On a free frost-point problem in non-isothermal film casting


Film casting is an engineering process for the manufacture of thin sheets or films of polymeric melts. In this work, we study existence and uniqueness of stationary solutions for a standard membrane model of non-isothermal film casting of viscous fluids. In contrast to previous work, our model assumes that the fluid film is cooled down and, as soon as it reaches its solidification temperature, freezes somewhere along the fluid curtain instead of at a fixed downstream position. Hence this model exhibits a free frost point which has to be found as part of the solution. The questions of existence and uniqueness of stationary solutions are of fundamental importance for the quality of the model. They are studied for prescribed-force and prescribed-velocity boundary conditions. While the first type of boundary condition gives rise to a parameter range for the governing equations where – surprisingly – no solutions exist, the second type leads to a shooting problem and requires a separate uniqueness argument for solutions, akin to a comparison principle.

Publication Title

ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik