New Jersey Institute of Technology
Department of Mathematical Sciences

Capstone Laboratory

Diffusion Limited Aggregation and Saffman-Taylor Instability in Non-Newtonian Hele-Shaw Flow

Supported by NSF Grant NO. DMS-1211713; PI: Cummings; Co-PI: Kondic

Instructor: Lou Kondic

Lab Assistant: Michael-Angelo Lam

Experimental Setup

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.

  • A) Plexiglass plates making up the top and bottom of HS cell.
  • B) Thin spacer to separate HS plates (A). Spacer thickness is controlled by gluing together multiple strips of transparency paper. A micrometer is used to measure the total thickness accurately.
  • C) A white sheet of paper wrapped in wax paper. Provides proper contrast for video capture.

Diagram of edge if Hele-Shaw Cell

Photo viewing bottom of Hele-Shaw cell

On the left is a photo viewing the Hele-Shaw cell from the bottom.

  • D) The inner fluid contained by HS cell plates (A).
  • E) Injection port on the bottom plate.
  • F) A ruler to get lengthscale of the experiment for imaging processing.

On the left is a diagram of the entire experimental setup.

  • G) Hele-Shaw cell.
  • H) Stands to elevate HS cell. Done to allow injection of inner fluid from the bottom of the HS cell. Removes tube (J) from obstructing camera view (K).
  • I) The syringe used to control injection rate of the inner fluid into HS cell.
  • J) A tube attached to the syringe (J) and tipped with a blunt fill needle. The needle is placed into injection port (E) of the HS cell.
  • K) Mount for camera.

Photo of experimental setup

Photo of complete experimental setup

Photo of experimental setup

Experimental Results

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.

Glycerol Experiment

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

PEO Experiment

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

NLC Experiment

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.

Photo of NLC experiment