Models for the Average of the Solutions to a P.D.E. with Stochastic Coefficients
In this study we investigate the successive approximation procedure allowed us to derive new equation, valid for the transport of dissolved matter in porous media, or in random force fields, on the macroscopic scale. To do this the diagram technique was exploited.
In this paper the numerical simulation of surfactant dynamics in the topographically trapped long waves over a cylindrical shelf is described. Numerical modeling is based on the balance equation of the surface concentration. The dynamics of impurities was considered in the advection - diffusion - relaxation model. The comparison of different models of the shelf: endless slope, shelf - step concave exponential shelf has been made. It was established that the transverse bottom topography does not signifi cantly affect the geometry of the distribution of the pellicle, but it has an impact on the quantitative parameters of concentration. The infl uence of the number of mode on the concentration level for various models of the shelf was studied. The growth of the modes number increases the derivative concentration extremes from the equilibrium level.
A semiphenomenological model of the transport processes under the action of power energy sources is proposed. To explain the observed deviations of the linear system response to an external perturbation in the transport processes induced by intense energy fluxes, it is proposed to take into account the effect of inertia of the medium. The semiphenomenological model of processes is reduced to a system with two basis states. The techniques of the theory of microscopic objects for the solution of the system are applied. It is shown that the inertia of the medium is due to the finite time of establishing the equilibrium between the basis states.
This classic survey considers passive scalar and vector transport processes in a random nonstationary medium, which are described by linear parabolic equations. Integration over random paths is used, along with the asymptotic behavior of the product of a large number of independent identically distributed random matrices. The most interesting effect is the appearance of concentrated structures (intermittency) of a smooth initial distribution of the transported quantity. The occurrence of intermittent distributions in the linear problem is due to the fact that the coefficients of the transport equation are stochastic. The intermittency shows itself in the rates of exponential growth of the successive moments (Lyapunov exponents) as the moment number increases. Moment equations are obtained for the scalar and vector, and are used to study temperature evolution and magnetic-field generation in a random fluid flow. These equations are differential in a medium with short time correlations and integral in the general case. The range of application of the diffusion description is analyzed. The behavior of the diffusion coefficients in the case of time reversal is examined. The properties of an individual realization of a scalar and vector are also explained, and a dynamo theorem is given on the exponential growth of the magnetic field in a random flow with renewal.
Cooling of tokamak boundary plasma owing to radiation of non-fully stripped lithium ions is considered as a promising way for protection of plasma facing elements (PFE) in tokamak. It may be effectively realized when the main part of lithium ions are involved in the closed circuit of migration between plasma and PFE surface. Such an approach may be implemented with the use of lithium device whose hot (500-600 °C) area to be effected by plasma serves as a Li-emitter and the cold part (∼180 °C) as a Li-collector in the shadow. Capillary-pore system (CPS) provides the returning of collected and condensed lithium to emitting zone by capillary forces. The main goals of the last T-11M lithium experiments were investigating Li ions transport in the tokamak scrape of layer (SOL) and their collecting by different kinds of limiters. The design of devices based on lithium CPS with different ratio of emitting/collecting area is the main subject of this paper. © 2015 The Authors.
Long-term deep bed filtration in porous media with size exclusion particle capture mechanism is studied. For mono dispersed suspension and transport in porous media whit distributed pore sizes, the micro stochastic model allows for upscaling and the exact solution is derived for the obtained macro scale equation system. Results show that transient pore size distribution and nonlinear relation between the filtration coefficient and captured particle concentration during suspension filtration and retention are the main features of long-term deep bed filtration, which generalises the classical deep bed filtration model and its latter modifications. Furthermore, the exact solution demonstrates earlier breakthrough and lower breakthrough concentration for larger particles. Among all the pores with different sizes, the ones with intermediate sizes (between the minimum pore size and the particle size) vanish first. Total concentration of all the pores smaller than the particles turns to zero asymptotically when time tends to infinity, which corresponds to complete plugging of smaller pores.