We consider the discretization and subsequent model reduction of a system of partial differential-algebraic equations describing the propagation of pressure waves in a pipeline network. Important properties like conservation of mass, dissipation of energy, passivity, existence of steady states, and exponential stability can be preserved by an appropriate semidiscretization in space via a mixed finite element method and also during the further dimension reduction by structure preserving Galerkin projection which is the main focus of this paper. Krylov subspace methods are employed for the construciton of the reduced models and we discuss modifications needed to satisfy certain algebraic compatibility conditions; these are required to ensure the well-posedness of the reduced models and the preservation of the key properties. Our analysis is based on the underlying infinite dimensional problem and its Galerkin approximations. The proposed algorithms therefore have a direct interpretation in function spaces; in principle, they are even applicable directly to the original system of partial differential-algebraic equations while the intermediate discretization by finite elements is only required for the actual computations. The performance of the proposed methods is illustrated with numerical tests and the necessity for the compatibility conditions is demonstrated by examples.

This work deals with the curling behavior of slender viscous jets in rotational spinning processes. In terms of slender-body theory, an instationary incompressible viscous Cosserat rod model is formulated which differs from the approach of Ribe et al.,18 in the incompressibility approximation and reduces to the string model of Marheineke and Wegener13 for a vanishing slenderness parameter. Focusing exclusively on viscous and rotational effects on the jet in the exit plane near the spinning nozzle, the stationary two-dimensional scenario is described by a two-point boundary value problem of a system of first-order ordinary differential equations for jet's center-line, tangent, curvature, velocity, inner shear and traction force and couple. The numerical analysis shows that the rod model covers the string model in an inertia-dominated jet regime. Beyond that it overcomes the limitations of the string model studied by Götz et al.10 and enables even the handling of the viscous-inertial jet regime. Thus, the rod model shows its applicability for the simulation of industrially relevant parameter ranges and enlarges the domain of validity with respect to the string approach.

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SUMMARYThis paper presents a slender body theory for the dynamics of a curved inertial viscous Newtonian fiber. Neglecting surface tension and temperature dependence, the fiber flow is modeled as a three-dimensional free boundary value problem in terms of instationary incompressible Navier-Stokes equations. From regular asymptotic expansions in powers of the slenderness parameter, leading-order balance laws for mass (cross-section) and momentum are derived that combine the unrestricted motion of the fiber centerline with the inner viscous transport. The physically reasonable form of the one-dimensional fiber model results thereby from the introduction of the intrinsic velocity that characterizes the convective terms. For the numerical investigation of the viscous, gravitational and rotational effects on the fiber dynamics, a finite volume approach on a staggered grid with implicit upwind flux discretization is applied.