For any fluid flow that can be written in terms of a dispersion relation, disturbances to the flow may be classified as either convective or absolute. Convective instabilities grow either completely downstream or upstream, leaving behind a non-growing wave. Absolute instabilities grow outward from the origin of the disturbance, eventually contaminating all regions of the domain. The distinction between convective and absolute instability is important to manufacturing process such as coating (where convective instability is desirable) and atomization of sprays (where absolute instability is desirable). The prediction of such instabilities is determined by examining the dispersion relation in the complex wave-number plane and identifying saddle points, whose associated growth-rates determine how fast the disturbance is growing and whether it is absolute or convective.
Our work is motivated by the need to characterize the stability of systems that are important to the sustainable operation of processes that have exacting tolerances. The aim is to develop a simple method to predict long-time algebraic growth in linear systems, a type of instability that has been largely unexamined in the prior literature. To accomplish this task, the Fourier-Laplace integral solutions of disparate classes of partial differential equations (PDEs) are examined for structural commonality via nonstandard long-time asymptotic methods.
Power series arise in virtually all applications of mathematical physics. However, limitations generally inherent to power series solutions often inhibit their direct use. For instance, a Taylor series representation of an unknown function may not converge, as it may have a finite radius of convergence arising from singularities (often complex) in the function it represents. Even when singularities are not a concern, higher-order terms of the series may be exceedingly difficult to compute, which is especially problematic if the series converges slowly. While “re-summation” methods such as Pade Approximants typically lead to an implementation improvement compared with the original series, global accuracy is not always guaranteed and the problem becomes one of choosing the ‘ best’ re-summation technique.
Our goal is to identify relevant problems that can be solved using an asymptotic approximant: a function whose Taylor expansion (about x=ax=a) matches the exact Taylor expansion to some order and whose limit as x→bx→b matches some known asymptotic behavior. Divergent, truncated, and/or slowly converging series can be replaced by asymptotic approximants if some behavior is known away from the series expansion point. Nonlinear ODEs that require the "shooting method" are well suited for asymptotic approximants, applied to the (typically divergent) power series solution to the ODE. So far, we have successfully applied asymptotic approximants to find closed-form analytic solutions to problems in thermodynamics, fluid mechanics, and astrophysics.
Below are 4 examples of how asymptotic approximants (denoted A in the figures) can be used to bridge the behavior between two physical regions. In all figures, symbols represent numerical solutions. top left: 2nd, 4th, and 6th order approximants, whose zero-density expansion matches the square-well virial series (V) to 2nd, 4th, and 6th order, while also limiting to non-classical scaling behavior near the thermodynamic critical point (Barlow et al, JCP, 2015). top right: Approximant matches known weak (W)- and strong (S)-field limits for the bending angle of a Kerr black hole (Barlow, Weinstein, & Faber, CQG, 2017). bottom left: Approximant bridges the series solution (S) of the Blasius ODE (describing boundary layer flow over a plate) with its far-field behavior (Barlow et. al., QJMAM, 2017). bottom right:Approximant bridges the series solution of the Flierl-Petviashvilli ODE (describing vortex solitons like Jupiter's red spot) with the far field behavior (Barlow et. al., QJMAM, 2017).
Barlow, Nathaniel S, Brian T Helenbrook, and Sung P Lin. 2009. “A Numerical Investigation Of The Stability Of Expanding Liquid Sheets And The Influence Of Boundary Conditions”. Computers & Fluids 38. Computers & Fluids: 552–563.
Byrne, C. J, Steven J Weinstein, and P. H Steen. 2006. “Capillary Stability Limits For Liquid Metal In Melt Spinning”. Chem. Eng. Sci. 61. Chem. Eng. Sci.: 8004-8009.
Jiang, W. Y, B. T Helenbrook, S. P Lin, and Steven J Weinstein. 2005. “Low Reynolds Number Instabilities In Three-Layer Flow Down An Inclined Wall”. J. Fluid Mech. 539. J. Fluid Mech.: 387-416.
Weinstein, Steven J, and K. J Ruschak. 2004. “Coating Flows”. Ann. Rev. Fluid Mech. 36. Ann. Rev. Fluid Mech.: 29-53.
Finnicum, D. S, Steven J Weinstein, and K. J Ruschak. 1993. “The Effect Of Applied Pressure On The Shape Of A Two-Dimensional Liquid Curtain Falling Under The Influence Of Gravity.”. J. Fluid Mech. 255. J. Fluid Mech.: 647-665.
Weinstein, Steven J, J. M Baumlin, and J. Servant. 1993. “The Propagation Of Surface Waves In Flow Down An Oscillating Inclined Plane”. Aiche J. 39 (7). Aiche J.: 1113-1123.
Weinstein, Steven J, E. B Dussan, and L. H Unger. 1990. “A Theoretical Study Of Two-Phase Flow Through A Narrow Gap With A Moving Contact Line: Viscous Fingering In A Hele-Shaw Cell”. J. Fluid Mech. 221. J. Fluid Mech.: 53-76.
