Finite volume approach for the instationary Cosserat rod model describing the spinning of viscous jets

The spinning of slender viscous jets can be described asymptotically by one-dimensional models that consist of systems of partial and ordinary differential equations. Whereas the well-established string models possess only solutions for certain choices of parameters and set-ups, the more sophisticated rod model that can be considered as $\epsilon$-regularized string is generally applicable. But containing the slenderness ratio $\epsilon$ explicitely in the equations complicates the numerical treatment. In this paper we present the first instationary simulations of a rod in a rotational spinning process for arbitrary parameter ranges with free and fixed jet end, for which the hitherto investigations longed. So we close an existing gap in literature. The numerics is based on a finite volume approach with mixed central, up- and down-winded differences, the time integration is performed by stiff accurate Radau methods

Three flow regimes of viscous jet falling onto a moving surface

A stationary viscous jet falling from an oriented nozzle onto a moving surface is studied, both theoretically and experimentally. We distinguish three flow regimes and classify them by the convexity of the jet shape (concave, vertical and convex). The fluid is modeled as a Newtonian fluid, and the model for the flow includes viscous effects, inertia and gravity. By studying the characteristics of the conservation of momentum for a dynamic jet, the boundary conditions for each flow regime are derived, and the flow regimes are characterized in terms of the process and material parameters. The model is solved by a transformation into an algebraic equation. We make a comparison between the model and experiments, and obtain qualitative agreement

Falling of a viscous jet onto a moving surface

We analyze the stationary flow of a jet of Newtonian fluid that is drawn by gravity onto a moving surface. The situation is modeled by a third-order ODE on a domain of unknown length and with an additional integral condition; by solving part of the equation explicitly we can reformulate the problem as a first-order ODE, again with an integral constraint. We show that there are two flow regimes, and characterize the associated regions in the three-dimensional parameter space in terms of an easily calculable quantity. In a qualitative sense the results from the model are found to correspond with experimental observations.Comment: 16 pages, 11 figure

Curved jets of viscous fluid : interactions with a moving wall

The processes where a jet of viscous fluid hits a moving surface arise in various industrial and everyday-life applications. A simple example is pouring honey onto a pancake. Similar processes are used in the production of glass wool, thermal isolation, three-dimensional polymeric mats, and para-aramid fibers. In all these processes a liquid jet emerges from a nozzle and is driven by gravity and possibly centrifugal and Coriolis forces towards a moving surface. The performance of the processes depends strongly on the properties of the jet between the nozzle and the moving surface. Very often experimental study of the jet is very difficult or sometimes even impossible. Therefore, modeling can give some insight into the process and describe the influence of the parameters on the performance. The parameters one can think of are: flow velocity at the nozzle, surface velocity, distance between the nozzle and the moving surface, and fluid properties such as viscosity. One of the simplest examples one can look at is the viscous jet falling under gravity from an oriented nozzle onto a moving belt. There is a vast amount of literature on jets hitting a stationary surface, but only very few publications involving a moving one. In our experiments we identify three stationary regimes: i) a concave shape aligned with the nozzle orientation (comparable to a ballistic trajectory), ii) a vertical shape, or iii) a convex shape aligned with the belt. The convexity or concaveness of the shape characterizes the three flow regimes. In addition to this overall structure, stationary or instationary boundary effects can be observed at the nozzle and near the belt. Moreover, when the nozzle does not point vertically down the whole jet can be instationary. To describe the jet we use a model which takes into account the effects of inertia, viscosity, and gravity, and disregards bending. This allows us to focus on the large-scale jet shape while avoiding the modeling of bending and buckling regions at the jet ends. Also, we neglect surface tension and assume the fluid to be isothermal and Newtonian. The key issue for this model are boundary conditions for the jet shape. They follow from the conservation of momentum equation which is a hyperbolic equation for the shape. The correct boundary conditions follow from consideration of the characteristic directions of that equation at each end. This also provides a criterion for partitioning the parameter space into the three regimes. The physical quantity which characterizes the three flow regimes is the momentum transfer through a jet cross-section, which has contributions from both inertia and viscosity. In a concave jet the momentum transfer due to inertia dominates the viscous one everywhere in the jet, and therefore the nozzle orientation is relevant. In the vertical jet the momentum transfer due to viscosity dominates at the nozzle and due to inertia at the belt, and in the point where they are equal the stationary jet should be aligned with the direction of gravity. From this the vertical shape follows. In the convex jet the viscous momentum transfer dominates in the jet and the tangency with the belt becomes important. This gives an alternative characterization of the three flow regimes in which the jet can be inertial, viscous-inertial, and viscous respectively. Moreover, for this model we prove existence and investigate uniqueness. When we have non-uniqueness, up to three stationary solutions are possible, which explains the instationary behaviour observed experimentally. The comparison between our theory and experiments shows a qualitative agreement. A similar process of rotatory fiber spinning is modeled using the same approach. In this process the jet is driven out from a rotating rotor by centrifugal and Coriolis forces towards a cylindrical surface (the ‘coagulator’). The parameter space contains four possible situations. Two correspond to the inertial and the viscous-inertial jets discussed before. The two others correspond to different types of non-existence of stationary jets, one because no stationary jet can reach the coagulator (causing real-world jets to wind around the rotor), and one because a stationary jet can not match velocities at the coagulator. An interesting fact is that the viscous jet situation is not possible; this would require the coagulator to rotate in the same direction as the rotor with at least half of its angular velocity

