On a non-isothermal model of free fluid films

The drawing of thin fluid films and fibers from highly viscous fluid melts is a common engineering process in the chemical and textile industry. A standard procedure for the manufacture of free thin films is film casting. The widely used, one-dimensional model of film casting considered here is based on a slender body approximation of the Navier-Stokes equations with moving boundaries, paired with kinematic assumptions. The model equations describe the flow of the viscous fluid between the die exit and a take-up point and permit variations in film width and film thickness. A heat transfer equation accounts for the heat loss due to cooling. We will address the following two objectives here: First we put existence and uniqueness of stationary solutions for the equations of non-isothermal film casting on a rigorous analytical basis. This objective is tackled with the help of continuity arguments and a variant of the maximum principle. Our second objective is the study of the linearized equations. We will prove semigroup results for the linearization about steady state and shed light on the long-term regularity of solutions. Among other things we obtain the validity of the spectral mapping theorem for the semigroup and the spectrally determined growth property. These results form the basis for computational studies in the literature about the linear stability of stationary solutions and the potential onset of physical instabilities.