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We deal with the dynamic and thermal characteristics of the rarefied flow between two eccentric isothermal cylinders one of which is rotating. A numerical program of direct simulation which is based on a Monte Carlo (DSMC) procedure was adopted and the main results concern the influence of the Knudsen and Mach numbers, and of the eccentricity on the location and extent of the flow separation. Part of the flow domain is shown to be cooler than the wall temperature, and the extent of this region decreases with an increasing gas rarefaction.

The propagation of small perturbation in a gas filled porous matrix is investigated. The skeleton is supposed rigid and governed by the energy balance equation, where the heat exchanged between the two phases is taken into account. The Boltzmann equation is written for the gas where the integrals of the collisions between gas and solid particles are evaluated as those for the particles of a mixture. Different choices of the time and space scales lead to models equations which hold for different rarefaction regimes. The wave propagation characteristics are then dealt with in various situations.

The impact of a wedge-shaped body on the free surface of a weightless inviscid incompressible liquid is considered. Both symmetrical and unsymmetrical entries at constant velocity are dealt with. The differential problem corresponds to the physico-mathematical model of a distribution of potential singularities and, in particular, the flow singularities at the ends of the wetted regions are represented by sinks. A conformal transformation of the flow field is adopted and the unknown intensities of the discontinuities are found by an optimization procedure, together with the solution of the nonlinear free-surface problem. The flow separation at a sideslip is also considered., Comment: 26 pages, 10 figures

The unsteady compressible boundary layer equations over an impulsively started flat plate are firstly solved subjected to velocity slip and temperature jump conditions. Comparisons are carried out with existing solutions. Then the conjugate problem for a slab of finite thickness is dealt, when the fluid dynamic field is coupled with the thermal field in the solid.

The gas e ow between two concentric rotating cylinders is investigated over a wide range of governing dimensionless products. Analytic solutions are given for the free molecular e ow, for the Bhatnagar ‐Gross‐Krook model up to the e rst-order approximation, and for the hydrodynamic limit in the Chapman ‐Enskog expansion. The slip e ow boundary conditionsareextensively discussed, and a constructivesolution to the transitional e ow is proposed. Numerical direct simulation data are obtained and adopted as exact reference experimental results.

The equations which govern a number of models for flows between rotating cylinders at different Knudsen numbers are solved numerically by means of the direct simulation Monte Carlo method (DSMCM) to show their limitations. The DSMC code was firstly tested and validated against existing experimental data and then its results represented the reference data base for evaluating the characteristics of each model.

A numerical experiment was carried out on the gas flow field between two eccentric cylinders, one of which is rotating. Attention was paid to the presence of separated recirculating regions from the continuum to the rarefied regimes. The direct simulations were performed by means of a Monte Carlo (DSMC) method and bi-polar co-ordinates were adopted. The calculations were relative to isothermal walls at the same temperature. Streamlines and velocity profiles were evaluated as functions of the Knudsen number, of the Mach number and of the geometric parameters. The gas considered was argon. Copyright © 2000 John Wiley & Sons, Ltd.

The axial characteristics of axisymmetric under-expanded sonic jets are dealt with, both experimentally and numerically. Experiments were carried out in a test rig where a sonic nozzle discharged into an ambient which could be kept at different pressures. The density measurements were taken by means of a differential interferometer. The numerical simulation was obtained by adapting a code by Bird (1994) which is based on the MonteCarlo method. The comparison between experimental data and numerical results was carried out by considering the density distribution along the axis of the jet. The direct simulation was then applied to explore the jet characteristics in a rather wide range of the dimensionless products upon which the phenomenology depends. New expressions are presented for the density distribution along the jet axis and for the location of the Mach disk. Sommario. Si trattano sia sperimentalmente che numericamente le caratteristiche assiali di getti assialsimmetrici sonici sottoespansi. Gli esperimenti sono stati eseguiti con ugelli sonici in ambienti a pressione variabile. La densita e stata misurata mediante un interferometro differenziale e le simulazioni sono state condotte mediante un codice basato sul metodo MonteCarlo. Il confronto tra dati sperimentali e numerici e basato sulla distribuzione di densita lungo l'asse del getto. La simulazione diretta e stata poi applicata allo studio del getto in un ampio campo di valori dei numeri caratteristici. Si discutono la distribuzione della densita lungo l'asse e la posizione del disco di Mach, ottenute sia dai dati sperimentali che dai risultati numerici e si propongono per esse nuove espressioni.

