Index. Boundary integral method, 27 Boundary location method, 255 Brinkman number, 14, 158

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1 Index ABFIND,275 Adaptive control, 304, 312 Adaptive process model, 315 Air drag, 221 Alternating Direction Implicit (ADI) technique, 30 Axisymmetric flow, 48 die entry, 58 Boundary integral method, 27 Boundary location method, 255 Brinkman number, 14, 158 Cauchy-Green tensor, 23 Celanese program, 200 Channel depth, 101 Channel flow equations, 97 Characteristic velocity, 12, 25 Balance zone, 313 Checkerboard oscillations, 48 Barrel temperature, 101 Closed signal loop, 304 Bead coating, 281 Coating flows, 15, Blow moulding, 222, applications, 243 basic process, essential features, 243 contact with mould, finite-element method, parison and sheet inflation, further developments, parison formation, illustrative effects, process control, rheological complications, 295 Boundary conditions,s, 10, 26, 35, selected operations, , 38,43,49,63--4,68,69, 96, theory, 243-7, , 101, 102, 106, 131, 169, three-dimensional, , 172, 174, , 193, two-dimensional, ,205,212,220,221,228, Collocation method, , 237, 238, 239, 248, 252, Collocation points, , 254, 285, 286, 288 Compression profile, 330 Boundary element discretisation, 75 Computer-aided design packages, 17 Boundary element method, 65, Computer application,

2 338 INDEX Computer simulation fibre spinning, , moulding, Conjugate Gradients (GC) technique, 30 Conservation equations, 2-5, 170 Constitutive equations, 12, 59, 220, 232, 236, 237 Contact angle, 250, 251, 290, 291 Continuity equations, 2, 10, 97, 248 Control engineering functions, Control strategy, 17 Convergence criterion, 31 Converging flow channel, 132 Corner points, 32 Coulomb friction mechanism, 110 Crank- Nicholson method, 105, 168 Creeping flow, 75, 82 approximation, 3 Cross-linking, 153, 176, 177 Curtain coating, 284, Curtain trajectory, 289 Data acquisition, 303 editing, 303 logging, 306 Deborah number, 13, 16, 82, 295 Demoulding temperature, 321 Denier variability, 16 Developing zones, 245 Diagonal dominance property, 31 Die flow analysis, 127 non-uniformities, 15 Die geometry effects, 85-6 Difference operators, 28, 29 Differential equations, 34, 212, 213, 214 Differential operators, 28 Dimensionless filament radius, 12 Dimensionless film thickness, 12 Dimensionless formulation, Dimensionless ratio, 12 Dip coating, 281 Direct digital control, 304 Dirichlet boundary conditions, 49 Discretisation, 212 errors, 34 Displacement functions, 23, 35, 36 DPID (Differential, Proportional, Integral, Differential closed loop control), Drag coefficient, 185, 198 Draw ratio, 179, 191 Draw resonance, 182 Dynamical approximations, 9-12 Eigenvalue problem, 212 EISPACK,294 Elastic effects, 7 Elastic liquids, 21 Elastic models, 12 Elastic viscous models, 12 Energy conservation, 148 equation, 4, 13,97,98, 105, 106, 149,166,168 Equilibrium equations, 97, III, 132 ESTIFM,275 Expansion coefficient, Extra-stresses, 51, 52, 53, 54, 56, 59, 194 Extrudate swell, factors affecting, free surface of jet, 63 material non-linearity, 63 mechanisms of, Extruder screw geometry, 95 Extrusion, overall process simulation, blow moulding process control, dies, 94 melt flow in, see also Die Feedback control, 304 Fibre spinning, 15, 16,58, asymptotic solutions, basic equations, dynamic simulation, finite-element solutions, flows, 14

