The singular kernel coagulation equation with multifragmentation

In this article we prove the existence of solutions to the singular coagulation equation with multifragmentation. We use weighted $L^1$-spaces to deal with the singularities and to obtain regular solutions. The Smoluchowski kernel is covered by our proof. The weak $L^1$ compactness methods are applied to suitably chosen approximating equations as a base of our proof. A more restrictive uniqueness result is also given.Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1210.150

Regular Solutions to the Coagulation Equations with Singular Kernels

In this article we prove the existence of solutions to the coagulation equation with singular kernels. We use weighted L^1-spaces to deal with the singularities in order to obtain regular solutions. The Smoluchowski kernel is covered by our proof. The weak L^1 compactness methods are applied to suitably chosen approximating equations as a base of our proof. A more restrictive uniqueness result is also mentioned.Comment: 19 page

Application of space–time CE/SE method to shallow water magnetohydrodynamic equations

AbstractIn this article we apply a space–time conservation element and solution element (CE/SE) method for the approximate solution of shallow water magnetohydrodynamic (SMHD) equations in one and two space dimensions. These equations model the dynamics of nearly incompressible conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium in the third direction. In this article we are using a variant of the CE/SE method developed by Zhang et al. [A space–time conservation element and solution element method for solving the two-dimensional unsteady Euler equations using quadrilateral and hexahedral meshes, J. Comput. Phys. 175 (2002) 168–199]. This method uses structured and unstructured quadrilateral and hexahedral meshes in two and three space dimensions, respectively. In this method, a single conservation element at each grid point is employed for solving conservation laws no matter in one, two, and three space dimensions. The present scheme use the conservation element to calculate flow variables only, while the gradients of flow variables are calculated by central differencing reconstruction procedure. We give both one- and two-dimensional test computations. A qualitative comparison reveals an excellent agreement with previous published results of wave propagation method and evolution Galerkin schemes. The one- and two-dimensional computations reported in this paper demonstrate the remarkable versatility of the present CE/SE scheme

On balance laws for mixture theories of disperse vapor bubbles in liquid with phase change

We study averaging methods for the derivation of mixture equations for disperse vapor bubbles in liquids. The carrier liquid is modeled as a continuum, whereas simplified assumptions are made for the disperse bubble phase. An approach due to Petrov and Voinov is extended to derive mixture equations for the case that there is a phase transition between the carrier liquid and the vapor bubbles in water. We end up with a system of balance laws for a multi-phase mixture, which is completely in divergence form. Additional non-differential source terms describe the exchange of mass, momentum and energy between the phases. The sources depend explicitly on evolution laws for the total mass, the radius and the temperature of single bubbles. These evolution laws are derived in a prior article and are used to close the system. Finally numerical examples are presented

Exact solutions to the Riemann problem for compressible isothermal Euler equations for two phase flows with and without phase transition

We consider the isothermal Euler equations with phase transition between a liquid and a vapor phase. The mass transfer is modeled by a kinetic relation. We prove existence and uniqueness results. Further, we construct the exact solution for Riemann problems. We derive analogous results for the cases of initially one phase with resulting condensation by compression or evaporation by expansion. Further we present numerical results for these cases. We compare the results to similar problems without phase transition

Convergence of a splitting scheme applied to the Ruijgrok–Wu model of the Boltzmann equation

AbstractThis paper deals with upwind splitting schemes for the Ruijgrok-Wu model (Physica A 113 (1982) 401–416) of the kinetic theory of rarefied gases in the fluid-dynamic scaling. We prove the stability and the convergence for these schemes. The relaxation limit is also investigated and the limit equation is proved to be a first-order quasi-linear conservation law. The loss of quasi-monotonicity of the present model makes it necessary to give a more careful analysis of its structure. We also obtain global error estimates in the spaces Ws,p for −1⩽s⩽1/p,1⩽p⩽∞ and pointwise error estimates for the approximate solution. The proof naturally uses the framework introduced by Nessyahu and Tadmor (SIAM J. Numer Anal. 29 (1992) 1505–1519) due to the convexity of the flux function

On phase change of a vapor bubble in liquid water

We consider a bubble of vapor and inert gas surrounded by the corresponding liquid phase. We study the behavior of the bubble due to phase change, i.e. condensation and evaporation, at the interface. Special attention is given to the effects of surface tension and heat production on the bubble dynamics as well as the propagation of acoustic elastic waves by including slight compressibility of the liquid phase. Separately we study the influence of the three phenomena heat conduction, elastic waves, and phase transition on the evolution of the bubble. The objective is to derive relations including the mass, momentum, and energy transfer between the phases. We find ordinary differential equations, in the cases of heat transfer and the emission of acoustic waves partial differential equations, that describe the bubble dynamics. From numerical evidence we deduce that the effect of phase transition and heat transfer on the behavior of the radius of the bubble is negligible. It turns out that the elastic waves in the liquid are of greatest importance to the dynamics of the bubble radius. The phase transition has a strong influence on the evolution of the temperature, in particular at the interface. Furthermore the phase transition leads to a drastic change of the water content in the bubble, so that a rebounding bubble is only possible, if it contains in addition an inert gas. In a forthcoming paper the equations derived are sought in order to close equations for multi-phase mixture balance laws for dispersed bubbles in liquids involving phase change. Also the model is used to make comparisons with experimental data on the oscillation of a laser induced bubble. For this case it was necessary to include the effect of an inert gas in the thermodynamic modeling of the phase transitio