Numerical Simulation of Pollution Dispersion over Real Terrain

The paper presents a mathematical and numerical investigation of the atmospheric boundary layer (ABL) flow over coal depot. Two mathematical models have been mentioned based upon: 1) the RANS equations in the conservative form and 2) the Boussinesq approximation of RANS equations in the non–conservative form, both formulated for an incompressible flow with a simple algebraic turbulence closure and given stationary boundary conditions. Also pollution dispersion of passive pollutants has been considered.     

On The Atmospheric Boundary Layer Flow Modelling

The paper presents a mathematical and numerical investigation of the flow in the atmospheric boundary layer (ABL) over complex topography. The flow is supposed to be viscous, incompressible, turbulent and stationary. Two different mathematical–numerical approaches are briefly mentioned. Both models have been used to simulate a flow and pollution dispersion over a complex surface coal field in the North Bohemia which is supposed to be partially covered by a high forest stand. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A compact local one‐dimensional scheme for solving a 3D N‐carrier system with Neumann boundary conditions

We consider a mathematical model for thermal analysis in a 3D N‐carrier system with Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for micro heat transfer. To solve numerically the complex system, we first reduce 3D equations in the model to a succession of 1D equations by using the local one‐dimensional (LOD) method. The obtained 1D equations are then solved using a fourth‐order compact finite difference scheme for the interior points and a second‐order combined compact finite difference scheme for the points next to the boundary, so that the Neumann boundary condition can be applied directly without discretizing. By using matrix analysis, the compact LOD scheme is shown to be unconditionally stable. The accuracy of the solution is tested using two numerical examples. Results show that the solutions obtained by the compact LOD finite difference scheme are more accurate than those obtained by a Crank‐Nicholson LOD scheme, and the convergence rate with respect to spatial variables is about 2.6. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A mathematical model for impulse resistance welding

We present a mathematical model of impulse resistance welding. It accounts for electrical, thermal and mechanical effects, which are non‐linearly coupled by the balance laws, constitutive equations and boundary conditions. The electrical effects of the weld machine are incorporated by a discrete oscillator circuit which is coupled to the field equations by a boundary condition. We prove the existence of weak solutions for a slightly simplified model which however still covers most of its essential features, e.g. the quadratic Joule heat term and a quadratic term due to non‐elastic energy dissipation. We discuss the numerical implementation in a 2D setting, present some numerical results and conclude with some remarks on future research. Copyright © 2003 John Wiley & Sons, Ltd.     

Towards a numerical simulation of a high temperature flow

Two mathematical models of unsteady plasma flows based on 3D finite volume methods for Navier‐Stokes equations are presented. In the first, cold jet model, the nondimensional NS equations are solved by the artificial compressibility method, and the results include time‐varying velocity fields for different Reynolds numbers. The second formulation is a compressible Stegger‐Warming flux vector splitting scheme that takes into account the non‐perfect character of the plasma fluid. Temperature and composition dependent gas properties are to be evaluated from real data.     

Mathematical modelling of compressible viscous fluid flow in a male rotor‐housing gap of screw machines

This contribution is devoted to the mathematical modelling of a compressible viscous fluid flow through a 2‐D model of the male rotor‐housing gap in screw machines. Numerical solution of the nonlinear conservative system of the compressible Navier‐Stokes equations is obtained by means of the cell‐centred finite volume formulation of the explicit two‐step TVD MacCormack scheme proposed by Causon on a structured quadrilateral grid using the own developed numerical code. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiresolution schemes for strongly degenerate parabolic equations in one space dimension

An adaptive finite volume method for one‐dimensional strongly degenerate parabolic equations is presented. Using an explicit conservative numerical scheme with a third‐order Runge‐Kutta method for the time discretization, a third‐order ENO interpolation for the convective term, and adding a conservative discretization for the diffusive term, we apply the multiresolution method combining two fundamental concepts: the switch between central interpolation or exact computing of numerical flux and a thresholded wavelet transform applied to cell averages of the solution to control the switch. Applications to mathematical models of sedimentation‐consolidation processes and traffic flow with driver reaction, which involve different types of boundary conditions, illustrate the computational efficiency of the new method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Velocity and Concentration Boundary Layers on Spinning Fibers

In this paper a mathematical model for the viscose wet spinning process is presented: We consider a single fibre, which is produced by pressing a basic solution of viscose into a bath containing sulphuric acid. H₂SO₄ diffuses into the viscose solution and reacts with the natrium hydroxide so that a solidifying fibre is formed which is pulled through the bath by drives. Due to the movement of the fibre and of diffusive transport of sulphuric acid into the fibre velocity and concentration boundary layers develop. Starting from the laminar boundary layer equations we investigate the flow and concentration fields in the bath induced by the fibre utilizing the Local Non‐Similarity method. Mass transfer in the fibre is modelled by transport equations for sulphuric acid and natrium hydroxide taking into account the neutralization reaction. The model of the fibre is coupled to the bath phase model by appropriate boundary conditions for the mass flow density and the chemical potential of sulphuric acid. The non‐constant diameter of the fibre is taken into account by a perturbation approach.     