Fall of viscous jet into a moving surface

A fall of the thin jet of viscous fluid onto the moving surface is considered. The jet is described by the effects of elongational viscosity, inertia and gravity. For the model equations we derive the boundary conditions allowing us to show existence for all the parameters, and investigate uniqueness. For the jet fall we distinguish three flow regimes, which are characterized by the convexity of the jet shape, or by an equivalent characterization of the dominant effect in the momentum transfer through the jet cross-section

Some studies on the deformation of the membrane in an RF MEMS switch

Radio Frequency (RF) switches of Micro Electro Mechanical Systems (MEMS) are appealing to the mobile industry because of their energy efficiency and ability to accommodate more frequency bands. However, the electromechanical coupling of the electrical circuit to the mechanical components in RF MEMS switches is not fully understood. In this paper, we consider the problem of mechanical deformation of electrodes in RF MEMS switch due to the electrostatic forces caused by the difference in voltage between the electrodes. It is known from previous studies of this problem, that the solution exhibits multiple deformation states for a given electrostatic force. Subsequently, the capacity of the switch that depends on the deformation of electrodes displays a hysteresis behaviour against the voltage in the switch. We investigate the present problem along two lines of attack. First, we solve for the deformation states of electrodes using numerical methods such as finite difference and shooting methods. Subsequently, a relationship between capacity and voltage of the RF MEMS switch is constructed. The solutions obtained are exemplified using the continuation and bifurcation package AUTO. Second, we focus on the analytical methods for a simplified version of the problem and on the stability analysis for the solutions of deformation states. The stability analysis shows that there exists a continuous path of equilibrium deformation states between the open and closed state

On modeling of curved jets of viscous fluid hitting a moving surface

A jet of Newtonian fluid can fall from the oriented nozzle onto the moving surface in three regimes. A flow regime depends on the process parameters and is characterized by the dominant effect in themomentumtransfer through the jet crosssection. To model the three jet flow regimes we describe the jet by the effects of inertia, longitudinal viscosity, and gravity. The key issue is to prescribe the boundary conditions for the jet orientation, which follow from the conservation of momentum for the dynamic jet. If the jet is under tension, the principal part of the conservation of momentum equation is of hyperbolic type, and the boundary conditions for the jet shape follow from the directions of characteristics. From this we find that the boundary conditions for the jet orientation are determined by the dominant effect in the momentum transfer through the jet cross-section, which can be due to inertia, or due to viscosity. This choice of boundary conditions allows us to find the solution to the steady jet model for all parameters, and partition the parameter space between the three jet flow regimes

On error estimation in the fourier modal method for diffractive gratings

The Fourier Modal Method (FMM, also called the Rigorous Coupled Wave Analysis, RCWA) is a numerical discretization method which is often used to calculate a scattered field from a periodic diffraction grating. For 1D periodic gratings in FMM the electromagnetic field is presented by a truncated Fourier series expansion in the direction of the grating periodicity. The grating’s material properties are assumed to be piece-wise constant (called slicing), and next per slice the scattered field is approximated by a truncated Fourier series expansion. The truncation representation of the scattered field and the piece-wise constant approximation of the grating’s material properties cause the error in FMM. This paper presents an analytical estimate/bound for the FMM error caused by slicing

A model of rotary spinning process

A rotary spinning process is used to produce aramide fibers. In this process thin jets of polymer solution emerge from the nozzles of the rotating rotor and flow towards the cylindrical coagulator. At the coagulator the jets hit the water curtain in which they solidify forming fibers. The rotary spinning is described by a steady jet of viscous Newtonian fluid between the rotor and the coagulator. The jet model includes the effects of inertia, longitudinal viscosity, and centrifugal and Coriolis forces. For the jet model the specific type of the boundary conditions depends on the balance between the inertia and viscosity in the momentum transfer through the jet cross-section. Based on that we find two possible flow regimes in rotary spinning: 1) viscous-inertial, where viscosity dominates at the rotor and inertia at the coagulator, 2) inertial, where inertia dominates everywhere in the jet. Moreover, there are two situations where spinning is not possible, either due to lack of a steady-jet solution or because the jet wraps around the rotor. Finally, we characterize the parameter space

The meandering instability of a viscous thread

A viscous thread falling from a nozzle onto a surface exhibits the famous rope-coiling effect, in which the thread buckles to form loops. If the surface is replaced by a belt moving with speed $U$, the rotational symmetry of the buckling instability is broken and a wealth of interesting states are observed [See S. Chiu-Webster and J. R. Lister, J. Fluid Mech., {\bf 569}, 89 (2006)]. We experimentally studied this "fluid mechanical sewing machine" in a new, more precise apparatus. As $U$ is reduced, the steady catenary thread bifurcates into a meandering state in which the thread displacements are only transverse to the motion of the belt. We measured the amplitude and frequency $\omega$ of the meandering close to the bifurcation. For smaller $U$, single-frequency meandering bifurcates to a two-frequency "figure eight" state, which contains a significant $2\omega$ component and parallel as well as transverse displacements. This eventually reverts to single-frequency coiling at still smaller $U$. More complex, highly hysteretic states with additional frequencies are observed for larger nozzle heights. We propose to understand this zoology in terms of the generic amplitude equations appropriate for resonant interactions between two oscillatory modes with frequencies $\omega$ and $2\omega$. The form of the amplitude equations captures both the axisymmetry of the U=0 coiling state and the symmetry-breaking effects induced by the moving belt.Comment: 12 pages, 9 figures, revised, resubmitted to Physical Review