Department of Mathematical Sciences and Center for Applied Mathematics and Statistics

The Department of Mathematical Sciences (DMS) and the Center for Applied Mathematics and Statistics (CAMS) have developed the Capstone Laboratory. This Laboratory has been supported through the years by multiple equipment grants from the National Sciences Foundation, as well as by NJIT and individual research grants.

The breakup of viscous filaments has, and is being studied experimentally, theoretically, and numerically. In this study, we focus on the breakup of finite size liquid filaments on substrates, using direct numerical simulations. Although there are many parameters involved when determining whether a liquid filament breaks up, we illustrate the effects of three parameters: Ohnesorge number, the ratio of the viscous forces to inertial and surface tension surfaces, the liquid filament aspect ratio, and a measure of the fluid slip on the substrate, i.e. slip length. Through these parameters, we are able to determine whether a liquid filament breaks up into one or multiple droplets or collapse into a single droplet on the substrate. We compare our results with the results for free standing liquid filaments. We show that the presence of the substrate promotes breakup of the filament. We also discuss the effect of the degree of slip on the break up. We comprehensively explore the parameter domain regions when including the slip effects.

Two-phase flow in quasi 2D geometry is relevant to a number of applications, in particular related to the flow in porous media. This relevance serves as one important motivation for considering fluid instabillities in the so-called Hele-Shaw geometry (flow between two glass plates). The participating students have carried out experimental, theoretical, and computational study of flow stability in such geometry, focusing in particular on the setup where a less viscous fluid is injected into a more viscous one, in a setup known to lead to instability carrying the name of Saffman-Taylor. The aim of this project has been to study the instability when the more viscous fluid is non-Newtonian, with viscosity that depends on shear rate. To characterize the properties of the emerging patterns, the students have used several methods to calculate the fractal dimension based on the data collected from experimental trials and extensive simulations of diffusion limited aggregation type. Both experimental and computational results suggest that the fractal dimensions between Newtonian and non-Newtonian setups differ. This results, if confirmed, will be of relevance to further work in this field.

This project concentrated on the force networks that form as a system of photoelastic particles is compressed. Photoelasticity allows to visualize these force networks and to analyze them using a variety of techniques. In the project, the students carried out physical experiments and produced sets of images (an example is shown). The brightness of a given particle is at least approximately proportional to the total stress exerted. These images were then processed using image processing techniques to extract the information such as size, position, and stress of each particle.