A rigorous definition of mass in special relativity

The axiomatic definition of mass in classical mechanics, outlined by Mach in the second half of 19th century and improved by several authors, is simplified and extended to the theory of special relativity. According to the extended definition presented here, the mass of a relativistic particle is independent of its velocity and coincides with the rest mass, i.e., with the mass defined in classical mechanics. Then, force is defined as the product of mass and acceleration, both in the classical and in the relativistic framework.Comment: to be published in Il Nuovo Cimento

Spatially Developing Modes: The Darcy–Bénard Problem Revisited

In this paper, the instability resulting from small perturbations of the Darcy-Benard system is explored. An analysis based on time-periodic and spatially developing Fourier modes is adopted. The system under examination is a horizontal porous layer saturated by a fluid. The two impermeable and isothermal plane boundaries are considered to have different temperatures, so that the porous layer is heated from below. The spatial instability for the system is defined by taking into account both the spatial growth rate of the perturbation modes and their propagation direction. A comparison with the neutral stability condition determined by using the classical spatially periodic and time-evolving Fourier modes is performed. Finally, the physical meaning of the concept of spatial instability is discussed. In contrast to the classical analysis, based on spatially periodic modes, the spatial instability analysis, involving time-periodic Fourier modes, is found to lead to the conclusion that instability occurs whenever the Rayleigh number is positive

A three-dimensional study of the onset of convection in a horizontal, rectangular porous channel heated from below

The onset of convection is studied in a rectangular channel filled with a fluid saturated porous medium, bounded above and below by impermeable isothermal walls at unequal temperatures and laterally by partially conducting walls. A three-dimensional linear stability analysis is carried out under the assumption of an infinite longitudinal channel length. Then, this assumption is relaxed in order to determine the threshold length for the three-dimensional convection to be the preferred mode at onset. Sensible parameters influencing the conditions for the instability are the aspect ratio of the transverse cross-section and the Biot number associated with the sidewall heat transfer to the external environment. The neutral stability is investigated by expressing the Darcy-Rayleigh number as a function of the longitudinal wave number, for assigned values of the transverse aspect ratio and of the Biot number

Asymptotic behaviour for convection with anomalous diffusion

We investigate the fully nonlinear model for convection in a Darcy porous material where the diffusion is of anomalous type as recently proposed by Barletta. The fully nonlinear model is analysed but we allow for variable gravity or penetrative convection effects which result in spatially dependent coefficients. This spatial dependence usually requires numerical solution even in the linearized case. In this work we demonstrate that regardless of the size of the Rayleigh number the perturbation solution will decay exponentially in time for the superdiffusion case. In addition we establish a similar result for convection in a bidisperse porous medium where both macro and micro porosity effects are present. Moreover, we demonstrate a similar result for thermosolutal convection.Comment: 9 page

Thermoconvective instability and local thermal non-equilibrium in a porous layer with isoflux-isothermal boundary conditions

The effects of lack of local thermal equilibrium between the solid phase and the fluid phase are taken into account for the convective stability analysis of a horizontal porous layer. The layer is bounded by a pair of plane parallel walls which are impermeable and such that the lower wall is subject to a uniform flux heating, while the upper wall is isothermal. The local thermal non-equilibrium is modelled through a two-temperature formulation of the energy exchange between the phases, resulting in a pair of local energy balance equations: one for each phase. Small-amplitude disturbances of the basic rest state are envisaged to test the stability. Then, the standard normal mode procedure is adopted to detect the onset conditions of convective rolls. Beyond the Darcy-Rayleigh number, playing the role of order parameter for the transition to instability, the relevant dimensionless parameters are the inter-phase heat transfer parameter and the thermal conductivity ratio. The disturbance governing equations, formulated as an eigenvalue problem, are solved numerically by a shooting method. Results are reported for the neutral stability curves and for the critical values for the onset of instability