Macroscopic description of arbitrary Knudsen number flow using Boltzmann–BGK kinetic theory

We derive, without approximation, a closed-form macroscopic equation for finite Knudsen number flow using the Boltzmann–BGK kinetic theory with constant relaxation time. This general closed-form equation is specialized into a compact integro-differential equation for time-dependent isothermal unidirectional flows and results are presented for channel flow. This equation provides a clear demonstration of the effects of finite Knudsen number, and it also illustrates the limitations of the Boltzmann–BGK theory with constant relaxation time and bounce-back boundary conditions.

On the macroscopic–mesoscopic mixture of a magnetorheological fluid

This paper is concerned with the modelling of a magnetorheological (MR) fluid in the presence of an applied magnetic field as a twofolded mixture—a macroscopic fluid continuum and mesoscopic multi-solid continua. By assigning to each solid particle a vectorial mesoscopic variable, which is defined as the nearest relative position vector with respect to other particles, the solid medium of the MR fluid is further treated as a mixture consisting of different components, specified by these mesoscopic variables. The treatment of multi-solid continua is similar to that in the mesoscopic theory of liquid crystals. However, the key difference lies in the fact that the time-discontinuity of the defined vectorial mesoscopic variable will give rise to a ‘pseudo’ chemical reaction in the solid continuum. The equation of the phenomenological mesoscopic distribution function of the solid continuum then has an additional production term from the pseudo chemical reaction, analogous to the collision term appearing in the Boltzmann equation. The mesoscopic and macroscopic balance equations are then derived and by assuming the special constitutive relations the macroscopic equation for the second moment of the distribution function can be obtained.

MATHEMATICAL TOPICS ON THE MODELLING COMPLEX MULTICELLULAR SYSTEMS AND TUMOR IMMUNE CELLS COMPETITION

This paper deals with a critical analysis and some developments related to the mathematical literature on multiscale modelling of multicellular systems involving tumor immune cells competition at the cellular level. The analysis is focused on the development of mathematical methods of the classical kinetic theory to model the above physical system and to recover macroscopic equation from the microscopic description. Various hints are given toward research perspectives, with special attention on the modelling of the interplay of microscopic (at the cellular level) biological and mechanical variables on the overall evolution of the system. Indeed the final aim of this paper consists of organizing the various contributions available in the literature into a mathematical framework suitable to generate a mathematical theory for complex biological systems.

NONLOCAL BILINEAR EQUATIONS: EQUILIBRIUM SOLUTIONS AND DIFFUSIVE LIMIT

This paper deals with the qualitative analysis of a class of bilinear systems of equations describing the dynamics of individuals undergoing kinetic (stochastic) interactions. A corresponding evolution problem is formulated in terms of integro-differential (nonlocal) system of equations. A general existence theory is provided. Under the assumption of periodic boundary conditions and the interaction rates expressed in terms of convolution operators two classes of equilibrium solutions are distinguished. The first class contains only constant functions and the second one contains some nonconstant functions. In the scalar case (one equation) under suitable scaling, related to the shrinking of interaction range of each individual, the limit to the corresponding "macroscopic" equation is studied. The limiting equation turns out to be the (nonlinear) porous medium equation.