The stationary Boltzmann equation is considered for the Lebowitz model gas of sticks in a slab in the presence of a nonconstant external field. It is proved that the solution is close in L∞ norm to a local “Maxwellian” distribution, the “parameters” of which satisfy the equations of hydrodynamics.

In this paper the unsteady coupling of conduction and convection for a thin body in a high speed stream is considered. The body is modeled as a strip on one side of which the fluid is impulsively started in motion whereas on the other side two different thermal boundary conditions are given : constant temperature or vanishing heat flux. In the first part of the analysis an approximate solution of the energy equation in the solid enables us to obtain relations between the temperature and the heat flux at solidfluid interface which are more accurate than those currently used in the literature. In the second part the exact solution of the two problems that arise from the coupling of the thermofluid-dynamic equations and the relations between the temperature and its derivatives at the interface are presented.

In this paper we prove a theorem of local existence and uniqueness for the solution of the hyperbolic heat equation in the case where the coefficient of the diffusive term is unknown. The problem is one-dimensional in space and the ratio of the two characteristics times, upon which the physics depends, is small. The demonstration relies on a constructive solution.

This paper proposes a method for the solution of two inverse problems which are governed by the nonlinear heat equation in one space dimension. In the first case phase transition occurs at the moving interface which divides two media. In the second one a random heat source is placed at a moving point. In both cases the temperature is assigned, as a function of time and within a random error, at a given fixed point. The solution procedure leads to quantitative results and is based on the Stochastic Adaptative Interpolation method.

The dynamics of gases in permeable media is approached both experimentally and by numerical simulations. The experiments were performed in matrices made of packed beds of spheres in rarefied conditions and a model for the direct simulation of the molecular kinetics is proposed. Comparisons between experimental data and numerical results show the influence of the main parameters of the gas–solid interaction and the range of validity of the model. Moreover it is shown that there is a flow condition for the minimum permeability of the medium to the gas flow. Such a minimum depends upon the Knudsen number, and can be explained by the molecular dynamics as in the well-known Knudsen’s experiment on capillaries.

In this paper we present experimental results for flows through packed bed of spheres, in cases where the flow characteristics can be classified in a range from the velocity slip regime to the transition and free molecular flows. A simple simulation model is also proposed and solved by means of a MonteCarlo code, and comparisons between results of numerical calculations and experimental data are also reported.

The Poiseuille–Couette gas flow in a channel and the gas flow through an adjacent porous medium are considered when the governing equations are obtained via a molecular kinetic approach based on the Boltzmann equation. The mass continuity, momentum balance and energy conservation are written for the gas in the contiguous regions, whereas the behavior of the solid matrix obeys to the heat diffusion equation. Two different space scalings lead to different forms of the equations for the steady flow through the fully saturated matrix. The boundary conditions at the interface between the two domains are investigated via a matching procedure.

The heat and mass transfer at the surface of a finite thickness ablating piston is considered. The piston is either impulsively put in motion or its speed follows an assigned time law. In both cases the wall reaches an hypersonic speed and its surface at the gas/solid interface is diffusive. The coupled problem of the temperature distribution in the two phases is dealt with by solving the parabolic heat conduction equation in the slab by means of an integral boundary layer approach and by adopting a Direct Simulation MonteCarlo Method for the gas phase. The solid and the gas regions are coupled by proper boundary conditions. A time scale analysis was necessary to take into account the more rapidly changing gas characteristics with respect to those of the solid.

In this paper, the effects of the observation frame on momentum transport and energy transfer in gases are considered. A numerical method of direct simulation is adopted for evaluating the shear stress distribution and the heat flux in the fluid-dynamic field between two rotating walls in a range of Knudsen numbers from free molecular flow to compressible continuum. The numerical solutions are compared with the analytical results where existing. It is shown that the effects of the reference frame increase either with the gas rarefaction or with the angular speed of the frame. With respect to the theoretical debate on the material frame indifference of the transport characteristics in a gaseous continuum, the simulations confirm that the frame dependence vanishes (below the numerical error) as the Knudsen number tends to zero.