Weinstein, Steven J. 1990. “Wave Propagation In The Flow Of Shear-Thinning Fluids Down An Incline”. Aiche J. 36 (12). Aiche J.: 1873-1889.
Convectively unstable wave due to an oscillating point source.
Exponentially growing absolute instability (dimension into the page is time).
Algebraically decaying wave (dimension into the page is time).
For more info on the PDE for the two models above, see: Barlow, Nathaniel S, B. T Helenbrook, S. P Lin, and S. J Weinstein. 2010. “An Interpretation Of Absolutely And Convectively Unstable Waves Using Series Solutions”. Wave Motion 47: 564-582.
Algebraically growing absolute instability (dimension into the page is time).
For more info on the PDE for the above model, see: King, Kristina R, Steven J Weinstein, Paula M Zaretzky, Michael Cromer, and Nathaniel S Barlow. 2016. “Stability Of Algebraically Unstable Dispersive Flows”. Phys. Rev. Fluids 1 (7).
The pressure-temperature-density diagram of a square-well fluid at its critical region and above, showing the correct non-classical scaling from all directions.
The equation of state used to produce the above model is given in: Barlow, Nathaniel S, Andrew J Schultz, Steven J Weinstein, and David A Kofke. 2015. “Communication: Analytic Continuation Of The Virial Series Through The Critical Point Using Parametric Approximants”. The Journal Of Chemical Physics 143 (071103).
10/29/18: Logan Melican, 3rd Year Math & ChemE major - won the Third Place award for the National Student Paper Competition at the National AIChE conference in Pittsburgh for his talk: “Application of Asymptotic Approximants to Fluid Equations of State.” . 9 students who won from respective regional competitions around the country competed. Logan did his work in collaboration with Stephen Kronenberger, 3rd Year ChemE major, Nate Barlow, and Steve Weinstein.
10/1/18: Ryne Beachley (B.S. Applied Math, 2018) and Morgan Mistysyn (Industrial & Systems Engineering, 4th year BS/ME student) were co-authors on a recent paper that enhanced our predictive ability for light deflection around black holes. The study began with Ryne during an NSF REU in the summer of 2017 at RIT, advised by Dr. Josh Faber. Ryne generated new coefficients for the weak-field limit of a Kerr black hole, and began exploring an extended parameter space that ultimately enabled the full trajectory of light around the black hole to be determined in closed-form. During the Fall of 2017, Drs. Nate Barlow and Steve Weinstein applied their method of asymptotic approximants to construct the closed-form expression, accurate to any order far from the black hole but only zeroth order in the limit as the black hole's critical radius (the radius below which light is absorbed) is approached. In the Spring of 2018, Morgan joined the team and developed a novel integral expansion method that enabled the approximant to retain accuracy to any order in this limit. The resulting closed-form expression is on the order of 10 times faster than alternative numerically-based methods and is thus a significant advance. The aricle can be found here.
6/04/18: This year, Nate is the recipient of two awards, Richard and Virginia Eisenhart Provost’s Award for Excellence in Teaching and the Innovative Teaching with Technology Award. Read the full story here. photo credit: A. Sue Weisler
4/7/18: Logan Melican (2nd year APPMTH/ CHME-BS) & Stephen Kronenberger (2nd year CHME-BS) won the student presentation competition at the AIChE (American Institute of Chemical Engineers) Eckhardt Northeast Student Regional Conference!
The work involved the use of series expansions and the Gamma function to predict critical exponents that govern fluid phase behavior.
12/8/17: Math undergraduate Meaghan Hoitt presented her work, "Development towards a methodology for predicting long-time algebraic growth in linear wave equations" at the SIAM Conference on Analysis of Partial Differential Equations, held in Baltimore. The research was done over the summer, as part of Meaghan's Summer Undergraduate Research Fellowship.
5/19/17: Math graduating senior Kristina King was awarded two research scholar awards - one from the School of Mathematical Sciences and one from the College of Science, for her publication in Physical Review Fluids, "Stability of algebraically unstable dispersive flows". Kristina will be entering the Math PhD program at University of Virginia in the Fall. Photo taken by Tina Williams.
5/7/17: "Kimberlee Keithley is the 2017 winner of the Norman A. Miles Scholarship, which is given to a student maintaining a perfect 4.0 GPA. Going into her final year at RIT, she is pursuing two bachelor’s degrees: One in chemical engineering, another in applied mathematics. As part of the scholarship recognition, she shared the award with faculty-researcher and her mentor, Steven Weinstein. He is the department head of chemical engineering in RIT’s Kate Gleason College of Engineering." - Michelle Cometa, University News. Photo taken by A. Sue Weisler.
4/6/17: Nicole Hill received an Outstanding Undergraduate Scholar Award for maintaining a high GPA while pursuing research. Her work resulted in a paper recently accepted to the Quarterly Journal of Mechanics and Applied Mathematics.