Fractional calculus and neural networks team up to model memory-driven fluid flow

Engineers studying complex fluids have long struggled with a deceptively simple question: how does a material remember its own past? For viscoelastic fluids such as polymer melts, paints, slurries, and even blood, the answer matters enormously, because the stress inside the fluid at any instant depends on its entire deformation history. A new theoretical study published in Results in Engineering tackles this memory problem head-on by combining fractional calculus with a classical non-Newtonian fluid model, and then checking the resulting mathematics with an artificial neural network.

Jeffrey fluids are a well-established class of non-Newtonian materials whose viscosity depends on how the shear rate changes over time, making them useful stand-ins for many real industrial and biological flows. The team layered additional physics onto the base model: the fluid moves through a homogeneous, isotropic porous medium, thermal radiation is included, the fluid can generate or absorb heat volumetrically, and a first-order chemical reaction acts on the species concentration field. Buoyancy forces arising from both temperature and concentration gradients drive the motion, under the Boussinesq approximation.

Unlike classical calculus, which is strictly local in time, fractional operators weight the entire past history of a quantity, allowing the equations to encode hereditary and nonlocal temporal effects. The fractional order, denoted beta-one and restricted to values between zero and one, acts as a continuous dial for memory strength: lower values mean stronger memory and greater deviation from classical behavior, while as beta-one approaches one the equations smoothly revert to the familiar integer-order Jeffrey fluid model, a limiting case the authors verify explicitly.

To solve the coupled system of fractional momentum, energy, and concentration equations, the researchers applied Laplace transforms, deriving closed-form expressions for the transformed velocity, temperature, and concentration fields. Because the resulting expressions involve functions that do not invert neatly by hand, the physical-space solutions were obtained through a numerical inverse Laplace transform. The authors note that the analytical expressions can be written in terms of Mittag-Leffler functions, and the solutions were computed and visualized using MATLAB software across a range of dimensionless parameters including the Prandtl number, Schmidt number, magnetic parameter, radiation parameter, chemical reaction parameter, and the Jeffrey fluid parameters.

The parametric study yielded a detailed picture of how each physical effect shapes the flow. The velocity profile rises with increasing thermal Grashof number, solutal Grashof number, radiation parameter, and the relaxation-to-retardation time ratio, because each of these either strengthens buoyancy or enhances momentum transport. Increasing the Prandtl number suppresses velocity, as stronger viscous effects overwhelm thermal diffusion, and a higher Schmidt number likewise slows the flow by limiting mass diffusivity. The magnetic parameter reduces velocity through the retarding Lorentz force that arises when an electrically conducting fluid moves through a magnetic field, a hallmark of magnetohydrodynamic resistance. Higher permeability of the porous medium, conversely, reduces drag and accelerates the flow.

The fractional parameter itself proved most influential in the early stages of the motion. At a dimensionless time of 0.5, increasing beta-one from 0.1 to 0.7 raised the velocity near the boundary and slowed its decay away from the wall, reflecting a weakening of the memory effect as the model approaches classical behavior. By a dimensionless time of 1, the curves for different fractional orders had largely collapsed onto one another, indicating that the influence of temporal memory fades as the transient response matures. Temperature profiles increased with the heat absorption parameter, the radiation parameter, and beta-one, while concentration profiles fell with a stronger destructive chemical reaction and a higher Schmidt number but rose with the kernel parameter delta.

To quantify the difference between fractional and classical descriptions, the authors tabulated side-by-side velocity, temperature, and concentration values for both models across the spatial domain. The fractional-order solutions consistently fell below their integer-order counterparts at the sampled points, with the gap widening away from the wall. A separate comparison against a previous YAC-based study of a related problem showed substantial percentage differences in velocity profiles at larger distances from the plate, which the authors attribute to the richer set of physical effects and memory-driven behavior captured by the present formulation.

The second pillar of the study is a data-driven surrogate: an artificial neural network trained to reproduce the analytical solutions. The network uses a 2-20-20-1 architecture, taking the spatial coordinate and transformed time as its two inputs and predicting one physical quantity, with two hidden layers of twenty neurons each. Separate networks were trained for velocity, temperature, and concentration, using the Levenberg-Marquardt algorithm to minimize mean squared error. The data were split 70 percent for training, 15 percent for validation, and 15 percent for testing. The results were striking: regression coefficients of approximately 0.99999 for the velocity network and essentially 1 for the temperature and concentration networks, with mean squared errors on the order of ten to the minus ten for the latter two fields and maximum absolute errors for temperature and concentration around ten to the minus three.

The authors are careful about what this means. Because the neural network was trained exclusively on data generated from the analytical solutions, with no independent experimental or numerical data involved, it serves as a computational approximation and consistency check of the derived equations rather than a validation against physical reality. The study also relies on simplifying assumptions, including a common fractional order across all three transport fields, which the authors describe as a modeling choice rather than a mathematical necessity. They state plainly that further verification through independent physical or experimental observations is needed before the model can be applied to practical engineering problems.

Even so, the work sketches a promising template for future research. The authors suggest extending the fractional model to other geometries, additional non-Newtonian fluids, and microfluidic configurations, and adding effects such as velocity slip, variable viscosity, and particle suspensions. The neural network could be retrained with deep learning techniques or tested against independent benchmark data to assess its ability to extrapolate beyond its training domain. If fractional operators genuinely capture the memory-dependent character of viscoelastic transport better than classical equations, as this and prior studies argue, the combination of fractional modeling and machine-learning surrogates could become a standard computational toolkit for everything from polymer extrusion and electronics cooling to biomedical fluid transport, where knowing how a fluid remembers may ultimately determine how well we can predict, and control, its behavior.

Subject of Research: Fractional-order modeling of unsteady MHD Jeffrey fluid heat and mass transfer with artificial neural network prediction

Article Title: Fractional modeling of MHD Jeffrey fluid flow with Rabotnov memory kernel: Heat and mass transfer analysis and artificial neural network prediction

Article References: Prabu, D., Fatima Ayub, S., Bayram, M., Hafiz Ullah, R., Jan, R., Begaliev, A., & Abdullaeva, B. (2026). Fractional modeling of MHD Jeffrey fluid flow with Rabotnov memory kernel: Heat and mass transfer analysis and artificial neural network prediction. Results in Engineering, 32, Article 113739. https://doi.org/10.1016/j.rineng.2026.113739

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113739

Keywords: Jeffrey fluid, fractional calculus, YAC operator, Rabotnov kernel, magnetohydrodynamics, heat transfer, mass transfer, porous medium, artificial neural network, Laplace transform, non-Newtonian fluid, thermal radiation