From the period Fall 2012 to Spring 2016, I was the Math Capstone Lab (link) assistant at New Jersey Institute of Technology (NJIT). During this time, I have focused on the Saffman-Taylor instability (link) in the Hele-Shaw cell (HS) (link). In this two-semester course, we introduced students to this instability by injecting water into the center a HS cell filled with a thin layer of glycerol, a solved problem. We then moved on to experiments with non-Newtonian fluids and filled HS cells with nematic liquid crystals (NLC) (link) and a polyethylene oxide solution (PEO) (link).
The goals of the experimental portion of this course are to:
Below is a list of NJIT undergraduates that have participated in the Capstone Lab course and have submitted posters the annual Frontiers in Applied & Computation Mathematics (FACM) conference at NJIT (link). All credits of the images and videos related to the Capstone Lab shown on my website are attributed to the students below and the exact credits may be found in the linked FACM posters.
Ibin Abdul-Hakeem, Jimmie Adriazola, Andres Alban, Hardik Darji, Zouhair Draben, Nick Hale, Jacob Moorman, Armando Rosa, Enkhsanaa Sommers, Thomas Tu, Bryan Valerio
Allen Cameron, Xizhi Cao, Antonio Jurko, Lucas Lamb, Paul Lorenz, Daniel Meldrim, Eric Motta, Alexander Pinho, Julia Porrino, Andrea Roeser, Fremy Santana, Chen Shu, Enkhsanaa Sommers, Sarp Uslu
E. Guerino, Y. Othman, M. Petretta, M. Sanghavi
The purpose of the Hele-Shaw cell is to confine a liquid between two thinly spaced plates. The length scale of the spacing (z direction) is much smaller than the width and height of the HS cell (x and y directions). Therefore, the thin film approximation (link) is applied the Navier-Stokes (NS) equation (link), simplifying the spatial dimensions of the NS equations from three (x,y,z) to two (x,y). Using linear stability analysis, it can be shown that a circular interface between two fluids is unstable (Saffman-Taylor instability) if a less viscous fluid (e.g. water) is forced (injected) into a more viscous fluid (e.g. honey, glycerol). In the case of non-Newtonian fluids, the viscosity anisotropic.
To the right is a diagram of the edge of the Hele-Shaw cell.
On the left is a diagram of the entire experimental setup.
In our experiments, there are two control parameters, the cell spacing, and the weight placed onto the upright syringe (controlled injection). Altering these parameters will lead to a different number of fingers developing during the experiments. The volume of the injecting fluid is fixed across all experiments.
In the four videos below, colored water (green) is injected into a HS cell filled with glycerol (a much more viscous fluid). These videos demonstrate the classical Saffman-Taylor instability. More glycerol experiments may be found here.
Cell Spacing: 350 micrometer, Weight: 500 grams
Cell Spacing: 400 micrometer, Weight: 500 grams
Cell Spacing: 750 micrometer, Weight: 400 grams
Cell Spacing: 850 micrometer, Weight: 800 grams
In the two videos below, colored water (red) is injected into a HS cell filled with a polyethylene oxide solution (green), a shear thinning (link) fluid. Shear-thinning fluids are a class of non-Newtonian fluids whose viscosity decreases under shear stress, which leads to longer and thinner fingers developing in our experiments. More polyethylene oxide solution experiments may be found here.
Cell Spacing: 400 micrometer, Weight: 500 grams
Cell Spacing: 880 micrometer, Weight: 700 grams
Special acknowledgment goes to Mykhailo Pevnyi (link) and Dr. Peter Palffy-Muhoray (link) at the Liquid Crystal Institute (link) at Kent State University (link). For without their assistance, our experiments with NLC would not be possible.
NLC molecules are typically rod-like with a dipole moment along its long axis. The dipole moments prefer to be in a uniform state and induce an elastic response in the bulk of the film when deformed. At an interface, there is a preferred orientation (boundary condition) often called the anchoring condition. Here the inner surface of the HS cell is treated with polyvinyl alcohol as to induce planar anchoring i.e. the long axis of NLC molecule lies in the plane of the surface. The remaining degree of freedom in the surface anchoring is fixed by rubbing the surface with a felt cloth, forcing NLC molecules to orient themselves in the direction of rubbing. Depending on the orientation of the plates (direction of rubbing), either a uniform state or a twisted nematic (link) state can be enforced on the orientation of NLC molecules.
Applying an electrical field across the HS cell, NLC molecules attempt to align their long axis parallel to the plates. By controlling the potential difference across the HS cell, we hoped to control the orientation of NLC molecules and thus the viscosity. Before the steps in the previous paragraph are performed, thin sheets coated with indium tin oxide (ITO) are attached to the plate surfaces facing the interior of the HS cell. By offsetting the two plates, electrical leads are connected to the plates, with little risk of tripping the circuit.
Below is an example of an early experiment of injecting air into a HS cell filled with 4-Cyano-4'-pentylbiphenyl (5CB) (link), a type of NLC. The red curve denotes the NLC-air interface.