On a Linearized Problem Arising in the Navier-Stokes Flow of a Free Liquid Jet

In this work, we analyze a Stokes problem arising in the study of the Navier-Stokes flow of a liquid jet. The analysis is accomplished by showing that the relevant Stokes operator accounting for a free surface gives rise to a sectorial operator which generates an analytic semigroup of contractions. Estimates on solutions are established using Fourier methods. The result presented is the key ingredient in a local existence and uniqueness proof for solutions of the full nonlinear problem

Navier-Stokes Flow for a Fluid Jet with a Free Surface

The three-dimensional Navier-Stokes flow of a viscous fluid jet bounded by a moving free surface under isothermal conditions and without surface tension is considered. The fluid domain is assumed to be periodic in the axial direction and initially axisymmetric. A local-in-time existence and regularity result is proven for the full governing equations using a contraction argument in an appropriate function space. Here a Lagrangian specification of the flow field is employedin order to mitigate the difficulties involved in dealing with an evolving fluid domain. It is also shown that the associated linear problem gives rise to an analytic semigroup of contractions on the space of divergence-free Lebesgue-square-integrable vector fields

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

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

The effect of shear in fiber spinning

We study the equations of isothermal fiber spinning under the assumption that viscous friction in the fiber is balanced by shear stresses. Our discussion gives a rather complete picture of the existence and nonexistence of stationary solutions. The linearization about steady state of the governing equations is analyzed by semigroup methods and shown to have the spectrally determined growth property. Both linear and nonlinear stability of stationary solutions is investigated numerically. © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim