishop to parametrize $W^{2,2}$-curves. Next, we derive the geometric invariants, curvature and torsion, with the RPAF associated to the curve. Finally, we discuss applications of the two approaches in variational problems.

math.AP↗
arXiv2021

Variational analysis of inextensible elastic curves

We minimize elastic energies on framed curves which penalize both curvature and torsion. We also discuss critical points using the infinite dimensional version of the Lagrange multipliers' method. Finally, some examples arising from the applications are discussed and also numerical experiments are presented.

math.AP↗
arXiv2017

Soap film spanning an elastic link

We study the equilibrium problem of a system consisting by several Kirchhoff rods linked in an arbitrary way and tied by a soap film, using techniques of the Calculus of Variations. We prove the existence of a solution with minimum energy, which may be quite irregular, and perform experiments confirming the kind of surface predicted by the model.

math-ph↗
arXiv2014

Steady free fall of one-dimensional bodies in a hyperviscous fluid at low Reynolds number

The paper is devoted to the study of the motion of one-dimensional rigid bodies during a free fall in a quasi-Newtonian hyperviscous fluid at low Reynolds number. We show the existence of a steady solution and furnish sufficient conditions on the geometry of the body in order to get purely translational motions. Such conditions are based on a generalized version of the so-called Reciprocal Theorem for fluids.

math-ph↗
arXiv2014

Nonlinear free fall of one-dimensional rigid bodies in hyperviscous fluids

We consider the free fall of slender rigid bodies in a viscous incompressible fluid. We show that the dimensional reduction (DR), performed by substituting the slender bodies with one-dimensional rigid objects, together with a hyperviscous regularization (HR) of the Navier--Stokes equation for the three-dimensional fluid lead to a well-posed fluid-structure interaction problem. In contrast to what can be achieved within a classical framework, the hyperviscous term permits a sound definition of the viscous force acting on the one-dimensional immersed body. Those results show that the DR/HR procedure can be effectively employed for the mathematical modeling of the free fall problem in the slender-body limit.

math-ph↗