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.
|Qualification||Doctor of Philosophy|
|Award date||16 Sep 2009|
|Place of Publication||Eindhoven|
|Publication status||Published - 2009|