Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows

In this paper we present a finite difference method for solving two-dimensional viscoelastic unsteady free surface flows governed by the single equation version of the eXtended Pom-Pom (XPP) model. The momentum equations are solved by a projection method which uncouples the velocity and pressure fields. We are interested in low Reynolds number flows and, to enhance the stability of the numerical method, an implicit technique for computing the pressure condition on the free surface is employed. This strategy is invoked to solve the governing equations within a Marker-and-Cell type approach while simultaneously calculating the correct normal stress condition on the free surface. The numerical code is validated by performing mesh refinement on a two-dimensional channel flow. Numerical results include an investigation of the influence of the parameters of the XPP equation on the extrudate swelling ratio and the simulation of the Barus effect for XPP fluids

Numerical solutions for Cauchy integral equations and applications

SIGLEAvailable from British Library Document Supply Centre- DSC:D81530 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Numerical solutions for Cauchy integral equations and applications

SIGLEAvailable from British Library Document Supply Centre- DSC:D81530 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

A review of linear and nonlinear Cauchy singular integral and integro-differential equations arising in mechanics

This study is primarily concerned with the presentation of a review of a collection (which could be regarded as a "test set") of linear and nonlinear singular integro-differential equations with Cauchy kernels, all of which arise from practical applications in Applied Mathematics and Mathematical Physics. The main objective of this review is to provide numerical analysts and researchers interested in algorithm development with model problems of genuine scientific interest on which to test their algorithms. Brief details of the methodology of derivation of the equations are provided and, where possible, existence, uniqueness and asymptotic results are discussed. References are also given to other studies that have dealt with similar problems. The importance of carrying out the necessary mathematical analysis is emphasized for one class of problems where it is shown that the solution abruptly ceases to exist as a parameter is varied. It is further shown that developing asymptotic estimates for the behavior of the solutions is very often a crucial component in the design of effective numerical methods. The importance of regularization is discussed for a class of problems, specific conclusions are drawn and recommendations are discussed. An appendix contains further related problems that may be used for further comparison purposes

GENSMAC 3D: Implementation of the Navier-Stokes equations and boundary conditions for 3D free surface flows

In the present work we describe a method which allows the incorporation of surface tension into the GENSMAC2D code. This is achieved on two scales. First on the scale of a cell, the surface tension effects are incorporated into the free surface boundary conditions through the computation of the capillary pressure. The required curvature is estimated by fitting a least square circle to the free surface using the tracking particles in the cell and in its close neighbors. On a sub-cell scale, short wavelength perturbations are filtered out using a local 4-point stencil which is mass conservative. An efficient implementation is obtained through a dual representation of the cell data, using both a matrix representation, for ease at identifying neighbouring cells, and also a tree data structure, which permits the representation of specific groups of cells with additional information pertaining to that group. The resulting code is shown to be robust, and to produce accurate results when compared with exact solutions of selected fluid dynamic problems involving surface tension

A singular integro-differential equation model for dryout in LMFBR boiler tubes

A 2D steady model for the annular two-phase flow of water and steam in the steam-generating boiler pipes of a liquid metal fast breeder reactor is proposed. The model is based on thin-layer lubrication theory and thin aerofoil theory. The exchange of mass between the vapour core and the liquid film due to evaporation of the liquid film is accounted for using some simple thermodynamics models, and the resultant change of phase is modelled by proposing a suitable Stefan problem. Appropriate boundary conditions for the flow are discussed. The resulting non-linear singular integro-differential equation for the shape of the liquid film free surface is solved both asymptotically and numerically (using some regularization techniques). Predictions for the length to the dryout point from the entry of the annular regime are made. The influence of both the traction provided by the fast-flowing vapour core on the liquid layer and the mass transfer parameter on the dryout length is investigated

An implicit technique for solving 3D low Reynolds number moving free surface flows

This paper describes the development of an implicit finite difference method for solving transient three-dimensional incompressible free surface flows. To reduce the CPU time of explicit low-Reynolds number calculations, we have combined a projection method with an implicit technique for treating the pressure on the free surface. The projection method is employed to uncouple the velocity and the pressure fields, allowing each variable to be solved separately. We employ the normal stress condition on the free surface to derive an implicit technique for calculating the pressure at the free surface. Numerical results demonstrate that this modification is essential for the construction of methods that are more stable than those provided by discretizing the free surface explicitly. In addition, we show that the proposed method can be applied to viscoelastic fluids. Numerical results include the simulation of jet buckling and extrudate swell for Reynolds numbers in the range [0.01, 0.5]

A finite difference technique for simulating unsteady viscoelastic free surface flows

This work is concerned with the development of a numerical method capable of simulating viscoelastic free surface flow of an Oldroyd-B fluid. The basic equations governing the flow of an Oldroyd-B fluid are considered. A novel formulation is developed for the computation of the non-Newtonian extra-stress components on rigid boundaries. The full free surface stress conditions are employed. The resulting governing equations are solved by a finite difference method on a staggered grid, influenced by the ideas of the marker-and-cell (MAC) method. Numerical results demonstrating the capabilities of this new technique are presented for a number of problems involving unsteady free surface flows

Recent advances in the marker and cell method

In this article recent advances in the MAC method will be reviewed. The MAC technique dates back to the early sixties at the Los Alamos Laboratories and this paper starts with a historical review,and then a summary of related techniques. Improvements since the early days of MAC (and the Simplified MAC -SMAC) include automatic time-stepping, the use of the conjugate gradient method to solve the Poisson equation for the corrected velocity potential, greater efficiency through stripping out all particles (markers) other than those near the free surface , more accurate approximations of the free surface boundary conditions, the addition of a bounded high accuracy upwinding for the convected terms (thereby being able to solve higher Reynolds number flows), and a (dynamic) flow visualization facility. This article will concentrate, in the main, on a three-dimensional version of the SMAC method. It will show how to approximate curved boundaries by onsidering one configurational example in detail; the same will also be done for the free surface. The article will avoid validation, but rather focus on many of the examples and applucations that the MAC method can solve from turbulent flows to rheology. It will conclude with some speculative comments on the future direction of the methodology