J. Molenaar

Dynamics of vibrating atomic force microscopy

An atomic force microscope which is operated in the oscillating mode is an example of an impact oscillator. The description of such dynamical systems can be reduced to a mathematical mapping, which displays a square-root singularity. A direct consequence of this property is the emergence of an infinite series of period-adding bifurcations. This extremely characteristic phenomenon should be observed in atomic force microscopes. We consider an atomic force microscope in which the tip-substrate forces are modelled by a liquid-bridge interaction. By integrating the dynamical equations we show that the atomic force microscopy (AFM) dynamical behaviour has the same characteristic bifurcation scenario as the square-root map. We point to the remarkable role of the energy that is dissipated upon impact. We finally suggest ways to improve the operation of AFM.

In-phase and out-of-phase break-up of two immersed liquid threads under influence of surface tension

The dynamical behaviour of two infinitely long adjacent parallel liquid threads immersed in a fluid is considered under influence of small initial perturbations. Assuming all fluids to behave Newtonian, we used the creeping flow approximation, which resulted in Stokes equations. Applying cylindrical coordinates and separation of variables, and writing the dependence on the azimuthal direction in the form of a Fourier expansion, we obtained general representations of the equations for both the threads and the surrounding fluid. Substitution of these expressions into the boundary conditions leads to an infinite set of linear equations for the unknown coefficients. Its solutions for the lowest two orders of the Fourier expansion, the so-called zero- and first-order solutions, are presented. Much attention is paid to the (in)stability of the configuration, in terms of the so-called growth rate of the disturbance amplitudes. The growth rate of these amplitudes determines the behaviour of the break-up process of the threads. It turns out that this breaking up occurs either in-phase or out-of-phase. This depends on the viscosity ratio of the fluids and on the distance between the threads. These findings agree with experimental observations. The results of the present work also show that the zero-order solution yields the qualitatively correct insight in the break-up process. The extension to a one order higher expansion only leads to relatively small quantitative corrections.

Postponing Polymer Processing Instabilities

In this report a short overview is given of the instabilities that may occur during extrusion of a polymer. For the so—called spurt phenomenon a model is discussed that makes use of the JSO constitutive equation. It is shown that this model lead to a differential-algebraic-integral equation. The structure of the the equations is sketched in terms of time scales. For the stability analysis a new method is proposed that is based on discretization of the integral equation. This approximation reduces the model to a set of (singularly perturbed) ODEs, which can be analysed with standard methods.