3 INDEX 339 Fibre spinning- contd. frequency response analysis, internal variables, 187 kinematics, 182, 192 mechanics of, 184 methods of lines, multi-filament, 200 non-linear transients, process details, processing transients, 181 processing variables, 180 rheology, sensitivity, 211 stability, steady-state simulation, uniform uniaxial extension, Filament breakage, 16 flow, II Film blowing elastic, flows, mechanical equations, 221 mechanics of idealised, modelling, 219 modelling heat transfer, process details, schematic diagram, 218 viscoelastic, viscous, Film extrusion, 312 Finger deformation tensor, 23 Finger strain tensor, 58 Finite-difference (FD) techniques, 27-37,65-67, 130, 131, 135, 136, 173 Finite-element interpolation, 51 Finite-element (FE) techniques, 27, 37-59,67-73,76,129,130,131, 135, 136, 173, 192-3,254-69, 281-3, Flow analysis, 106, 127 channel, 9-10, 12 equation, 313 field, I, 7, II, 12 geometry, 22, 24, 35 processes, I Flowrate-pressure drop curve, 162 Fluid flow, 2 Fluid model, 10, 13 Force balance, 247 Forming zones, 245, 247, 287, 289, 292 Forward roll-coating, 283 Fracture mechanics, 67 Free boundary location, 77-8 Free surfaces, 257, 260, 262, 263, 264, 269,272, 275, 276, 277, 279, 295 Freeze-line height, 222 position, temperature, 229 Frequency response, Friction coefficient, III, 112 Galerkin form of Navier-Stokes equations, 43, 44, 52, 71 Gauss- Laguerre quadrature, 36 Gauss- Legendre quadrature, 57 Gauss-Seidel iteration, 30 Gauss- Seidel (GS) technique, 30, 129 Godet rolls, 179 Graetz number, 13, 149, 158 Graetz- Nusselt problem, 160 Green's function, 73 Green's theorem, 74 Heat-transfer coefficient, 5, 14 Incompressibility condition, 2, 3 Incremental loading, 44 Inelastic non-newtonian fluids, 81-2 Inertial effects, 13 Inflation pressure, 225 Injection moulding, 14,17,139, approximations dynamical, geometrical and topological, kinematical, rheological, ,

4 340 INDEX Input/output dialogue, Integral equation, 34, 35 Integral model, 35 Irrotational extensional flows, 12 Isoparametric transformation, 47 Iterative convergence, 34 Iterative technique, 30, 35 Jacobian matrix, 271, 272, 279 Kase-Matsuo correlation, 192 Kaye-BKZ constitutive equation, 236 Kaye-BKZ model, 187 Kinematic approximations, 9-12 Knife coating, 282 Kronecker delta, 22 Lagrange polynomials, 208 Laplacian operator, 25 Leakage flow, 100 Legendre polynomial, 208 Local coordinates, 45 Long-range memory, 22, 35 Lubrication approximation, 9, 10, II, 15,21,98,146 Mass balance, 117 equation, 124 conservation equation, 2, 71 flow rate, 117, 119, 121 Maxwell fluid, 53, 56 Maxwell model, 8, 9, 11,22,23,25, 32,59 Melt fracture, 15, 86 Mixed formulation, 38 Momentum conservation equation, 3 Multi-grid method, 27 Nahme-Griffith number, 14, 85 Nahme number, 98, 100, 101, 158 Navier-Stokes equations, 6, 37, 38, 42,49, 50, 53, 56, 247, 252, 259, 261, 269 see also Galerkin; Petrov-Galerkin Neumann conditions, 36,49, 252 Newton iteration, 270, 272, 273, 278, 286, 292 Newton-Raphson method, 44, 54, 72-3,87,88, 103, 120, 123, 196,238-9 Newtonian behaviour, 7 Newtonian creeping flow, 81 Newtonian equation, 28 Newtonian flow, 133 Newtonian fluid, 68, 74, 79-81, 133, 182,191,192,195,198,207 mechanics, 30, 32, 33 Newtonian model, 6, 7, 8, 13,230 isothermal, 223 non-isothermal, Newtonian swelling ratio, 66 Nodal pressure, 43 Nodal values, 39, 40, 43, 44 Nodes, 39 Non-Newtonian elastic liquids, 21 Non-Newtonian flow simulation, 27 Non-Newtonian fluid flow, 58 Non-Newtonian fluid mechanics, 30, 32, 33, 37, 38, 39 Non-Newtonian fluids, 69 Non-Newtonian non-isothermal model, Nusselt number, 5, 14, 85, 185,200 Parent element, 45 Parison, formation, inflation, Partial differential equations, 59, 70, 99 Peelet number, 13, 85, 98, 99, 101, 149, 169 Penalty formulation, 39 Performance values, 307 Petrov-Galerkin form of Navier- Stokes equations, 43 Phan-Thien/Tanner model, 8, 9, 22, 23,59,186,191,196,213,214

