THE (2+1)-DIMENSIONAL COUPLED CUBIC-QUINTIC COMPLEX GINZBURG-LANDAU EQUATIONS IN BINARY FLUID CONVECTION AND SOLITARY WAVE SOLUTIONS

Many features of binary fluid thermal convection can be modeled by the coupled cubic-quintic complex Ginzburg-Landau equations(CC-QGLEs) including complex physical coef-ficients

On the theory of multiple encapsulated microbubbles interaction: Effect of lipid shell thickness

Encapsulated microbubbles are empty, micrometer-sized, spherical bubbles, typically micron which are used in many applications as lipid-shells in soft tissues. Here, this paper proposes the theoretical and mathematical modelling of the interaction of lipid-encapsulated microbubbles in soft tissues, as well as an examination of the effect of the thickness of the lipid-encapsulated microbubbles envelope located between the inner and outer radius of the microbubble. The mathematical models are formulated for a single encapsulated microbubble and interacting microbubbles in lipid shells. The first models are placed while considering the effect of shell-thickness on microbubbles in soft tissues. The second one has weak shell-thickness layers on it. In encapsulated single microbubbles and interacting microbubbles, the modified Church equations of shell-thickness microbubbles are formulated and solved analytically. Under the effect of layer thickness, the radii of outer microbubble dynamics are larger than those of inner radii of encapsulated microbubble dynamics, and the radii of bubbles with weak shell thickness in soft tissue are smaller. The clusterization of microbubbles reduces the process of microbubble growth, and interparticle interaction enhances it. In addition, the approximate value of the inner and outer microbubble radius was calculated at the different time intervals when the thickness of the shell, δ=0,15,150,250,350nm. Moreover, the growth process has been studied in different materials such as polymers, albumin, lipids, and liquids, and it turns out the results are consistent with previous works

Physico-mathematical models for interacting microbubble clouds during histotripsy

This work proposes physico-mathematical models for cavitation of microbubble clouds under the influence of bubble–bubble interaction during histotripsy. The mathematical models are formulated to non-interacting and interparticle interacting microbubble clouds on histotripsy under considering the effect of variable surface tension. The governing equations of Keller–Miksis (KM) based on Neo-Hookean (NH) and the Quadratic Law Kelvin–Voigt (QLKV) models are transformed into ordinary differential equations using the non-dimension variables methodology, which are then solved analytically by the modified Plesset–Zwick method. The generalized case of variable surface tension is derived and evaluated for both cases of non-interacting and interacting microbubbles during histotripsy. The effects of the viscoelastic medium on the dynamics of a single microbubble dynamic and interactions between microbubbles through the histotripsy are investigated. From the analysis of the results, the behavior of single bubble growth is bigger than in the case of interaction of multi-bubbles under considering the effect of viscoelastic tissue of Young modulus, viscosity, and stiffening factor on histotripsy. Moreover, the study reveals that, when an increase in the number of cavitation microbubbles occurs, a decrease of the behavior of cavitation microbubbles occurs, on the contrary, increasing of the distance between microbubbles leads to increasing in the growth process; these processes for growth are playing a significant role during the process of histotripsy of cancerous tissues

Charged Cavitation Multibubbles Dynamics Model: Growth Process

The nonlinear dynamics of charged cavitation bubbles are investigated theoretically and analytically in this study through the Rayleigh–Plesset model in dielectric liquids. The physical and mathematical situations consist of two models: the first one is noninteracting charged cavitation bubbles (like single cavitation bubble) and the second one is interacting charged cavitation bubbles. The proposed models are formulated and solved analytically based on the Plesset–Zwick technique. The study examines the behaviour of charged cavitation bubble growth processes under the influence of the polytropic exponent, the number of bubbles N, and the distance between the bubbles. From our analysis, it is observed that the radius of charged cavitation bubbles increases with increases in the distance between the bubbles, dimensionless phase transition criteria, and thermal diffusivity, and is inversely proportional to the polytropic exponent and the number of bubbles N. Additionally, it is evident that the growth process of charged cavitation bubbles is enhanced significantly when the number of bubbles is reduced. The electric charges and polytropic exponent weakens the growth process of charged bubbles in dielectric liquids. The obtained results are compared with experimental and theoretical previous works to validate the given solutions of the presented models of noninteraction and interparticle interaction of charged cavitation bubbles

On the Theory of Methane Hydrate Decomposition in a One-Dimensional Model in Porous Sediments: Numerical Study

The purpose of this paper is to present a one-dimensional model that simulates the thermo-physical processes for methane hydrate decomposition in porous media. The mathematical model consists of equations for the conservation of energy, gas, and liquid as well as the thermodynamic equilibrium equation for temperature and pressure (P−T) in the hydrate stability region. The developed model is solved numerically by using the implicit finite difference technique on the grid system, which correctly describes the appearance of phase, latency, and boundary conditions. The Newton–Raphson method was employed to solve a system of nonlinear algebraic equations after defining and preparing the Jacobean matrix. Additionally, the proposed model describes the decomposition of methane hydrate by thermal catalysis of the components that make up the medium through multiple phases in porous media. In addition, the effect of thermodynamic processes during the hydrate decomposition on the pore saturation rate with hydrates a7nd water during different time periods was studied in a one-dimensional model. Finally, in a one-dimensional model over various time intervals, t=1, 10, 50 s, the pressure and temperature distributions during the decomposition of methane hydrates are introduced and investigated. The obtained results include more accurate solutions and are consistent with previous models based on the analysis of simulations and system stability