Angiolo Farina

On Darcy's law for growing porous media

Abstract The flow of a Newtonian fluid in porous media can be described using Darcys law when inertial effects and deformations in the solid can be neglected and no mass interchange occur between the solid and the fluid components. Having in mind bio-medical applications, we analyze the correction to be considered when non-negligible mass exchanges between the constituents are present. This is done both on a thermodynamical basis and using a symmetry and frame indifferent argument.

BIOMASS GROWTH IN UNSATURATED POROUS MEDIA: HYDRAULIC PROPERTIES CHANGES

We present a model to describe the biomass growth process taking place in an unsaturated porous medium during a bioremediation process. We focus on the so-called column experiment. At the initial time biomass and polluted water is inoculated in the column. The subsequent changes of hydraulic properties are analyzed. We also show some preliminary simulations.

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.

Non-isothermal injection molding with resin cure and preform deformability

In this paper, a non-isothermal model to simulate some injection molding processes used to fabricate composite materials is deduced. The model allows the solid constituent in both the dry and the wet region to deform during infiltration. The dry porous material is assumed to behave elastically, while the mixture of resin and preform is assumed to behave as a standard linear solid. The model also takes into account the fact that the liquid undergoes an exothermic cross-linking reaction during infiltration and eventually gels stopping the infiltration process. Focusing then on one-dimensional problems it is shown that the integration of the mechanical problem in the uninfiltrated region can be reduced to the integration of an ordinary differential equation defining either the space-independent volume ratio or the location of the infiltration front, depending on whether the flow is driven by a given infiltration velocity or by a given inlet pressure. The remaining system of partial differential equations in the two interfaced and time-dependent domains is then posed with the proper interface and boundary conditions. After writing the problem in a Lagrangian formulation fixed on the solid constituent, domain decomposition techniques are used for the simulation.

Deformable Porous Media and Composites Manufacturing

A growing number of industrial activities demands advanced materials that satisfy stringent requirements and lower costs. These requirements, which involve a combination of many properties, can often be satisfied by using a composite material, whose constituents act synergically to solve the needs of application. Modelling the behavior of such a heterogeneous material during its production is a very hard task, but it is very useful for the optimization of the manufacturing process itself. This chapter focuses on the deduction of mathematical models of deformable porous media and on their application to composite materials manufacturing as a first step toward the understanding of this complex process.

Waxy Crude Oils: Some Aspects of their Dynamics

Waxy crude oils are highly non-Newtonian fluids known to cause pipelining difficulties because their rheological properties are strongly affected by paraffin crystallization. On the basis of experimental data, a physical model has been developed to describe the behavior of these crudes. The corresponding mathematical problem has been studied in planar geometry proving the existence and uniqueness of a classical solution. A condition on the pressure gradient has been found ensuring that the system do not come to a complete stop in finite time.

Flow in deformable porous media: Modelling and simulations of compression moulding processes

This paper deals with the mathematical modelling, based on the theory of flows in deformable porous media, and with some numerical simulations of a unidirectional compression molding process. The technological process which we have considered consists of a compression of a pile of preimpregnated layers operated by a piston acting on the top of the pile. The main idea towards the modelling is the use of Lagrangian coordinates fixed on the solid skeleton. This technique allows us to simulate the system evolution either when the dynamics is controlled by the velocity of the piston and when it is controlled by the pressure applied on the piston.

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.

Continuous dependence on the constitutive functions for a class of problems describing fluid flow in porous media

In this paper we consider the PDE describing the fluid flow in a porous medium, focusing on the solution’s dependence upon the choice of the saturation curve and the hydraulic conductivity. Basically, we consider two different saturation curves (say θ1 and θ2) and two different hydraulic conductivities (K1 and K2) which are both “close” in the Lloc-norm. Then we find estimates to prove a constitutive stability for the solutions of the corresponding problems with the same boundary and initial conditions.

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.