Recent advancements have seen the integration of nanocomposites, composed of clay minerals and polymers, into cementitious materials to enhance their mechanical properties. This investigation focuses on the dynamics of clay-based cementitious nanofluids along a vertical plate, adopting a Jeffrey fluid model to encompass various phenomena. The effects of a first-order chemical reaction and heat generation/absorption are considered, alongside slip velocity and Newtonian heating conditions. The governing equations, represented as partially coupled partial differential equations, have been extended using a constant proportional Caputo (CPC) fractional derivative. Exact solutions were derived employing the Laplace transform technique. A detailed graphical analysis was conducted to elucidate the influence of pertinent flow parameters on the velocity, temperature, and concentration profiles. It was observed that the incorporation of clay nanoparticles results in a reduction of the fluid's heat transfer rate by 10.17%, and a decrease in the mass transfer rate by 1.31% at a nanoparticle volume fraction of 0.04. These findings underscore the nuanced role of nanoparticle concentration in modifying fluid dynamics under the studied conditions, providing a validated and precise understanding of nanofluid behavior in construction-related applications. This research not only supports the potential of nanotechnology in improving cementitious materials but also contributes to the broader field of fluid mechanics by integrating complex heating and slip conditions into the study of nanoparticle-enhanced fluids.

Nanofluids have improved thermophysical properties compared to conventional fluids, which makes them promising successors in fluid technology. The use of nanofluids enables optimal thermal efficiency to be achieved by introducing a minimal concentration of nanoparticles that are stably suspended in conventional fluids. The use of nanofluids in technology and industry is steadily increasing due to their effective implementation. The improved thermophysical properties of nanofluids have a significant impact on their effectiveness in convection phenomena. The technology is not yet complete at this point; binary and ternary nanofluids are currently being used to improve the performance of conventional fluids. Therefore, this work aims to theoretically investigate the ternary nanofluid flow of a couple stress fluid in a vertical channel. A homogeneous suspension of alumina, cuprous oxide, and titania nanoparticles is formed by dispersing trihybridized nanoparticles in a base fluid (water). The effects of pressure gradient and viscous dissipation are also considered in the analysis. The classical ternary nanofluid model with couple stress was generalized using the fractal–fractional derivative (FFD) operator. The Crank–Nicolson technique helped to discretize the generalized model, which was then solved using computer tools. To investigate the properties of the fluid flow and the distribution of thermal energy in the fluid, numerical methods were used to calculate the solution, which was then plotted as a function of various physical factors. The graphical results show that at a volume fraction of 0.04 (corresponding to 4% of the base fluid), the heat transfer rate of the ternary nanofluid flow increases significantly compared to the binary and unary nanofluid flows.

Fractal-fractional derivatives are more general than the fractional derivative and classical derivative in terms of order. Fractal-fractional derivative is used in those models where the classical continuum hypothesis theory fails. More precisely, these derivative operators are used where the surface or space is discontinuous, e.g., porous medium. Fractal-fractional derivative is considered advance tool to analyze the fluid dynamic model more than fractional and classical model. Given the extensive applicability of fractal-fractional derivatives, the current analysis focuses on investigating the behavior of a non-linear Walter’s-B fluid model under the influence of time-varying temperature and concentration During the simulation process, we have also taken into account the effects of first-order chemical reactions, Soret numbers, thermal radiation, Joule heating, and viscous dissipation of energy. A magnetic field with a strength of B0 was applied to the left plate in the transverse direction. The classical mathematical model was first developed using relative constitutive equations and later generalized with the fractal-fractional derivative operator. Numerical solutions to the generalized model have been obtained using the finite difference method. Various graphs are drawn from the obtained numerical solutions to study the influence of physical parameters on the rheology of Walter’s-B fluid. It has been observed that by varying the fractional and fractal order of the generalized model, one can easily derive fractal, fractional, and classical models.

This article aims to examine Jeffery nanofluid with joint effects of mass and heat transfer in a horizontal channel. The classical model is transferred to the Caputo fractional model by using the generalized Fourier’s and Fick’s laws. The nanofluids are formed by dispersing two different nanoparticles, silver and copper, into a based fluid. A novel transformation has been applied to the mass and energy equation and then solved by using the sine Fourier and the Laplace transformation jointly. The exact solution is given in terms of a special function, that is, the Mittag-Leffler function. The Sherwood number and Nusselt number are calculated and displayed in the tabular form. The effect of embedded parameters on the velocity, concentration, and temperature profile is discussed graphically. It is noted that the heat transfer rate of EO is improved by 28.24% when the volume fraction of Ag nanoparticles is raised from 0.00 to 0.04.

The use of fractal–fractional derivatives has attracted considerable interest in the analysis of chaotic and nonlinear systems as they provide a unique capability to represent complex dynamics that cannot be fully described by integer-order derivatives. The fractal–fractional derivative with a power law kernel is used in this paper as an analytical tool to analyze the dynamics of a chaotic integrated circuit. Using coupled ordinary differential equations of classical order, the complexity of an integrated circuit is modeled. The classical order model is generalized via fractal–fractional derivatives of the power law kernel. Moreover, this paper is concerned with investigating the Ulam stability of the model and conducting theoretical studies in order to analyze equilibrium points, identify unique solutions, and verify the existence of such solutions. By examining the complex dynamics that result in chaotic behavior, these investigations shed light on the fundamental properties of integrated circuits. For the purpose of exploring the non-linear fractal–fractional order system, a numerical algorithm has been developed to facilitate our analysis. MATLAB software has been used to implement this algorithm, making it possible to carry out detailed simulations. Simulating solutions are accomplished using 2D and 3D portraits, which provide visual and graphical representations of the results. Throughout the simulation phase, particular attention is given to the impact of fractional order parameter and fractal dimension. As a result of this study, we have gained a comprehensive understanding of the behavior of the system and its response to variations in values.