Mathematics Colloquium 02/03/23

Abstract:  Flow down a wide inclined channel such as a canal or dam spillway is typically modeled by the inclined Saint Venant equations for shallow water flow, a system of hyperbolic balance laws w_t + f(w)_x + g(w)_y= r(w) in the form of a relaxation system.  Related in structure to combustion models, and in various asymptotic limits to KdV and Kuramoto-Sivashinsky equations, these provide fascinating examples of pattern formation in quasilinear hyperbolic flow.  In particular, they give rise to dramatic 1d hydraulic shocks or bores and to ``rogue'' periodic patterns known as roll waves, which can reach heights several times that of a smooth laminar flow with equivalent fluid throughput, both relevant to design and safety issues in hydraulic engineering.

The 1d existence and stability theory for these waves has a long and fascinating history, now essentially complete.  Here, we initiate a new chapter with a first study of transverse, or 2d stability of these waves, with the twin goals to (i) refine the 1d stability picture and thus the description of sustainable rogue waves, and (ii) to look for steady transverse bifurcation to genuinely multi-d waves, in particular the familiar crosshatch or ``herringbone'' flow readily found in conditions when roll waves are observed.  We indeed find a much narrower zone of stability when transverse effects are included; and, as hoped, we observe transition to herringbone flow.  However, curiously, the transition is for hydraulic shocks rather than roll waves and of a previously unknown ​``2d essential bifurcation'' type.  There do occur also transverse bifurcations of roll waves, but these are of a quite different ``Floquet-Hopf'' type that we have dubbed ``flapping waves,'' and so far as we know have not previously been observed.

Biosketch: Prof. Zumbrun received his PhD degree in 1990 from the Courant Institute at New York University. He was a visiting professor and NSF postdoctoral fellow at SUNY Stony Brook and Stanford University before joining the IU Bloomington in 1992. He was promoted to associate professor in 1996 and professor in 1999. During 2006-2008 he served as director of graduate studies for the Math Department and from 2009-2014 he served as chairperson. Among others, he was a recipient of the Navy Young Investigator Award (1994), elected as an AMS fellow (class of 2014) and SIAM fellow (class of 2017) for “his contributions in traveling wave stability and his exceptional mentoring of graduate students and post-doctoral researchers”. In 2020, he received the IU Bicentennial Medal.

For more information, please contact:  Dr. Vu Thai Luan (luan@math.msstate.edu).