The only application I know is "Non-Fickian Transport" and "Anomalous Diffusion" in porous media (groundwater contamination) or polymer flows or very high gradients of concentration or heat.

The other application I have seen is turbulence subjected to a magnetic field .

Also meandering of rivers, in geomorphology can be described in fractional derivative

Examples to applications of fractional calculus (i.e. derivatives, integrals) in fluid dynamics:

- Two phases fluid flows in porous medium (e.g. instability phenomenon called fingering which occurs at the water injection into oil saturated porous medium),

- Turbulence free shear flows in natural and engineering environments,

- Hydrodynamic instabilities (e.g. the Rayleigh-Taylor, the Kelvin-Helmholtz and other)

Regards

The field of fractional calculus has a wide range of popularity among researchers due its robust and precise modelling of problems that integer order calculus cannot handle. There are many applications of fractional differential equations of studies can be found in recent time, in spite of the long aged contributions of Niels Henrik Abel in 1823 who was considered the “father of the complete fractional-order calculus framework” [1]. Fractional differential equations have been used for modelling problems that are related to anomalous diffusion in the oil drilling sector [2], tilt control in rail vehicles [3], romantic and interpersonal relationships [4], and in financial economics [5].

[1] Podlubny, I.; Magin, M.; Trymorush, I. Historical survey Niels Henrik Abel and the birth of fractional calculus. Fract. Calc. Appl. Anal. 2017, 20, 1068–1075, doi:10.1515/fca-2017-0057.

[2]. Sergei, F.; Vladimir, C.; Toshiyuki, H. Application of Fractional Differential Equations for Modeling the Anomalous Diffusion of Contaminant from Fracture into Porous Rock Matrix with Bordering Alteration Zone. Transp. Porous Media 2010, 81, 187–205, doi:10.1007/s11242-009-9393-2.

[3]. Hassan, F.; Zolotas, A. Impact of fractional order methods on optimized tilt control for rail vehicles. Fract. Calc. Appl. Anal. 2017, 20, 765–789, doi:10.1515/fca-2017-0039.

[4]Singh, J.; Kumar, D.; AlQurashi, M.; Baleanu, D. A Novel Numerical Approach for a Nonlinear Fractional Dynamical Model of Interpersonal and Romantic Relationships. Entropy 2017, 19, 375, doi:10.3390/e19070375.

[ 5]. Fallahgoul, H.A.; Fabozzi, S.M.; Frank, F.J. Fractional partial differential equation and option pricing. In Fractional Calculus and Fractional Processes with Applications to Financial Economics; Fallahgoul, H.A., Focardi, S.M., Fabozzi, F.J., Eds.; Academic Press: Cambridge, MA, USA, 2017; pp. 59–80, doi:10.1016/B978-0-12-804248-9.50006-1.

The fractional derivative describes the influence of viscous and thermal losses on pressure wave propagation in tubes at high shear numbers, when the bulk of the flow is frictionless. It is in fact a convolution integral describing the memory of the boundary layers. See for example W. Chester (

• Volume 18, Issue 1
• January 1964 , pp. 44-64

Resonant oscillations in closed tubes

• W. Chester (a1)
  • https://doi.org/10.1017/S0022112064000040)

Following!

Most of the fluid are not linear in Nature and Fractional order derivative are very suitable to handle such kinds of fluids. Further, the impact of the thermophysical properties of nanofluids are more visible using the fractional model rather than the classical one.

Fractional calculus is applicable to the solution of time-dependent, viscous diffusion fluid mechanic problems.