Introduction: Recent research has demonstrated that nanofluids exhibit a nonlinear relationship between their shear stress and shear rates due to the differing particle characteristics. Some complex classical mechanical models have been employed to describe the rheological properties of nanofluids. Given the heterogeneity of nanofluids, anomalous diffusion, Brownian motion-induced nano-convection, nanoparticle clustering, and thermophoresis often occur, leading to a large disparity between simulation results from classical mathematical models and experimental measurements. The traditional rheological constitutive equations may not be suitable for describing these anomalous phenomena; however, they can be effectively addressed using fractional calculus.
Methods: Justification for the use of fractional calculus in nanofluid modelling is discussed from the viscoelastic mechanical viewpoint. The well-established models, including Newtonian, Maxwell, Kelvin-Voigt, and anti-Zener models, are summarised and generalised to the fractional setting. Several classes of convolution kernel functions are derived, which are suitable for describing the rheological behaviour of viscoelastic nanofluids. The effects of using different fractional operators in the constitutive equation of nanofluids are also investigated.
Results: It is found that the Maxwell constitutive equation with the Caputo-Fabrizio or Atangana-Baleanu-Caputo derivative is a special case of the generalised constitutive model with the fractional Riemann-Liouville derivative. Validation of the fractional Maxwell constitutive model against experimental data of nanofluids demonstrates that the proposed fractional constitutive formulation can effectively capture the rheology of non-Newtonian nanofluids exhibiting shear-thinning type curves. Based on a novel development of the nonlocal constitutive relationship, a unifying generalised nanofluid model incorporating rheological and heat relaxation is proposed.
Conclusions: This study provides useful insights into treating nanofluids as a class of viscoelastic fluids using fractional calculus and highlights the potential of employing nonlocal mathematical models to simulate nanofluid flow in real applications.