On the Turbulence Modelling of 3D-Atmospheric Boundary Layer Flow

The paper presents a mathematical and numerical investigation of the 3D–flow in the atmospheric boundary layer (ABL) over complex relief. The two–equation k - ε model is applied to account for the turbulence. The flow is also supposed to be viscous, incompressible and stationary. The boundary conditions are realized through the wall-functions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hydromagnetic rotating flow of a fourth‐order fluid past a porous plate

The rotating flow in the presence of a magnetic field is a problem belonging to hydromagnetics and deserves to be more widely studied than it has been to date. In the non‐linear regime the literature is scarce. We develop the governing equations for the unsteady hydromagnetic rotating flow of a fourth‐order fluid past a porous plate. The steady flow is governed by a boundary value problem in which the order of differential equations is more than the number of available boundary conditions. It is shown that by augmenting the boundary conditions based on asymptotic structures at infinity it is possible to obtain numerical solutions of the nonlinear hydromagnetic equations. Effects of uniform suction or blowing past the porous plate, exerted magnetic field and rotation on the flow phenomena, especially on the boundary layer structure near the plate, are numerically analysed and discussed. The flow behaviours of the Newtonian fluid and second‐, third‐ and fourth‐order non‐Newtonian fluids are compared for the special flow problem, respectively. Copyright © 2004 John Wiley & Sons, Ltd.     

Classroom Notes

This note gives a simple‐minded approach to the two‐dimensional boundary layer equations. The pressure is eliminated from the equations of motion and the resulting equation is simplified by assuming that certain derivatives in the direction of the boundary are small compared with those at right angles to it. The simplified equation is then integrated to give a single boundary layer equation which, together with the stress rate of strain law and the continuity equation, is sufficient (in theory at least) to predict the flow.

The boundary layer equation as given does not depend on a particular form for the stress rate of strain law and could possibly form the basis for a non‐Newtonian investigation. The viscous boundary layer is given as a special case.     

New explicit approximate solution of MHD viscoelastic boundary layer flow over stretching sheet

In this paper, the study the momentum and heat transfer characteristics in an incompressible electrically conducting non‐Newtonian boundary layer flow of a viscoelastic fluid over a stretching sheet. The partial differential equations governing the flow and heat transfer characteristics are converted into highly nonlinear coupled ordinary differential equations by similarity transformations. The resultant coupled highly nonlinear ordinary differential equations are solved by means of, homotopy analysis method (HAM) for constructing an approximate solution of heat transfer in magnetohydrodynamic (MHD) viscoelastic boundary layer flow over a stretching sheet with non‐uniform heat source. The proposed method is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiry parameter, which provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics. Copyright © 2012 John Wiley & Sons, Ltd.     

Solutions for Initial-Boundary-Value Problems Representing Gravity Currents Arising from Variable Inflow

In this article, we report on theoretical and numerical studies of models for suddenly initiated variable inflow gravity currents in rectangular geometry. These gravity currents enter a lighter, deep ambient fluid at rest at a time‐dependent rate from behind a partially opened lock gate and their subsequent dynamics is modeled in the buoyancy‐inertia regime using ½‐layer shallow water theory. The resistance to flow that is exerted by the ambient fluid on the gravity current is accounted for by a front condition which involves a non‐dimensional parameter that can be chosen in accordance with experimental observations. Flow filament theory is used to arrive at expressions for the variable inflow velocity under the assumptions of an inviscid and incompressible fluid moving through an opening of fixed area which is suddenly opened under a lock gate at one end of a large rectangular tank. The fluid in the lock is subjected to a (possibly) time varying pressure applied uniformly over its surface and the finite movement of the free surface is accounted for. Finding this time‐dependent inflow velocity, which will then serve as a boundary condition for the solution of the shallow‐water equations, involves solving forced non‐linear ordinary differential equations and the form of this velocity equation and its attendant solutions will, in general, rule out finding self‐similar solutions for the shallow‐water equations. The existence of self‐similar solutions requires that the gravity currents have volumes proportional to t ^α , where α≥ 0 and t is the time elapsed from initiation of the flow. This condition requires a point source of fluid with very special properties for which both the area of the gap and the inflow velocity must vary in a related and prescribed time‐dependent manner in order to preserve self‐similarity. These specialized self‐similar solutions are employed here as a check on our numerical approach. In the more natural cases that are treated here in which fluids flow through an opening of fixed dimensions in a container an extra dimensional parameter is introduced thereby ruling out self‐similarity of the solutions for the shallow‐water equations so that the previous analytical approaches to the variable inflow problem, involving the use of phase‐plane analysis, will be inapplicable. The models developed and analyzed here are expected to provide a first step in the study of situations in which a storage container is suddenly ruptured allowing a heavy fluid to debouch at a variable rate through a fixed opening over level terrain. They also can be adapted to the study of other situations where variable inflow gravity currents arise such as, for example, flows of fresh water from spring run‐off into lakes and fjords, flows from volcanoes and magma chambers, discharges from locks and flash floods.     

