Stability Analysis

A stable system will retain its initial behavior.  However, in an unstable system minor fluctuations in the system conditions can grow to dominate the system behavior.

For example, a vertically standing pencil is physically realizable, but a small fluctuation in the surrounding air pressure would topple it; the pencil is unstable.  Instabilities occur often in nature and industry alike.

The Kelvin-Helmholtz instability: An aesthetically pleasing unstable behavior seen in traditional fluids.

Click to play the numerical simulation of two fluids with opposing motion.*

The shear at the interface leads to an unstable, turbulent flow.








The Kelvin-Helmholtz instability generates internal ocean waves. The same process also creates wave patterns in clouds on windy days. 









Instabilities also occur in granular systems. Perturbations in particle velocity, number concentration, and pressure can grow to create clusters of particles as seen below.


Assessment of continuum models through their predictions of clustering instabilities

A system of elastic particle (i.e. a molecular gas) will remain with homogenous positions and a Gaussian distribution of velocity through time. Inelastic particles have more interesting behavior because the system in unstable to small perturbations. In the inelastic case, velocity alignment and regions of high number density, known as clusters, can develop even when beginning with homogeneous initial conditions.

These clusters are observed in industrial systems, such as fluidized beds and coal gasifiers and have a strong impact on process performance and efficiency. The clustering instability is qualitatively predicted through molecular dynamics simulations as well as the stability analysis and transient solutions of the continuum balances for granular materials. However, the quantitative ability of continuum models to predict clustering instabilities has not been assessed.


Peter Mitrano


Click to play simulation of clustering instability







Impact of Particle Size Distribution on Cluster Characteristics

Investigate experimentally impact of polydispersity on cluster characteristics in risers of circulating fluidized beds. 

Collaborators: PSRI


Relevant Papers:

Rice, R. B. and C. M. Hrenya , “Clustering in rapid granular flows of binary and continuous particle size distributions,” Physical Review E, 81, art. no. 021302 (2010). 


Rice, R. B. and C. M. Hrenya , “Characterization of clusters in rapid granular flows,” Physical Review E, 79, art. no. 021304 (2009). 


Past Students: 

Brent Rice, Jia Chew, Mike Pacella, Daniel Cromer, Andrew Hilger 


*This simulation was not produced by the Hrenya Research Group