A model for a continuum gas flowing through a porous matrix is proposed where the gas kinetics is governed by the Boltzmann equation and the solid phase by the energy equation. In the Boltzmann equation the integral relative to the gas–solid collisions is evaluated as for the collisions of hard spheres molecules against much heavier and longer straight particles (Lebowitz model of a sticks gas), randomly distributed in space according to a Maxwellian function with zero mean velocity. The mean flow is one-dimensional but the molecules are free to move in all three space dimensions. In the continuum limit, the moments of the Boltzmann equation provide the mass continuity, energy and momentum equations, the last one expressing the Darcy law for a compressible gas. The transport coefficients are analytically evaluated and a few examples are dealt with.

We show how the motion equations of an aircraft can be very properly dealt with after reference is made to the natural coordinates of its center of gravity. The time domain is first divided into intervals, the duration of which is of the order of magnitude of the characteristic time of the slow, flight path variables. Then an analytic solution for the center of gravity position and velocity is immediately obtained over the entire interval in the form of a series expansion. The second step is then to integrate, by a numerical procedure, the fast attitude variables in the same interval. A few applications are presented and discussed to show the reliability and accuracy of the met hod. The proposed procedure appears very promising for those applications where fast flight simulations are required and also for those problems where advantage has to be taken of the time-scale separation. (Author)

The inverse problem of the control actions which are necessary to keep a High Altitude Range Observatory Platform on an assigned trajectory is considered. The reference vehicle is similar to a Japanese airship designed for cruising over Antarctica at an altitude of about 18,000 m. The controls act on the tail empennage, on the thrust of a fan driven by a solar motor, on the mass of Helium released into the envelope, and, finally, on the location of the centre of gravity. Depending on the number of state parameters involved by the prescribed manoeuver, the inverse problem of determining the control actions is usually a redundant one. Among the various methods for solving the problem, a recently proposed procedure was adopted and the redundancy was eliminated by imposing that a local optimum criterion be satisfied by some of the state variables along the trajectory. In particular, helical descents were investigated with constraints on the indicated air speed and on the re-compressed Helium flow rate. The optimal solutions were obtained via a SQP based E04UCF IBM-NAG routine.

Our purpose is to provide, in this chapter, the introductory framework to the topics to be dealt with through the rest of this textbook. We will thus present some mathematical methods suitable for the analysis of nonlinear stochastic systems which are modelled by stochastic and random nonlinear partial differential equations. These methods will be developed with particular attention to their applications to the physics of continuum and to mechanics. In this respect, a partial differential equation can, in many cases, be regarded as a mathematical model of a real physical system, which governs the time and/or space evolution of its dependent variables. As such, the model equation describes the physical state of the system. When the state of the system is defined by more than one variable, the mathematical model is given in terms of a set of equations, the number of which is equal to the number of components of the state variable.

All the preceding chapters were essentially devoted to stochastic evolution problems which are described by partial differential equations with their proper initial and boundary conditions. However the SAI method seems flexible enough to be adopted for the solution of a larger class of mathematical problems.

The content of the preceding chapters was essentially devoted to the analysis of the first and second order moments of the state variable u = u(t,x,ω) at fixed values of the time t and space x variables. On the other hand, in the class of problems dealt with in this book, u is a time dependent random field and its behaviour can be properly described by its probability distribution as shown in the Appendix. This chapter deals with the analysis of the time-evolution of the probability densities associated to a dependent variable u at fixed values of the independent variables, i. e. time and space.

The mathematical modelling of stochastic systems in continuum physics and mechanics can be realized in such a fashion that the time and space behaviour of the dependent variable is defined by the superposition on a deterministic evolution (in space and time) of an additional weighted noise. This type of modelling was already announced in Chapter 1 when we presented Eq.(1.14).

In this chapter we consider systems of partial differential equations either in the whole space or in bounded domains, whereinto the randomness enters only through the initial conditions. In other words, we treat the “initial value problem”, as defined in Section 1.2, for which the initial condition u 0 is a vector-valued random function. Our aim is to provide a mathematical method which is suitable for the interpolation of the solution and which will be assumed to be continuous in the space variables. A proof of the existence of the solution will be given for particular problems.

