Lorenzo Fusi

Mathematical Modeling of a Solid–Liquid Mixture with Mass Exchange Between Constituents

The mechanical behavior of a mixture composed by an elastic solid and a fluid that exchange mass is investigated. Both the liquid flow and the solid deformation depend on how the solid phase has increased (diminished) its mass, i.e. on the mass conversion between constituents. The model is developed introducing a decomposition of the solid phase deformation gradient. In particular, exploiting the criterion of maximization of the rate of entropy production, we determine an explicit evolution equation for the so-called growth tensor which involves directly the solid stress tensor. An example of a possible choice of the constitutive functions is also presented.

Determining calcium carbonate neutralization kinetics from experimental laboratory data

In the framework of a research aimed at estimating the performance and lifetime of porous filters filled with marble powder and used to neutralize acid waters, we propose a mathematical model for determining the calcium carbonate reaction kinetics from some experimental data. In particular we show how to determine the order of the reaction and the reaction rate when calcium carbonate is immersed in a HCl solution. These parameters are evaluated by means of a fitting procedure based on least square methods. The experiments are performed using CaCO3 in the form of a slab and powder and measuring (by means of BET analysis) the specific reaction surface.

Retrieving the Bingham model from a bi-viscous model: Some explanatory remarks

Abstract In this note we prove some analytical results on the Bingham model. In particular we show how to derive some constitutive and kinematical properties through a limit procedure in which the visco-plastic model is retrieved from a linear bi-viscous model. We also prove that, assuming a no-slip condition at the interface separating the two viscous fluids, no source of entropy can be present on such interface.

On the stationary flow of a waxy crude oil with deposition mechanisms

In this paper we will present a model for the flow of a waxy crude oil in a test loop, taking into account deposition mechanisms due to the high content of paraffin. We will analyse the flow in a non-isothermal condition considering the main rheological parameters depending on the radial coordinate of the pipe. We will formulate the related mathematical problem, which will turn out to be a free boundary problem, and perform a quasi-steady approximation for some of the equations involved. For such approximated problem well posedness is proved.

MODELING MATERIALS WITH A STRETCHING THRESHOLD

We examine the dynamics of materials characterized by the presence of a deformation threshold beyond which no deformation is possible. The class of bodies that we are interested in studying are described by an implicit constitutive relationship between the Cauchy stress and the deformation gradient. A specific one-dimensional dynamical problem is studied, showing that the mathematical model takes the form of a hyperbolic free boundary problem in which the free boundary conditions can be of two different types, selected according to whether the stress is continuous at the interface (separating the deformable region from the fully strained region), or whether it is discontinuous. Both situations have been analyzed. Sample numerical computations are carried out using data that are relevant to biological materials. A comparison with the problem obtained from a limiting procedure for a constitutive model with a piecewise linear elastic response is performed, showing a very interesting feature, namely that the limit does not lead to the solution of the model with a threshold. This is however not surprising as the latter exhibits dissipative behavior.

Some new results on the flow of waxy crude oils in a loop

It is well known that the presence of paraffin in a crude oil can create difficulties in storage, extraction, pumping and all classical operations in a pipeline system. Oils with high content of paraffin are usually called Waxy Crude Oils (WCO’s). One of the most important problem occurring in WCO’s, for low temperatures, is the formation of paraffin deposits on pipeline walls. This is a very complex mechanism for which many models have been proposed ([1], [2], [3] [4], [5], [6]). On the basis of available experimental results we propose here a physical-mathematical isothermal model for the deposition of paraffin on the walls of a cylindrical circular pipe of radius R,like the ones used in experimental laboratory, usually called loops.

Short and Long Wave Peristaltic Flow: Modeling and Mathematical Analysis

In this paper, we present a mathematical model for the peristaltic flow of a Newtonian fluid in an axisymmetric channel with small aspect ratio. In particular, we study the effects of the wave length of the wall oscillation distinguishing between long wave length (same order of the vessels length) and short wave length (same order of the vessels radius). We prove that the oscillation produces flow even in the absence of a pressure gradient in case of long wave. In case of short wave length, peristalsis does not affect the flow. We also prove that, in both cases, the tube resistance increases as the oscillation amplitude increases.