Criterion of hyperbolicity for non‐conservative quasilinear systems admitting a partially convex conservation law

A system of conservation laws admitting an additional convex conservation law can be written as a symmetric t‐hyperbolic in the sense of Friedrichs system. However, in mathematical modeling of complex physical phenomena, it is customary to use non‐conservative hyperbolic models. We generalize the Godunov–Friedrichs–Lax approach to this new class of models. Copyright © 2011 John Wiley & Sons, Ltd.     

A dynamical approach to large eddy simulation of turbulent flows: existence of weak solutions

We consider a system of equations coming from turbulence models using a large eddy simulation (LES) technique. The idea of this approach bases on decomposing the velocity into a part containing large flow structures and a part consisting of small scales. The equations for large‐scale quantities are derived from the Navier–Stokes equations with an additional constitutive relation for the contribution of small eddies. The mathematical difficulties in this paper focus on the non‐linear and non‐local turbulent term. Copyright © 2005 John Wiley & Sons, Ltd.     

Analysis of a new multispecies tumor growth model coupling 3D phase-fields with a 1D vascular network

In this work, we present and analyze a mathematical model for tumor growth incorporating ECM erosion, interstitial flow, and the effect of vascular flow and nutrient transport. The model is of phase-field or diffused-interface type in which multiple phases of cell species and other constituents are separated by smooth evolving interfaces. The model involves a mesoscale version of Darcy’s law to capture the flow mechanism in the tissue matrix. Modeling flow and transport processes in the vasculature supplying the healthy and cancerous tissue, one-dimensional (1D) equations are considered. Since the models governing the transport and flow processes are defined together with cell species models on a three-dimensional (3D) domain, we obtain a 3D–1D coupled model.     

On entropy stable schemes for degenerate parabolic multispecies kinematic flow models

Entropy stable schemes for the numerical solution of initial value problems of nonlinear, possibly strongly degenerate systems of convection–diffusion equations were recently proposed in Jerez and Parés's study. These schemes extend the theoretical framework of Tadmor's study to convection–diffusion systems. They arise from entropy conservative schemes by adding a small amount of viscosity to avoid spurious oscillations. The main condition for feasibility of entropy conservative or stable schemes for a given model is that the corresponding first‐order system of conservation laws possesses a convex entropy function and corresponding entropy flux, and that the diffusion matrix multiplied by the inverse of the Hessian of the entropy is positive semidefinite. As a new contribution, it is demonstrated in the present work, first, that these schemes can naturally be extended to initial‐boundary value problems with zero‐flux boundary conditions in one space dimension, including an explicit bound on the growth of the total entropy. Second, it is shown that these assumptions are satisfied by certain diffusively corrected multiclass kinematic flow models of arbitrary size that describe traffic flow or the settling of dispersions and emulsions, where the latter application gives rise to zero‐flux boundary conditions. Numerical examples illustrate the behavior and accuracy of entropy stable schemes for these applications.     

Coupling 3D and 1D fluid-structure interaction models for blood flow simulations

Three-dimensional (3D) simulations of blood flow in medium to large vessels are now a common practice. These models consist of the 3D Navier-Stokes equations for incompressible Newtonian fluids coupled with a model for the vessel wall structure. However, it is still computationally unaffordable to simulate very large sections, let alone the whole, of the human circulatory system with fully 3D fluid-structure interaction models. Thus truncated 3D regions have to be considered. Reduced models, one-dimensional (1D) or zero-dimensional (0D), can be used to approximate the remaining parts of the cardiovascular system at a low computational cost. These models have a lower level of accuracy, since they describe the evolution of averaged quantities, nevertheless they provide useful information which can be fed to the more complex model. More precisely, the 1D models describe the wave propagation nature of blood flow and coupled with the 3D models can act also as absorbing boundary conditions. We consider in this work the coupling of a 3D fluid-structure interaction model with a 1D hyperbolic model. We study the stability of the coupling and present some numerical results. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotics of eigenfrequencies and eigenmodes of non‐homogeneous inextensible filament with an end load

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. We derive the spectral asymptotics for the aforementioned two‐parameter family of non‐selfadjoint operators. In the forthcoming papers, based on the asymptotical results of the present paper, we will prove the Riesz basis property of the eigenfunctions. The spectral results obtained in the aforementioned papers will allow us to solve boundary and/or distributed controllability problems for the filament using the spectral decomposition method. Copyright © 2001 John Wiley & Sons, Ltd.