When dealing with Problems 1 and 2 introduced in Section 2 of Chapter 1, the physical situation to be studied can be considered as an input-output system where the set of initial and boundary conditions represent the input and the actual solution of the problem is the output and the phenomenology is modelled by a partial differential equation.

This papers introduces a procedure for solving a non-linear system of partial differential equations of the parabolic type with random coefficients. An adaptive method transforms the system into a set of ordinary differential equations which is solved after a suitable expansion of the probability density of the unknown random field in terms of orthogonal polynomials.

This paper introduces a procedure for solving a non-linear system of a partial differential equations of the parabolic type with random coefficients. An adaptive method transform the system into a set of ordinary differential equations which is solved after a suitable expansion of the probability density of the unknown random field in terms of orthogonal polynomials.

The stability and the admittance analysis are considered for a Poiseuille flow running over a permeable slab. The case where a suction, due to a cross flow, is present through both the channel and the porous slab is also dealt with. An analytic solution is found for the basic flow in the entire field whereas the stability analysis and the evaluation of the admittance at the interface are numerically carried out. The errors made in the usually simplified analyses are fully discussed.

Anisotropic diffusion of a tracer in a turbulent medium is considered. Once the neighbour concentration has been calculated, the evaluation of the tracer's concentration may be obtained by series expansion in terms of orthonormal functions. The harmonic-oscillator wave functions are suggested as a suitable choise. The ambiguities arising about the sign of the series coefficients are discussed. Die anisotrope Diffusion einer Beimengung in einem turbulenten Medium wird betrachtet. Nachdem man die Nachbarkonzentration berechnet hat, kann man die Konzentration der Beimengung mit Hilfe einer Reihenentwicktung nach orthonormalen Funktionen erhalten. Die Wellenfunktionen des harmonischen Oszillators werden als geeignetes System vorgeschlagen. Die Mehrdeutigkeiten, die sich aus der Vorzeichenwahl der Reihenkoeffizienten ergeben, werden diskutiert.

The one-dimensional nonlinear heat equation is considered with stochastic coefficients and initial/boundary values in a finite strip and in the half-space. Approximated solutions are obtained by the stochastic adaptive interpolation method which corresponds first to transforming the original partial differential equation into a system of ordinary differential equations via a generalization to the stochastic case of the Bellman's differential quadrature method. The system of ordinary differential equations is then solved by a continuous approximation following by Adomian's decomposition method, and the solutions are compared with those obtained by more standard numerical techniques.

Heat conduction is considered in a semi-infinite solid subjected to a high step change in surface heat flux, such that melting occurs. A time-dependent relaxation model for the energy flux is assumed, leading to a non-Fourier, non-linear equation for the thermal field, which is solved under suitable conditions on the interface displacement.

Free convection is considered in a rectangular trench with an assigned temperature distribution at the bottom and radiative vertical walls. Following a domain perturbation technique, the problem reduces to the Stokes' equation. After proving the existence and uniqueness of the solution, from a functional point of view, the stream function is expressed in closed form via a bi-harmonic series and the shape of the free surface is also given. An application follows in order to show the convergence characteristics of the solution and the flow structure. Die freie Konvektion wird in einer rechteckigen Rinne bei Temperaturverteilung am Boden und strahlenden vertikalen Wanden betrachtet. Unter Ausnutzung eines Storungsansatzes reduziert sich das Problem auf die Stokes'-Gleichung. Nach dem funktionalanalytischen Beweis der Existenz und Einzigkeit der Losung wird die Stromfunktion in geschlossener Form durch eine biharmonische Reihe ausgedruckt. Die Form der freien Oberflache ist auch gegeben. Um die Konvergenzeigenschaften der Losung und die Struktur der Stromung zu demonstrieren, folgt ein Anwendungsbeispiel.

The equations of the unsteady, incompressible laminar boundary layer are considered in the case of two-dimensional flow and in the presence of a pressure gradient. The fundamental equations are cast into a more manageable form through the expansion of the unknown velocity by Taylor’s formula, in terms of a suitable variable. Consideration is also given to the truncation error. Several examples were carried out to show that even the first term in Taylor’s formula provides good accuracy in many circumstances.

