Fractional Navier-Stokes equations from first principles

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. Thus it is natural to introduce a fractional or a non-local operator. However, an 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. 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”.

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).