Mathematical model for calcium carbonate acid mine drainage reaction: a multiple time scale approach

— In this paper we present a mathematical model for the flow of an acid solution through a reacting porous medium. The solid matrix is supposed to be formed by families of spheres with di¤erent radii and the fluid is supposed to saturate the pores. The system is described by the evolution of the overall ion concentration and the radii of the spheres. The structure of the mathematical problem is multi-scale in time and for each time-scale di¤erent simplified problems can be obtained. We give some analytical results and display some numerical simulations to show the behavior of the solutions. The main practical application of this model is the flow of acid solution through neutralizing cartridges in which solid particles of CaCO3 are used to neutralize a given flow of an acid mine drainage.

The one-dimensional flow of a fluid with limited strain-rate

We present a model for a continuum in which the strain rate depends linearly on the stress, as long as the latter is below a fixed threshold, but it is frozen to a constant value when the stress exceeds such a threshold. The constitutive equation is given in an implicit form as the stress is a multi-valued function of the strain rate. We derive the model in a general 3D setting and we study the one-dimensional case of a pressure-driven flow between two parallel plates. We prove some analytical results and describe a procedure to determine the main physical parameters (stress threshold and viscosity) by means of a rotational viscometer. Finally we show that the model can be obtained as the limit case of a piecewise linear viscous model.

Thermostatted kinetic theory for complex systems: comment on "Thermostatted kinetic equations as models for complex systems in physics and life sciences" by Carlo Bianca.

The modelling of complex phenomena by means of equilibrium/nonequilibrium statistical mechanics has been the subject of a well developed research aimed at describing the macroscopic behaviour of a system from the knowledge of the microscopic interactions of the particles that form it [1]. The study of nonequilibrium systems is typically focused on transport/diffusive processes and rate of chemical reactions which keep such systems far from equilibrium. Nonequilibrium states can be roughly divided into two main categories (actually a more detailed classification can be made [6]): transient and steady. In the former case the system is initially in equilibrium and later a perturbation is applied, whereas in the latter case the system is driven away from equilibrium by external forces. The forces used to achieve nonequilibrium produce energy that must be dissipated, in order to prevent the system from heating up. This is obtained by removing heat (by conduction, convection or radiation) through a isothermal bath surrounding the system. The introduction of thermostats in a thermodynamical model then allows to neglect the complex bath-system interactions upon which energy dissipation is based. In his exhaustive survey [2], C. Bianca reviews the kinetic models describing inert and living systems with timereversible deterministic thermostats. These models (Gauss, Kac, Boltzmann, Jager–Segel) can be applied to a broad variety of practical problems including semiconductor devices, nanosciences, biological phenomena, vehicular traffic, economics, crowds and swarms dynamics. The first model reviewed is the so-called Gaussian isokinetic thermostat, which relies on the principle of least constraint. Initially the principle is introduced (and applied e.g. to three-particle Hooke’s law) and subsequently the Gaussian isokinetic and isoenergetic thermostats are derived by the extended velocity-rescaling method (temperature is kept constant with no fluctuations). Then the author reviews the existing literature on the coupling between the Kac kinetic equation (with and without cutoff) and the Gaussian isokinetic thermostat. The Kac equation is derived and then modified to allow the achievement of steady states by means of the Gaussian isokinetic thermostat. Some analytical results of this problem are reported together with some hints on the proofs – in particular convergence to the stationary state is proved exploiting Fourier transform method – and numerical simulations. Subsequently the author considers the thermostatted Boltzmann equation, reporting in particular the asymptotic analysis proposed by Degond and Wennberg [5] with the Bhatnagar–Gross–Krook collision operator and the