Below we give a review of one of Cafaro's papers. We also give summaries, abstracts and extracts from the introduction for 5 other Cafaro papers:

**1. **N Bellomo, E Cafaro and G Rizzi, On the mathematical modelling of physical systems by ordinary differential stochastic equations, *Mathematics and Computers in Simulation* **26** (4) (1984), 361-367.

**1.1. Review by: G Adomian.**

This very interesting paper considers some physical systems in mechanics and relativistic mechanics modelled by stochastic differential equations (equations with random coefficients as well as initial conditions). One system deals with the dynamics of a quasispherical body subject to a dissipative random force. The second is a system of charged particles in the field of a black hole. A constructive method determines the probability density and its evolution.

**2.**E Cafaro and V Bertola, Intermediate asymptotic behaviour of fluid flows by scale-size analysis,

*Proceedings of the Royal Society of London A*

**461**(2055) (2005), 755-760.

**2.1. Summary:**

A new approach to the estimation of intermediate asymptotic behaviours of fluid flows is presented. This method is based on a modified dimensional analysis in which fundamental quantities having well-distinguished orders of magnitude are accounted for with different dimensions. Thus, the effects of different physical mechanisms can be isolated inside dimensional equations. The new technique is applied to the analysis of heat and mass transfer correlations for convective flows, and to the interpretation of turbulent drag reduction by additives in wall-bounded flows.

**2.2. From the Introduction:**

Several problems in fluid mechanics and transport phenomena admit self-similar solutions, describing the so-called 'intermediate asymptotic' behaviour of the sys- tem, that is, the behaviour in a region of the variables space where the solution is no longer dependent on the details of initial and boundary conditions, but the system is still far from equilibrium (Barenblatt & Zel'dovich 1972). Very often, self-similarity of intermediate asymptotics can be derived from simple dimensional-analysis arguments. As a matter of fact, dimensional analysis turns out to be an extremely powerful tool in fluid dynamics. It can be used to approach a huge variety of problems, as reported, for instance, in the classic book by Sedov (1959): examples include fluid motion in pipes, convective heat transfer, boundary-layer flows, flame propagation, shock waves, and many more. Dimensional analysis allows one to express physical laws in the form of monomial relationships among dimensionless numbers, which are usually interpreted as ratios between dimensionally homogeneous quantities representing competing effects. ... In general, dimensional analysis alone cannot determine the values of coefficients and exponents in such relationships, so that very often their values must be determined empirically.

**3.**E Cafaro and V Bertola, On the speed of heat,

*Physics Letters A*

**372**(1) (2007), 1-4.

**3.1. Summary:**

A formalism based on the concept of transport velocity allows writing the energy equation for a rigid conductor with heat propagation in Liouvillean form. Such a formalism leads us to reconsider the well-known paradox of infinite propagation velocity in diffusing systems and suggests a deterministic particle approach for the heat conduction equation instead of the standard random walk description. Some straightforward thermodynamic consequences and fluid-dynamic analogies are derived.

**3.2. From the Introduction:**

Models of diffusion phenomena are currently defined in terms of parabolic differential equations for the characteristic fields. The parabolic model suffers the well-known infinite speed paradox, which is still a subject of debate and implies that a local perturbation of the characteristic field prop- agates instantaneously in the diffusion domain even at infinite distance. In the case of heat conduction, a solution of the paradox was attempted by replacing the Fourier constitutive equation by more complex constitutive relationships which allow one to derive heat conduction models defined by hyperbolic equations. However, discarding Fourier's law is not easy, because it is supported by general experimental evidence, and has been proven true in virtually any conditions. Furthermore, one can easily show that under the assumption of local thermodynamic equilibrium, which is perfectly reasonable for phenomena like heat conduction, hyperbolic conduction models may yield a negative entropy production rate, in open violation of the local equilibrium formulation of the second law of thermodynamics. According to some authors this violation is only apparent, because negative values of the local entropy production would mean that the local equilibrium scheme does not apply, so that temperature should be interpreted in a generalized sense rather than as the usual thermodynamic temperature. However, this argument appears weak: in fact, one can always measure the thermodynamic temperature in a given point of a solid, even where hyperbolic conduction models predict a negative entropy production. Furthermore, at present nobody knows whether the concept itself of generalized temperature has a physical meaning, which is a necessary condition to allow its measurement. An alternative approach attempts to disentangle the mathematical paradox of the heat equation from the physics of heat propagation. Because temperature can be measured only when it is larger than the instrument sensitivity threshold, if one attempts detecting temperature changes at a given point in a solid consequent to a local temperature variation in another point, no change will be detected until a certain amount of time has elapsed. Therefore, based on this observation, one could conclude that from the practical standpoint no difference exists between the propagation of heat and the propagation of waves.

