### Date:

**Speaker: **Pavan Pranjivan Mehta, SISSA, International School of Advanced Studies, Italy

**Time : **16:00 - 17:00 CEST (Rome/Paris)

**Hosted at: **SISSA, International School of Advanced Studies, Trieste, Italy

**YouTube** : https://youtu.be/d-sPRoG1Tm0

**Organizers :** Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

*Keywords: fractional Navier–Stokes equations, fractional Cauchy equations, fractional continuity equation, fractional Reynolds-averaged Navier–Stokes equations, turbulence*

**Abstract: **Turbulence is a non-local and multi-scale phenomenon. Resolving all scales implies non-locality is addressed implicitly. However, if spatially or temporally averaged fields are considered for computational feasibility, then addressing non-locality explicitly becomes important as a result of missing information of all scales.

Our previous work involved constructing fractional closure model [1,2] for Reynolds-averaged Navier–Stokes equations, which are temporally averaged. In [2] ”fractional stress-strain hypothesis” was introduced using a a variable-order (spatially-dependent) Caputo fractional derivative. Indeed, it addresses the amalgamation of local and non-local effects [1, 2]. The results of two-sided model [2] were very encouraging, where, we find a power-law behaviour of fractional order akin to logarithmic regime for velocity and also a law of wake. Further, we investigated tempered fractional definitions in [2]. Since turbulence is a decay process, it led to define a ”horizon of non-local interactions” in [2].

Despite the success, one overwhelming question remains, ”how do we derive a fractional conservation law from first principles?” Thus, in this talk, I shall introduce the recently developed control volume approach in [3] to derive fractional conservation law from first principles. The first step was to extend the fractional vector calculus developed in [4] to two-sided operators [3]. Subsequently, derive the fractional analogue of Reynolds transport theorem [3]. By virtue of this theorem, we derive the ”fractional continuity” and ”fractional Cauchy equations”, which follows conservation of mass and momentum, respectively [3]. The stress tensor of fractional Cauchy equation is treated with a fractional stress-strain relationship (which was developed in [2]) to get to the final form of ”fractional Navier–Stokes equations”.

**Biography: **Pavan Pranjivan Mehta is a PhD student within the mathematics area of SISSA, Italy. He hold’s two Master’s degree, namely, Thermal and Fluid Engineering from University of Manchester, UK and Applied Math from Brown University, USA; with an undergrad in Aeronautical Engineering. He has held research positions in France, USA, UK and India; with an internship at Airbus Group and visiting researcher at Newton Institute, Cambridge for two scientific programs, namely, turbulence and fractional differential equations. Pavan's organisational activities for fractional calculus, includes mini-symposiums at prestigious conferences and seminar series, such as JINX seminar’s at Newton Institute and weekly seminars at SISSA. His current research interests are non-local turbulence modeling and numerical methods for fractional PDE.

**Bibliography**

[1] Mehta, P. P., Pang, G., Song, F. and Karniadakis, G. E. (2019).Discovering a universal variable-order fractional model for turbulent couette flow using a physics-informed neural network. Fractional calculus and applied analysis, 22(6), 1675–1688.

[2] Pranjivan Mehta, P. (2023).Fractional and tempered fractional models for Reynolds- averaged Navier–Stokes equations. Journal of Turbulence, 24(11-12), 507–553.

[3] Pranjivan Mehta, P. (2024). Fractional vector calculus and fractional Navier-Stokes equations. (in preparation).

[4] Tarasov, V. E. (2008). Fractional vector calculus and fractional Maxwell’s equations. Annals of Physics, 323(11), 2756-2778

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