By J. R. A. Pearson (auth.), J. R. A. Pearson, S. M. Richardson (eds.)

Large, quick, electronic pcs were generic in engineering perform and their use has had a wide impression in lots of fields. Polymer processing isn't any exception, and there's already a great deal of literature describing ways that approaches may be analysed, designed or managed utilizing the prospects of contemporary desktops. The emphasis given varies with the appliance, and such a lot authors are inclined to quote the result of their calculations instead of describing in any element the way in which the calculations have been undertaken or the problems skilled in wearing them out. We objective to offer right here as invaluable and attached an account as we will of a large type of purposes, for the good thing about scientists and engineers who locate themselves engaged on polymer processing difficulties and think the necessity to adopt such calculations. the main program we now have in brain is the simulation of the dynamics ofthe quite a few actual phenomena which come up in a polymer strategy handled as a posh engineering process. This calls for that the process be quite good represented via a constrained variety of particularly basic subprocesses whose connections could be truly pointed out, that the domi nant actual results correct to every subprocess could be good outlined in an appropriate mathematical shape and that the units of equations and boundary stipulations built to explain the entire procedure will be effectively discretised and solved numerically.

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CHAPTER 2 Computational Techniques for Viscoelastic Fluid Flow M. J. CROCHET Unite de Mecanique Appliquee, Universite Catholique de Louvain, Belgium and K. 1 INTRODUCTION The materials used in polymer processing can usually be classified as nonNewtonian elastic liquids (see, for example, White, 1980). Such liquids are often processed in complex geometries. There is therefore a practical motivation for the current interest in the numerical simulation of nonNewtonian flow in complex geometries. Under some conditions, the peculiar memory effects associated with elastic liquids do not have a strong influence on the flow, an obvious example being those situations where the lubrication approximation can be employed (see, for example, Pearson, 1967).

A good overview may be obtained by inspecting the books edited by Gallagher (1975a,b; 1978), but the selection of the best method for solving a given type of problem remains a question of debate; a good example is given in the proceedings of a recent symposium devoted to the sole problem of the advective terms in the Navier-Stokes equations (Hughes, 1979), where papers alternate in favouring or dissuading the use of some new techniques for handling the non-linear terms. Finite elements have been developed by engineers within the context of linear elasticity theory where, fortunately, minimum principles provide means of evaluating the error, and uniqueness theorems give the necessary confidence; later, mathematicians were able to show that the engineers were right in their use of the finite-element concept, and built the theoretical background for linear elliptic problems (see, for example, Strang and Fix, 1973).

The evaluation of the error is a difficult problem which will not be treated here (see Girault and Raviart, 1981; Temam, 1977). 1 shows various combinations of shape functions. The first element is the simplest one might think of; it is easy to show that the incompressibility condition is so strong that no solution is possible. 1 FINITE ELEMENTS FOR SOLVING STOKES FLOW 2 3 4 5 6 Shape Velocity Pressure Remarks Triangle Triangle Triangle Quadrilateral Quadrilateral Quadrilateral pi_CO p2_ CO p2_ CO pi_CO p2_ CO p2_ CO pO-C - 1 pO-C- 1 pl_ CO pO-C- 1 pi_CO pl-C- I Impossible element Widely used Spurious pressure modes Widely used Elements 3 and 5 are the most widely used in engineering applications; for quadrilaterals, one may consider eight or nine nodes, but recent publications tend to prefer the nine-node element.