5 INDEX 341 Picard iterative procedure, 54, 87 Plane creeping (slit) extrusion, 76 Plane creeping flow, 87-8 Plane flow, 48 Plug flow, 113 analysis, 109, 112 solutions, Point SOR, 30 Poiseuille flow, 66 Poisson's equation, 49, 127 Polyethylene terephthalate (PEl), 180-2,186, 187, 196-8,204 Polymer melt mechanics, 1-19 Power-law fluids, 7, 82, 189 index, 6 model, 12 rule, 81 Predictor--corrector method, 172, 174, 211 Preplasticisation, 140 Pressure curve, 327 drop, II, 14, 162 gradient, 10,97,98, 101, 109, 119, 134 profile, 113 readings, Process control, computer applications, 303 engineering functions, input/output dialogue, equation, 313 models, 305 optimisation, p-v-t diagrams, 324 Quality aspects, 301 Quality variables, 312 Reaction injection moulding, Reciprocal theorem, 74 Regression equation, 320 Regula falsi method, 159, 162 Relaxation time, 8, 12, 183,230 Residuals, 41 Reverse roll-coating, 284 Reynolds number, 13, 16,25, 50, 74, 77,80,86,87,177,185,200, 247, 267, 269, 278, 289, 291 Rheological equations of state, 6-9, 22,33,220 Rimming flow, 279 Rivlin-Ericksen tensor, 69 Robin condition, 253 Roll coating, 283 Runge-Kutta method, 159, 168, 196, 201,213,226,267 Screw extrusion, 93 Second-order derivatives, 41 Second-Order Equivalent (SOE), 23-4 Sensitivity, Serendipity eight-node element, 46 Shape functions, 39, 40, 42, 44, 45, 47-9,71,72 Sharp-edged orifice, 65 Shear stress, , 185 thinning, 82 fluid flow, 50 viscosity, 6 Sheet inflation, thickness, 11 Simpson's rule, 78 Simulation packages, Single-screw extruder, 93 developing flow, fully developed flow, 99 melt flow in, melting in, one-dimensional plug flow solutions, solids feeding, Slide coating, 284, 287 Slip condition, 249, 251 Slip velocity distribution, 249 Slot coating, 283 Smoothing parameter, 31 Software packages, 18 Solid plug motion, 110 Solidification point, 203, 204

6 342 Solidification temperature, 188 Spectral method, 27 Spinline model, Spinneret, 179 St. Venant's principle, 66 Stability, 14-16, 86-8, analysis, 88 Stanton number, 196 Stefan-Boltzmann constant, 221 Stick-slip problem, 250 Stiffness matrix, 131 Stokes equations, 47 Stokes flow, 37, 48 Stokes number, 268 Stream function, 25, 26, 29, 49, 68, 132, 135 Streamlines, 10, 78 Stress equilibrium relation, 3 field, 58 levels, 21 Styrene-butadiene rubber, 236 Subparametric transformation, 47 Successive Over Relaxation (SOR) technique, 30, 31, 34 Surface tension, 81 Swelling ratio, 79 Tadmor model, 115 Taylor series expansion, 278 Teapot effect, 288 Temperature analysis, 105 control, curve, 327 effects, 4,6,9, II, 12, 14,84, 114, 150, 313 gradient, 4, 120, 121, 122 profile, 102, readings, Thermal conductivity, 149 Thermal effects, 84-5 Thermal swelling, 85 Thermoforming, 222, basic process, contact with mould, parison and sheet inflation, INDEX Thickness control, tolerances, 315 Thin-filament equation, 184 Toyobo simulation package, 200 Traction vector, 73 Transfer function, 211 Transient equations, 212 Transport coefficients, 184, 185 Upwinding techniques, 43 Velocity analysis, 102, 104, 106, 108 components, 24, 26, 53, 57 field, 21 gradient, 50, 147 ratio, 126 vector, 22 Viscoelastic effects, 82-4 Viscoelastic fluid flow, Viscoelastic fluids, 51, 83 Viscoelastic parameters, 9 Viscoelastic swelling, 67 Viscoelasticity, 83 Visco metric flow, 10 Viscosity, 84, 86, 183, 230 function, 11, 69 law, 58 ratio, 248 Viscosity-deformation rate equation, 189 Voigt model, 229 non-isothermal, Volumetric flow rate, 98, 101 Vorticity, 25, 26, 30, 32 equation, 29 Wall slip, 85 temperature, 14, 150, 158 control, thickness control, 319 Weighted momentum equations, 52 Weighted residuals, 41, 52

7 INDEX 343 Weighting functions, 41, 42 Weissenberg number, 13, 16,25,55, 58, 73, 82, 83, 84, 86, 90 White-Metzner fluid, 56 White-Metzner model, 232 Wire coating, throughput control, Yarn, 180