Fractional Dynamics in Non-Newtonian Fluid Flow

Fractional dynamics has emerged as a vital framework for capturing history-dependent and spatially non-local effects in non-Newtonian fluid flow. By replacing conventional integer-order derivatives with fractional operators, researchers can model anomalous transport, long-range interactions and memory effects that are intrinsic to complex fluids such as polymer solutions, suspensions and nanofluids. Fractional formulations extend classical constitutive relations—Maxwell, Burgers’ and Kelvin–Voigt models—by introducing fractional derivatives in time or space, thereby enabling more accurate representation of stress relaxation, transient diffusion and boundary-layer development. This approach finds applications across enhanced oil recovery, microfluidic devices, cooling technologies and biomedical flows, where deviations from Newtonian behaviour and finite propagation speeds of thermal or mechanical disturbances play a decisive role. Advances in analytical and semi-analytical methods, coupled with efficient numerical schemes, have opened the way to predictive simulations of flows under magnetic fields, oscillatory forcing or electrokinetic driving, with direct implications for industrial design and fundamental understanding of complex fluid mechanics.

Research from Nature Portfolio

Recent studies have elucidated how fractional operators modulate both momentum and heat transfer in Maxwell-type nanofluids. One investigation examined mineral-oil-based Maxwell nanofluids laden with multi-walled carbon nanotubes, revealing that increasing the order of the time-fractional derivative weakens the viscoelastic memory and alters the velocity profile, while nanoparticles enhance thermal conductivity and cooling performance in electrical systems. In a complementary work, the space-fractional form of Fourier’s law was applied to hybrid nanofluids flowing along a permeable plate under an inclined magnetic field. It was shown that increasing the fractional order elevates the Nusselt number and thickens the thermal boundary layer, and that hybrid nanoparticle dispersions yield higher temperatures compared with single-component nanofluids, offering design guidance for magnetic-field-assisted thermal management.

Fractional Dynamics in Non-Newtonian Fluid Flow publication trend

The graph below shows the total number of articles in fractional dynamics in non-newtonian fluid flow across all publications each year (not limited to Nature Index journals).

Technical terms

Fractional derivative: A generalisation of differentiation to non-integer orders, capturing material memory and spatial non-locality.

Maxwell fluid: A viscoelastic constitutive model with a spring and dashpot in series, representing fluid relaxation behaviour.

Burgers’ fluid: A combined Maxwell and Kelvin–Voigt model describing both stress relaxation and retardation in complex fluids.

Kelvin–Voigt viscoelastic fluid: A constitutive model with spring and dashpot in parallel, capturing immediate elastic and delayed viscous responses.

Caputo–Fabrizio derivative: A non-singular fractional operator with an exponential memory kernel, suited to modelling transient anomalous processes.

Electroosmotic flow: Fluid motion induced by an applied electric field acting on the charged double layer at a solid–liquid interface.

References

1. Interaction of multi-walled carbon nanotubes in mineral oil based Maxwell nanofluid. Scientific Reports (2022).
2. Space-fractional heat transfer analysis of hybrid nanofluid along a permeable plate considering inclined magnetic field. Scientific Reports (2022).
3. Heat Transfer in MHD Flow of Maxwell Fluid via Fractional Cattaneo-Friedrich Model: A Finite Difference Approach. Computers Materials & Continua (2020).
4. Electroosmotic flow of generalized Burgers’ fluid with Caputo–Fabrizio derivatives through a vertical annulus with heat transfer. Alexandria Engineering Journal (2020).
5. Magnetohydrodynamics flow and heat transfer of novel generalized Kelvin–Voigt viscoelastic nanofluids over a moving plate. Physics of Fluids (2024).