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

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

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

Catching gas with droplets : modelling and simulation of a diffusion-reaction process

The packaging industry wants to produce a foil for food packaging purposes, which is transparent and lets very little oxygen pass. To accomplish this they add a scavenger material to the foil which reacts with the oxygen that diffuses through the foil. We model this process by a system of partial differential equations: a reaction-diffusion equation for the oxygen concentration and a reaction equation for the scavenger concentration. A probabilistic background of this model is given and different methods are used to get information from the model. Homogenization theory is used to describe the influence of the shape of the scavenger droplets on the oxygen flux, an argument using the Fourier number of the foil leads to insight into the dependency on the position of the scavenger and a method via conformal mappings is proposed to find out more about the role of the size of the droplet. Also simulations with Mathematica were done, leading to comparisons between different placements and shapes of the scavenger material in one- and two-dimensional foils

Catching gas with droplets : modelling and simulation of a diffusion-reaction process

The packaging industry wants to produce a foil for food packaging purposes, which is transparent and lets very little oxygen pass. To accomplish this they add a scavenger material to the foil which reacts with the oxygen that diffuses through the foil. We model this process by a system of partial differential equations: a reaction-diffusion equation for the oxygen concentration and a reaction equation for the scavenger concentration. A probabilistic background of this model is given and different methods are used to get information from the model. Homogenization theory is used to describe the influence of the shape of the scavenger droplets on the oxygen flux, an argument using the Fourier number of the foil leads to insight into the dependency on the position of the scavenger and a method via conformal mappings is proposed to find out more about the role of the size of the droplet. Also simulations with Mathematica were done, leading to comparisons between different placements and shapes of the scavenger material in one- and two-dimensional foils

Mathematical problems for Moineau pumps

No abstract

Mathematical problems for Moineau pumps

No abstract

Catching gas with droplets : modelling and simulation of a diffusion-reaction process

The packaging industry wants to produce a foil for food packaging purposes, which is transparent and lets very little oxygen pass. To accomplish this they add a scavenger material to the foil which reacts with the oxygen that diffuses through the foil. We model this process by a system of partial differential equations: a reaction-diffusion equation for the oxygen concentration and a reaction equation for the scavenger concentration. A probabilistic background of this model is given and different methods are used to get information from the model. Homogenization theory is used to describe the influence of the shape of the scavenger droplets on the oxygen flux, an argument using the Fourier number of the foil leads to insight into the dependency on the position of the scavenger and a method via conformal mappings is proposed to find out more about the role of the size of the droplet. Also simulations with Mathematica were done, leading to comparisons between different placements and shapes of the scavenger material in one- and two-dimensional foils