A deterministic stochastic approach is successfully applied to the investigation of some problems of the fluid-dynamics of two-phase systems. The method follows the guidelines of the theory of differential equations with random initial conditions.

Die Losung des allgemeinen Problems der viskomagnetischen Grenzschicht wird mit Hilfe asymptotischer Entwicklungen des zugehorigen singularen Storungsproblems behandelt. Dabei werden diejenigen Situationen betrachtet, bei denen sich das Vorhandensein zweier Grenzschichtbereiche erkennen last. Die Durchfuhrbarkeit der Methode, die auf Reihenentwicklungen nach Potenzen geeigneter Parameter in jedem Bereich beruht, wird gezeigt. Im einzelnen wird das magnetofluiddynamische Feld einer ebenen dielektrischen Platte fur Grosenordnungen der charakteristischen Zahlen behandelt, die dem Fall entsprechen, das eine viskomagnetische Grenzschicht uber der Platte liegt, wahrend sich zwischen der viskomagnetischen Region und der auseren freien Stromung eine rein magnetische Grenzschicht erstreckt. Die zugehorigen Gleichungen werden gelost und die Ergebnisse in analytischer Gestalt oder numerisch angegeben. The solution of the general problem of the viscous magnetic boundary layer equations is dealt with by means of the asymptotic expansions of the related singular perturbation problem. Those situations are considered when the existence of two boundary layer regions is in general recognized. As a consequence, the feasibility of a solution method based on series expansions of powers of the suitable characteristic parameters of each region is demonstrated. The magnetofluiddynamic field on a dielectric flat plate in a uniform, parallel magnetic field is considered when the orders of magnitude of the characteristic numbers are such that a viscous magnetic boundary layer lies over the plate, and a purely magnetic boundary layer extends itself between the viscous magnetic region and the external free stream. The solution of the pertinent equations is then carried out and the results are given either in analytical or in numerical form.

A chain model is proposed for solving the nonlinear problem of a single-membrane sail wing, eventually in the presence of elasticity and porosity. The presented results are compared with existing data. The model deals with extended ranges of slack and angle of attack effectively and evaluates the corresponding lift and tension coefficients. The model is also extended to partially separated flows. Nomenclature b = length of the element in the chain over chord b0 = initial length of the element over chord c = chord length CL = lift coefficient ACp = pressure difference coefficient, = (pv -pL)/q CT = tension coefficient, = T/qc k = stretch coefficient, = (

Free convection in a rectangular well of infinite depth at low Rayleigh number is considered. A pair of opposite vertical walls are at different temperatures, whereas the other walls are adiabatic. The three-dimensional Stokes flow regime is analyzed and the boundary-layer function for the adiabatic walls is determined.

A method is described for solving steady-state fluid flow and heat transfer problems which are governed by elliptic-type differential equations. A contrived transient version of the steady-state problem is constructed by appending time derivatives to all the participating equations, regardless of whether or not such terms have physical reality. Each time derivative is multiplied by a fictive diffusivity coefficient which is varied during the course of an explicit marching procedure in order to achieve rapid, stable convergence to the steady state. The solution method is applied to a three-component laminar flow in a cylindrical enclosure having one rotating wall and coolant throughflow. Recirculation patterns are set up in the enclosure due to the shearing action of the throughflow and to the rotation of the disk. The surface heat transfer is found to decrease as the Reynolds number of the throughflow increases.

The elementary theory of lubrication is applied to some slider bearings having different film shapes and the minimisation of the friction coefficient is particularly considered. The inadequacy of the mathematical model is shown since it involves, in the usually dealt with situations, discontinuities not present in physics, and the characteristics are given of a particular profile for which this difficulty is overcome.

An approximate integral method is proposed for solving the unsteady laminar boundary-layer equations. The essence of the method is to assume a similar solution for the velocity profile even in those situations where similar solutions do not exist, and then to find a suitable scale function for the similarity variable. The scale function is the solution to a simple first-order linear partial differential equation. This solution is determined in closed form. Satisfactory results were obtained in comparison with more sophisticated and time-consuming procedures.