# Statistical Hydrodynamics

*Classical Statistical Field*in two contexts: In Turbulence where large magnitude, strongly correlated fluctuations abound, and in Interfacial (Marangoni) flows where the Non-equilibrium behavior of spreading surfactants and their nonlinear interaction with bulk fluids over which they spread opens up a diverse range of phenomena and questions.

**Turbulence:** Thermodynamics describes the macroscopic properties of equilibrium systems which are largely independent of microscopic details. No such description exists for driven, dissipative (non-equilibrium) systems. The high spatio-temporal character of non-equilibrium fluctuations and correlations strikes at the heart of the difference between the equilibrium and non-equilibrium regimes as they relate directly to detailed balance, or lack thereof. Fluids present a model system where the differences between the equilibrium and non-equilibrium regimes are brought out in stark relief. The state and properties of a quiescent fluid in thermal equilibrium are accurately described by Equilibrium Statistical Physics. A mathematically sophisticated theory also successfully describes the transition of a fluid (technically, a liquid) to other phases/states of matter and Linear response theory successfully captures fluid behavior when nominally removed from thermal equilibrium. However, when the fluid is driven far from thermal equilibrium into a turbulent regime characterized by strongly nonlinear and non-equilibrium behavior, the system exhibits intense and often, intermittent fluctuations with strong spatio-temporal correlations in the form of eddies, vortices or other coherent structures. Neither Fluid Mechanics nor Non-equilibrium Statistical Physics offer satisfactory theoretical explanation of turbulent flows and their intense fluctuations; it is no surprise this oldest, unsolved problem in all Classical Physics is infamous for being the *Graveyard of theory.* The existence of these fluctuations automatically implies most measured quantities in turbulent flows hold statistics of their own. Furthermore, extended spatio-temporal correlations indicate these statistics are usually Non-Gaussian in character. Understanding turbulence as a problem is of purely intellectual interest, but the statistics, spectra, and dynamics of various turbulent quantities are also relevant in the natural and engineered realms. Our studies have been geared towards understanding energy flux fluctuations, entropy production rate and its fluctuations, the statistics of injected power etc.

- MM Bandi, WI Goldburg, JR Cressman Jr. and A Pumir, "Energy flux fluctuations in a finite volume of turbulent flow",
*Phys. Rev. E.***73**, 026308 (2006). - MM Bandi, WI Goldburg and JR Cressman Jr., "Measurement of the Entropy Production Rate in compressible turbulence",
*Europhys. Lett.***76**, 595 (2006). - MM Bandi, JR Cressman Jr. and WI Goldburg, "Test of the Fluctuation Relation in Lagrangian turbulence on a free surface",
*J. Stat. Phys.***130,**27 (2008). - MM Bandi and CP Connaughton, "Craig's XY-distribution and the statistics of Lagrangian power in two-dimensional turbulence",
*Phys. Rev. E.***77**, 036318 (2008). - MM Bandi, SG Chumakov and CP Connaughton, "Probability distribution of power fluctuations in turbulence",
*Phys. Rev. E.***79**, 016309 (2009). - J Larkin, MM Bandi, WI Goldburg and A Pumir, "Power-law distribution of particle concentration in free-surface flows",
*Phys. Rev. E.***80**, 066301 (2009). - J Larkin, WI Goldburg and MM Bandi, "Time evolution of a fractal distribution: particle concentrations in free-surface turbulence",
*Physica D***239**, 1264 (2010). - F Paraz and MM Bandi, "Second order structure functions for higher powers of turbulence velocity, J. Phys.: Condens. Matter DOI: 10.1088/1361-648X/ab38ca (2019).

**Interfacial Dynamics:** Surface tension is conceptually simple to understand, and its effects are easily observed with readily available materials in a kitchen. Yet, its quantitative analysis, both theoretical and experimental, is fraught with difficulties, particularly when it involves Marangoni flows driven by surface tension gradients. Our other interest in interfacial flows has to do with flowing soap films, which simultaneously represent both a quasi two-dimensional fluid as well as an elastic membrane in the same system.

- MM Bandi, JR Cressman Jr., WI Goldburg, H Kellay, "Where surface physics and fluid dynamics meet: Rupture of an amphiphile layer at the air-water interface",
*J. Chem. Phys.***124**, 104701 (2006). - MM Bandi, T Tallinen and L Mahadevan, "Shock-driven jamming and periodic fracture of particulate rafts",
*Europhys. Lett.***96**, 36008 (2011). - MM Bandi, A Concha, R Wood and L Mahadevan, "A pendulum in a flowing soap film",
*Phys. Fluids.***25**, 041702 (2013). - C Peco, W Chen, Y Liu, MM Bandi , JE Dolbow, and E Fried, "Influence of surface tension in the surfactant-driven fracture of closely-packed particulate monolayers"
*Soft Matter***13**, 5832 (2017). - S Basu, A Yawar, A Concha and MM Bandi, "On angled bounce-off impact of a drop impinging on a flowing soap film"
*Fluid Dyn. Res.***49**, 065509 (2017). - MM Bandi, VS Akella, DK Singh, RS Singh and S Mandre, "Hydrodynamic signatures of stationary Marangoni-driven surfactant transport"
*Phys. Rev. Lett.***119**, 264501 (2017). - VS Akella, DK Singh, S Mandre and MM Bandi, "Dynamics of a camphoric acid boat at the air-water interface"
*Phys. Lett. A***382**, 1176 (2018). - MM Bandi, "Tension grips the flow"
*J. Fluid Mech.*846, 1 (2018).

Additionally, we take occasional diversions to perform fun experiments with theorist colleagues or explore biological fluid dynamics as outlined in the Quantitative Life Sciences section. In coming years, we plan to explore a different avenue at the intersection of thermodynamics and fluids, *viz.* phase change phenomena involving adsorption, condensation, and evaporation.

- M Roper, A Seminara, MM Bandi, A Cobb, HR Dillard and A Pringle, "Dispersal of fungal spores in a cooperatively generated wind",
*Proc. Nat. Acad. Sci.***107**, 104747 (2010). - Thierry Savin, MM Bandi and L Mahadevan, "Pressure-driven occlusive flow of a confined red blood cell",
*Soft Matter***12**, 562 (2015). - Ravi Singh, MM Bandi, Amala Mahadevan and Shreyas Mandre, "Linear stability analysis for
*monami*in a submerged sea grass bed",*J. Fluid. Mech*.**786**, R1