**4.**E Cafaro, L Stantero, R Taurino, A Capuani, A Mercalli and S Fantino, Nimbus Meta-plane, a New Aerodynamic Concept for Unmanned Hybrid Aerial Vehicle Application, Technical Paper

**2008-01-2243**(19 August 2008).

**4.1. Abstract:**

The work attains a new hybrid concept of a light unmanned aerial vehicle, called "Nimbus", based on an inflatable lifting body which combines the characteristics and the performances of Lighter-Than-Air and Heavy-Than-Air platforms. This hybrid design allows short take off and landing (STOL) capability due to its low wing loading, absolute stability reference, good employable payload capacity, high endurance, no runway or ground handling system requirements and easy piloting. The inflatable wing is a cylindrical pneumatic chamber, light weight, V shaped and filled with helium. The main aim of the paper is to describe the approach followed in the preliminary design of the Nimbus platform supported by evidences provided by both CFD simulations and experimental trials of a model prototype. The numerical simulations are performed in order to characterize the near flow field on the blimp lifting body balloon and to estimate the main performances of the aerial vehicle.

**5.**V Bertola and E Cafaro, Deterministic-stochastic approach to compartment fire modelling,

*Proceedings of the Royal Society of London*

**465**(2104) (2009), 1029-1041.

**5.1. Summary:**

A generalized Semenov model is proposed to describe the dynamics of compartment fires. It is shown that the transitions to flashover or to extinction can be described in the context of the catastrophe theory (or the theory of dynamical systems) by introducing a suitable potential function of the smoke layer temperature. The effect on the fire dynamics of random perturbations is then studied by introducing a random noise term accounting for internal and external perturbations with an arbitrary degree of correlation. While purely Gaussian perturbations (white noise) do not change the behaviour of the fire with respect to the deterministic model, perturbations depending on the model variable ('coloured' noise) may drive the system to different states. This suggests that the compartment fires can be controlled or driven to extinction by introducing appropriate external perturbations.

**5.2. From the Introduction:**

The study of compartment fires (i.e. fires occurring in spaces delimited by solid walls) received much attention in the past decades, driven by the continuously growing importance of fire safety issues in engineering design, and several deterministic or stochastic models were developed in order to understand the fundamental mechanisms of the fire growth and to assess fire hazard. Deterministic models can be sorted into zone models, which allow one to find the main parameters of the fire by solving a set of first-order ordinary differential equations derived from global balance equations supplemented by semi-empirical physical models, and field models, which simulate the fire evolution within extended geometric domains by solving a set of discretized partial differential equations using general-purpose CFD packages, where specific submodels have been introduced to describe the effects of buoyancy, radiation heat transfer and turbulence. Stochastic approaches view the fire growth as a percolation process, where the transition from non-propagating to propagating fire is described as a phase change phenomenon. One of the main objectives of such models is to predict whether a fire will extinguish spontaneously or grow until it reaches a point when the flame spreads almost instantaneously to occupy the whole enclosure, which is known as the flashover. However, even a qualitative description of the phenomenon is not easy to obtain, both because of the complexity of the physical model, which must take into account the radiative heat exchange between the flame, the fuel and the surroundings, and because most of the physical and environmental parameters (such as the burning surface and the ventilation conditions) are variable in time in a way that is generally not predictable a priori.

**6.**V Bertola and E Cafaro, Diffusion phenomena and thermodynamics,

*Physics Letters A*

**374**(34) (2010), 3373-3375.

**6.1. Summary:**

The established theory of linear diffusion phenomena is reformulated in the framework of non-Hamiltonian dynamical systems theory (Liouville's approach), through the introduction of a suitable characteristic velocity for the processes under consideration. The dissipative features of diffusion processes are studied in a Riemannian manifold which has flux densities and generalized forces as coordinates, and a tensorial metric defined by the phenomenological coefficients. Some relationships between diffusion phenomena and thermodynamics are explored. Finally, a preliminary extension of the proposed approach to the case of nonlinear constitutive equations is presented, and the effects of nonlinearity on the dynamics of the process are briefly discussed.