Abstract
The results of mathematical modelling of the dynamics of a mixture of the viscous incompressible liquid and gas, which fills a spherical layer with free boundaries and contains a gas bubble within itself, are presented in this paper. Spherical symmetry is assumed, and it is considered that the dynamics of the layer is determined by thermal, diffusive and inertial factors. On the basis of constructed numerical algorithm the studies of the formation of the liquid glass layers, which contain the carbon dioxide gas within themselves, have been conducted. The impact of the external thermal regime, external pressure and the density of gas in the bubble at the initial time on the dynamics of the layer, diffusion and heat-and-mass processes inside it is investigated. The results of numerical investigation of the full and simplified thermal problem statement, without consideration of gas diffusion, are compared.
Highlights
The study of spherical liquid layers is a task of current interest in connection with the investigation of properties of some new materials, such as spheroplast or sensitizers for emulsion explosives [1, 2]
V is the rate of change of the spherical layer's volume, V = r 2υ, where υ is the radial velocity of the fluid, ρg is the density of the gas in the bubble, T is the temperature, C is the gas concentration in the fluid layer, Pg, Pex are the pressure in gas bubble and external pressure, ν (T ), σ (T ) are the kinematic viscosity and surface tension coefficients, χ(T ), D(T ) are the thermal diffusivity and diffusion coefficients
Computation of the temperature T within the fluid layer for finite-difference analogue of equation (4) by Thomas algorithm complicated by a parameter, which is the unknown value of temperature at the internal free boundary [6]
Summary
The study of spherical liquid layers is a task of current interest in connection with the investigation of properties of some new materials, such as spheroplast or sensitizers for emulsion explosives [1, 2]. The mathematical and numerical modeling of the formation of spherical microballoons is held in [37]. The dynamics of a spherical layer with free boundaries, consisting of the viscous incompressible liquid and gas and containing a gas bubble within itself [3,4,5], is investigated numerically in this paper. It is assumed that the gas is insoluble in the liquid and all transfer coefficients depend on the temperature. The Navier-Stokes equations, as well as the equations of the heat transfer and diffusion of the gas form the basis of mathematical model, describing the processes within the liquid layer. Inside the gas bubble the pressure, density and absolute temperature of the gas satisfy the ideal gas law
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