Phase Plane Analysis of Linear Systems in Dynamic Mathematical Models

A plethora of dynamic mathematical models exist and to understand and master all of them would be a gargantuan task. The author had, nonetheless, attempted to outline some of the methods used to analyse linear systems in modeling. Systems techniques are fundamental to current research in molecular cell-biology. The systems-approach stands in stark contrast to the historically, reductionist paradigm of molecular biology. Field work can be very dangerous. The main purpose of this study was to come up with the best analysis that would be used without going to the real field and thus saving time, money and risks associated with remote field localities. This research showed that the best analysis depends on the nature of the objectives intended to be solved by the model. Phase plane analysis on linear systems assisted in gaining deeper knowledge on the characteristics of such systems. This work analysed some dynamic models looking at phase planes, bifurcation, sensitivity and stability. The research provided a qualitative analysis of the processes not a numerical analysis

Investigation of the airflow instability in the print gap of inkjet printers

Inkjet printer manufacturers face the challenge of designing systems that operate at elevated print gap heights to enable new applications for inkjet technology. At large print gap heights, the print quality tends to be compromised due to an airflow instability that misdirects the ink droplets, misplacing them on the paper. This phenomenon can lead to the printing defect commonly referred to as tiger-stripes. To characterise this instability, simultaneous high-velocity planar PIV measurements and printing tests were performed, together with numerical simulations using the dispersed-phase continuum (DPC) and particle-in-cell (P-in-C) droplet modelling techniques. The DPC technique is based on a predifined temporally averaged number density field to model the droplets. The comparison between both techniques indicated that the DPC approximation effectively computed the mean flow field with a reduction of up to 1000x in processing time. The predicted and measured flow-fields showed a pair of counter-rotating vortices located near the injection-zone that became unstable with an increase in both print gap heights and number density of droplets. The increase in droplet number density or print gap height induced a supercritical pitchfork bifurcation, which was characterised by a transition from a steady and uniform flow field across the domain to a standing wave regime with deformed vortex cores in the spanwise direction. With further increases in number density and print gap height, the flow field became oscillatory. These different flow dynamics were linked to specific printing patterns. The uniform flow field produced a defect-free print, whereas the standing wave regime created a print with straight dark stripes, and the oscillatory flow regime produced the tiger-stripe printing defect. The PIV measurements also showed the existence of tip vortices at the edges of the print-zone that can lead to secondary droplet misplacement

Asymmetric flows of complex fluids past confined cylinders: A comprehensive numerical study with experimental validation

Three non-Newtonian constitutive models are employed to investigate how fluid rheological properties influence the development of laterally asymmetric flows past confined cylinders. First, simulations with the shear-thinning but inelastic Carreau-Yasuda model are compared against complementary flow velocimetry experiments on a semidilute xanthan gum solution, showing that shear-thinning alone is insufficient to cause flow asymmetry. Next, simulations with an elastic but non-shear-thinning finitely extensible non-linear elastic dumbbell model are compared with experiments on a constant viscosity solution of poly(ethylene oxide) (PEO) in an aqueous glycerol mixture. The simulations and the experiments reveal the development of an extended downstream wake due to elastic stresses generated at the stagnation point but show no significant lateral asymmetries of the flow around the sides of the cylinder. Finally, the elastic and shear-thinning linear Phan-Thien-Tanner (l-PTT) model is compared with experimental velocimetry on a rheologically similar solution of PEO in water. Here, at low flow rates, lateral symmetry is retained, while the growth of a downstream elastic wake is observed, in qualitative similarity to the non-shear-thinning elastic fluids. However, above a critical flow rate, the flow bifurcates to one of the two stable and steady laterally asymmetric states. Further parameter studies with the l-PTT model are performed by varying the degrees of shear-thinning and elasticity and also modifying the confinement of the cylinder. These tests confirm the importance of the coupling between shear-thinning and elasticity for the onset of asymmetric flows and also establish stability and bifurcation diagrams delineating the stable and unstable flow states