Injection moulding of thermoplastics: Freezing of variable-viscosity fluids.

The injection moulding of thermoplastics involves, during mould filling, flows of hot polymer melts into mould networks, the walls of which are so cold that frozen layers form on them. An analytical study of such flows is presented here for the case when the Graetz number is small and the Nahme number is non-zero and can be large. Thus the flows are fully-developed and temperature differences due to heat generation by viscous dissipation are sufficiently large to cause significant variations in viscosity.Gz Graetz number - h half-height of channel or disc - h^* half-height of polymer melt region in channel or disc - L length of channel or pipe - m viscosity shear-rate exponent - Na_Q Nahme number based on flowrate - Na_P Nahme number based on pressure drop - Na_PL lower critical value of Nahme number based on pressure drop - Na_PU upper critical value of Nahme number based on pressure drop - Na_P Nahme number based on pressure gradient - p pressure - P pressure drop - Q volumetric flowrate - r radial coordinate in pipe or disc - R radius of pipe - Re Reynolds number - R_i inner radius of disc - R₀ outer radius of disc - R^* radius of polymer melt region in pipe - T temperature - T_m melting temperature of polymer - T₀ reference temperature - T_w wall temperature - u axial velocity in pipe or channel or radial velocity in disc - w width of channel - x axial coordinate in channel - y transverse coordinate in channel or disc - z axial coordinate in pipe - thermal conductivity of molten polymer - thermal conductivity of frozen polymer - heat capacity of molten polymer - viscosity temperature exponent - dimensionless transverse coordinate in channel or disc - ^* dimensionless half-height of polymer melt region in channel or disc - dimensionless temperature - ^* dimensionless wall temperature - µ viscosity of molten polymer - µ₀ consistency of molten polymer - dimensionless pressure drop - dimensionless pressure gradient - density of molten polymer - dimensionless radial coordinate in pipe or disc - _i dimensionless inner radius of disc - ^* dimensionless radius of polymer melt region in pipe - dimensionless velocity