Viscous dissipation effect on solidification of a liquid onto a tube wall

The effect of viscous dissipation on solidification onto the inner surface of a straight circular tube of a liquid flowing laminarly with a specified volumetric rate is analytically determined. Steady conditions and a constant tube wall temperature below the solidification temperature of the liquid are assumed. Solutions for the thickness of the solidified shell and for the pressure drop are obtained. The pressure gradient versus volumetric flow rate curve exhibits a minimum, suggesting that it might be possible to design runners for injection molding machines to have a minimum pressure drop if the volumetric flow rate is specified.