Dynamics of Slender Viscoelastic Free Jets

Forest, M. Gregory; Wang, Q.

doi:10.1137/s0036139992236761

Cited by 32 publications

(12 citation statements)

References 26 publications

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“…The mathematical properties of one-dimensional models have been studied extensively (Forest and Wang, 1990;Bechtel, Forest, and Lin, 1992;Renardy, 1994;Bechtel, Bolinger, Cao, and Forest, 1995) and simulations under idealized conditions have been performed as well (Forest and Wang, 1994;Renardy, 1995). By using only the leading-order curvature contribution, and for ␣ϭ s ϭ0, a linearization reveals that the model may enter a regime where it is ill posed (Forest and Wang, 1990).…”

Section: Polymeric Liquidsmentioning

confidence: 99%

Nonlinear dynamics and breakup of free-surface flows

1997

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Surface-tension-driven flows and, in particular, their tendency to decay spontaneously into drops have long fascinated naturalists, the earliest systematic experiments dating back to the beginning of the 19th century. Linear stability theory governs the onset of breakup and was developed by Rayleigh, Plateau, and Maxwell. However, only recently has attention turned to the nonlinear behavior in the vicinity of the singular point where a drop separates. The increased attention is due to a number of recent and increasingly refined experiments, as well as to a host of technological applications, ranging from printing to mixing and fiber spinning. The description of drop separation becomes possible because jet motion turns out to be effectively governed by one-dimensional equations, which still contain most of the richness of the original dynamics. In addition, an attraction for physicists lies in the fact that the separation singularity is governed by universal scaling laws, which constitute an asymptotic solution of the Navier-Stokes equation before and after breakup. The Navier-Stokes equation is thus continued uniquely through the singularity. At high viscosities, a series of noise-driven instabilities has been observed, which are a nested superposition of singularities of the same universal form. At low viscosities, there is rich scaling behavior in addition to aesthetically pleasing breakup patterns driven by capillary waves. The author reviews the theoretical development of this field alongside recent experimental work, and outlines unsolved problems. [S0034-6861(97)00303-6] CONTENTS

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Section: Polymeric Liquidsmentioning

confidence: 99%

Nonlinear dynamics and breakup of free-surface flows

1997

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“…A huge amount of literature about a wide variety of interesting theoretical, numerical and experimental investigations can be found, about, e.g. stability analyses of fiber drawing processes [3][4][5][11][12][13][14][15], break up behavior and drop forming for straight and curved viscous liquid flows [1,[16][17][18][19][20], dynamics and crystallization behavior of non-Newtonian (visco-elastic) flows [21][22][23], studies of effects due to viscosity, surface tension, rotation and gravitation [8,9,18,24,25] as well as derivations of asymptotic one-dimensional models. In general, such model reductions are based on the cross-sectional averaging of the threedimensional balance laws and asymptotic assumptions on geometry and profile quantities.…”

Section: Introductionmentioning

confidence: 99%

Systematic derivation of an asymptotic model for the dynamics of curved viscous fibers

Panda

Marheineke

Wegener

2007

Math Methods in App Sciences

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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.

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“…We note that a general discussion of boundary conditions for the hyperbolic systems arising in viscoelastic flows is given in [3]; in the context of that discussion the Newtonian case is degenerate, even if inertia is included.…”

Section: T) = Qb(q T) and After B Is Determined The First Equationmentioning

confidence: 99%

Draw resonance revisited

Renardy¹

2007

J. Phys.: Conf. Ser.

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Abstract. We consider the problem of isothermal fiber spinning in a Newtonian fluid with no inertia. In particular, we focus on the effect of the downstream boundary condition. For prescribed velocity, it is well known that an instability known as draw resonance occurs at draw ratios in excess of about 20.2. We shall revisit this problem. Using the closed form solution of the differential equation, we shall show that an infinite family of eigenvalues exists and discuss its asymptotics. We also discuss other boundary conditions. If the force in the filament is prescribed, no eigenvalues exist, and the problem is stable at all draw ratios. If the area of the cross section is prescribed downstream, on the other hand, the problem is unstable at any draw ratio. Finally, we discuss the stability when the drawing speed is controlled in response to changes in cross section or force.Key words. extensional flow, fiber spinning, draw resonance AMS subject classifications. 76E99 DOI. 10.1137/0506342681. Formulation of the problem. Fiber spinning is a manufacturing process used in making textile or glass fibers. A highly viscous fluid is extruded vertically from a nozzle. It is then cooled by the ambient air and solidifies. The solidified fiber is then wound on a spool at the end of the spinline.Many physical effects are potentially significant in the study of this problem: viscosity, inertia, gravity, surface tension, cooling, elasticity, and air drag may all be relevant. In this paper, we focus on the simplest model and study the influence of varying boundary conditions. We assume that the force in the fiber is purely due to viscous effects, and we ignore temperature dependence. We use a one-dimensional model based on slender geometry and cross-sectional averaging. Let u(x, t) denote the axial speed and A(x, t) the area of the cross section. The spinneret is located at x = 0 and the spool is at x = L. The conservation of mass implies thatIf only viscous forces contribute, the tension in the fiber is given by 3ηAu x , where η is the viscosity. The requirement of constant tension in the fiber leads toBoundary conditions in industrial processes are notoriously ill defined. It is customary to assume that A and u are given at the spinneret:This of course, is an idealization; in reality there is a transition to an upstream flow, which cannot be described by the one-dimensional model. At the spool, it is sensible to prescribe either the speed or the force with which the fiber is wound. One might also consider control strategies where the flow is monitored and the speed of the spool adjusted to achieve a given objective. Since the goal of the manufacturing process is *

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Dynamics of Slender Viscoelastic Free Jets

Cited by 32 publications

References 26 publications

Nonlinear dynamics and breakup of free-surface flows

Nonlinear dynamics and breakup of free-surface flows

Systematic derivation of an asymptotic model for the dynamics of curved viscous fibers

Draw resonance revisited

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