Daniela Vasileva

Numerical analysis of radically nonsymmetric blow-up solutions of a nonlinear parabolic problem

Abstract The process of combustion for a nonlinear heat conducting medium with a nonlinear volume source is considered. Blow-up self-similar solutions which describe the evolution of radially nonsymmetric waves — with complex symmetry and spiral waves — are realized numerically. Their asymptotic behaviour is analysed and their metastability is established. To solve the self-similar nonlinear problem the continuous analog of the Newton method and the finite element method are used. The semidiscrete Galerkin finite element method and an explicit difference scheme are used to solve the nonlinear parabolic problem. Special adaptive grids, consistent with the structure of the self-similar solutions, are utilized.

On the numerical simulation of unsteady solutions for the 2D Boussinesq paradigm equation

For the solution of the 2D Boussinesq Paradigm Equation (BPE) an implicit, unconditionally stable difference scheme with second order truncation error in space and time is designed. Two different asymptotic boundary conditions are implemented: the trivial one, and a condition that matches the expected asymptotic behavior of the profile at infinity. The available in the literature solutions of BPE of type of stationary localized waves are used as initial conditions for different phase speeds and their evolution is investigated numerically. We find that, the solitary waves retain their identity for moderate times; for larger times they either transform into diverging propagating waves or blow-up.

LUMPED-MASS FINITE-ELEMENT METHOD WITH INTERPOLATION OF THE NONLINEAR COEFFICIENTS FOR A QUASILINEAR HEAT TRANSFER EQUATION

Abstract A unified numerical technique for computing single-point, regional, and total blow-up solutions of quasilinear heat transfer equations in the radial symmetric case is proposed and realized. It is based on the lumped-mass finite-element method in space, with interpolation of the nonlinear coefficients and explicit methods for solving the system of ordinary differential equations. A special mesh adaptation, consistent with the structure of the known self-similar solution, is realized for the cases of single point and total blow-up. This gives a possibility to calculate the solution close to the blow-up time and so to analyze its asymptotic behavior.

Numerical realization of blow‐up spiral wave solutions of a nonlinear heat‐transfer equation

The problem of finding the possible classes of solution of different nonlinear equations seems to be of a great importance for many applications. In the context of the theory of self‐organization it is interpreted as finding all possible structures which arise and preserve themselves in the corresponding unbounded nonlinear medium. First, results on the numerical realization of a class of blow‐up invariant solutions of a nonlinear heat‐transfer equation with a source are presented in this article. The solutions considered describe a spiral propagation of the inhomogeneities in the nonlinear heat‐transfer medium. We have found initial perturbations which are good approximations to the corresponding eigen functions of combustion of the nonlinear medium. The local maxima of these initial distributions evolve consistent with the self‐similar law up to times very close to the blow‐up time.

Comparison of Two Numerical Approaches to Boussinesq Paradigm Equation

In order to study the time behavior and structural stability of the solutions of Boussinesq Paradigm Equation, two different numerical approaches are designed. The first one A1 is based on splitting the fourth order equation to a system of a hyperbolic and an elliptic equation. The corresponding implicit difference scheme is solved with an iterative solver. The second approach A2 consists in devising of a finite difference factorization scheme. This scheme is split into a sequence of three simpler ones that lead to five-diagonal systems of linear algebraic equations. The schemes, corresponding to both approaches A1 and A2, have second order truncation error in space and time. The results obtained by both approaches are in good agreement with each other.

Numerical simulation of drop coalescence in the presence of drop soluble surfactant

Numerical method is presented for simulation of the deformation, drainage and rupture of axisymmetric film (gap) between colliding drops in the presence of film soluble surfactants under the influence of van der Waals forces at small capillary and Reynolds numbers and small surfactant concentrations. The mathematical model is based on the lubrication equations in the gap between drops and the creeping flow approximation of Navier-Stokes equations in the drops, coupled with velocity and stress boundary conditions at the interfaces. A non-uniform surfactant concentration on the interfaces, related with that in the film, leads to a gradient of the interfacial tension which in turn leads to additional tangential stress on the interfaces (Marangoni effects). Both film and interface surfactant concentrations, related via adsorption isotherm, are governed by a convection-diffusion equation. The numerical method consists of: Boundary integral method for the flow in the drops; Finite difference method for the flow...

Numerical investigation of a new class of waves in an open nonlinear heat-conducting medium

The paper contributes to the problem of finding all possible structures and waves, which may arise and preserve themselves in the open nonlinear medium, described by the mathematical model of heat structures. A new class of self-similar blow-up solutions of this model is constructed numerically and their stability is investigated. An effective and reliable numerical approach is developed and implemented for solving the nonlinear elliptic self-similar problem and the parabolic problem. This approach is consistent with the peculiarities of the problems — multiple solutions of the elliptic problem and blow-up solutions of the parabolic one.

On a local refinement solver for coupled flow in plain and porous media

A local refinement algorithm for computer simulation of flow through oil filters is presented. The mathematical model is based on laminar incompressible Navier-Stokes-Brinkman equations for flow in pure liquid and in porous zones. A finite volume method based discretization on cell-centered, collocated, locally refined grids is used. Special attention is paid to the conservation of the mass on the interface between the coarse and the fine grid. A projection method, SIMPLEC, is used to decouple momentum and continuity equations. The corresponding software is implemented in a flexible way for arbitrary 3D geometries, approximated by an union of parallelepipeds with different sizes. Results from numerical experiments show that the solver could be successfully used for simulation of coupled flow in plain and in porous media.

Mathematical Modelling of the Effect of Biosurfactants on the Surface Tension

In this study a continuum mathematical model is developed to describe the effect of biosurfactants on the surface tension. The model includes generalized reaction-advection-diffusion equations for the concentration of the substrate, biomass and the product (biosurfactant). The reaction (bioprocess) is modeled using classical Monod kinetics for the growth of the biomass and biosurfactant. In the reaction-advection-diffusion equation for the biomass the bacterial chemotaxis is taken into account using KellerSegel model [1]. The evolution of the concentration of biosurfactants on the surface is governed by advection-diffusion equation where the flux bulk-surface is given by Fick’s low. The dependence bulk-surface concentration of biosurfactants is given by adsorption isotherm. The dependence of the surface tension on the biosurfactant concentration is modeled by the corresponding equation of state. Different nonlinear adsorption isotherms, respectively equations of state, are considered. In the case of flat surface in the absence of fluid flow and initially uniformly distributed substrate and biomass the model can be considered as one-dimensional. In this case the evolution of the surface tension is given by a generalization of the Ward-Tordai integral equation. Numerical results for the 1D model are presented. The authors are partially supported by Grant DFNI-I02/9/12.12.2014 from the Bulgarian National Science Fund.

Numerical Simulation of Drop Coalescence in the Presence of Inter-Phase Mass Transfer

A numerical method for simulation of the deformation and drainage of an axisymmetric film between colliding drops in the presence of inter-phase solute transfer at small capillary and Reynolds numbers and small solute concentration variations is presented. The drops are considered to approach each other under a given interaction force. The hydrodynamic part of the mathematical model is based on the lubrication equations in the gap between the drops and the Stokes equations in the drops, coupled with velocity and stress boundary conditions at the interfaces. Both drop and film solute concentrations, related via mass flux balance across the interfaces, are governed by convection-diffusion equations. These equations for the solute concentration in the drops and the film are solved simultaneously by a semi-implicit finite difference method. Tests and comparisons are performed to show the accuracy and stability of the presented numerical method.