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Whilst a modest number of investigations have been undertaken concerning nanofluids (NFs), the exploration of fluid flow under exponentially stretching velocities using NFs remains comparatively uncharted territory. This work presents a distinctive contribution through the comprehensive examination of heat and mass transfer phenomena in the NF ND–Cu/H₂O under the influence of an exponentially stretching velocity. Moreover, the investigation delves into the intriguing interplay of gyrotactic microorganisms and convective boundary conditions within the system.

Similarity transformations have been used on PDEs to convert them into dimensionless ODEs. The solution is derived by using the homotopy analysis method (HAM). The pictorial notations have been prepared for sundry flow parameters. Furthermore, some engineering quantities are calculated in terms of the density of motile microbes, Nusselt and Sherwood numbers and skin friction, which are presented in tabular form.

The mixed convection effect associated with the combined effect of the buoyancy ratio, bioconvection Rayleigh constant and the resistivity due to the magnetization property gives rise to attenuating the velocity distribution significantly in the case of hybrid nanoliquid. The parameters involved in the profile of motile microorganisms attenuate the profile significantly.

The current simulations have uncovered fascinating discoveries about how metallic NFs behave near a stretched surface. These insights give us valuable information about the characteristics of the boundary layer close to the surface under exponential stretching.

The novelty of the current investigation is the analysis of NF ND–Cu/H₂O along with an exponentially stretching velocity in a system with gyrotactic microorganisms. The investigation of fluid flow at an exponentially stretching velocity using NFs is still relatively unexplored.

This study aims to examine the flow of unsteady mixed convective hybrid nanofluid over a rotating sphere with heat generation/absorption. The hybrid nanofluid contains different shapes of nanoparticles (copper [Cu] and aluminium oxide [Al₂O₃]) in the base fluid (water [H₂O]). The influence of different shapes (sphere, brick, cylinder, platelets and blades) of nanoparticle in water-based hybrid nanofluid is also investigated.

To analyse the nanomaterial, the flow model is established, and in doing so, the Prandtl’s boundary layer theory is incorporated into the present model. The bvp4c approach, i.e. finite difference method, is used to find the numerical solution of differential equations that is controlling the fluid flow. The effect of relevant flow parameters on nanofluid temperature and velocity profile is demonstrated in detailed explanations using graphs and bar charts, whereas numerical results for Nusselt number and the skin’s coefficient for various form parameters are presented in tabular form.

The rate of heat transfer is least for spherical-shaped nanoparticle because of its smoothness, symmetricity and isotropic behaviour. The rate of heat transfer is highest for blade-shaped nanoparticles as compared to other shapes (brick, cylindrical and platelet) of nanoparticles because the blade-shaped nanoparticles causes comparatively more turbulence flow in the nanofluid than other shapes of nanoparticle. Heat generation affects the temperature distribution and, hence, the particle deposition rate. The absorption of heat extracts heat and reduce the temperature across the rotating sphere. The heat generation/absorption parameter plays an important role in establishing and maintaining the temperature around the rotating sphere.

The numerical study is valid with the exception of the fluctuation in density that results in the buoyancy force and the functional axisymmetric nanofluid transport has constant thermophysical characteristics. In addition, this investigation is also constrained by the assumptions that there is no viscosity dissipation, no surface slippage and no chemically activated species. The hybrid nanofluid Al₂O₃–Cu/H₂O is an incompressible and diluted suspension. The single-phase hybrid nanofluid model is considered in which the relative velocity of water (H₂O) and hybrid nanoparticles (Al₂O₃–Cu) is the same and they are in a state of thermal equilibrium.

Study on convective flow across a revolving sphere has its applications found in electrolysis management, polymer deposition, medication transfer, cooling of spinning machinery segments, spin-stabilized missiles and other industrial and technical applications.

The originality of the study is to investigate the effect of shape factor on the flow of electrically conducting hybrid nanofluid past a rotating sphere with heat generation/absorption and magnetic field. The results are validated and provide extremely positive balance with the recognised articles. The results of the study provide many appealing applications that merit further study of the problem.

Thermophoresis deposition of particles is a crucial stage in the spread of microparticles over temperature gradients and is significant for aerosol and electrical technologies. To track changes in mass deposition, the effect of particle thermophoresis is therefore seen in a mixed convective flow of Williamson hybrid nanofluids upon a stretching/shrinking sheet.

The PDEs are transformed into ordinary differential equations (ODEs) using the similarity technique and then the bvp4c solver is employed for the altered transformed equations. The main factors influencing the heat, mass and flow profiles are displayed graphically.

The findings imply that the larger effects of the thermophoretic parameter cause the mass transfer rate to drop for both solutions. In addition, the suggested hybrid nanoparticles significantly increase the heat transfer rate in both outcomes. Hybrid nanoparticles work well for producing the most energy possible. They are essential in causing the flow to accelerate at a high pace.

The consistent results of this analysis have the potential to boost the competence of thermal energy systems.

It has not yet been attempted to incorporate hybrid nanofluids and thermophoretic particle deposition impact across a vertical stretching/shrinking sheet subject to double-diffusive mixed convection flow in a Williamson model. The numerical method has been validated by comparing the generated numerical results with the published work.

The purpose of this study is to analyze the impact of inclined magnetohydrodynamics (MHD) and thermal radiation on the flow of a ternary micropolar nanofluid on a sheet that is expanding and contracting while applying mass transpiration and velocity slip conditions to the flow. The nanofluid, which is composed of Au, Ag and Cu nanoparticles dispersed in water as the base fluid, possesses critical properties for increasing the heat transfer rate and is frequently used in manufacturing and industrial establishments.

The set of governing nonlinear partial differential equations is transformed into a set of nonlinear ordinary differential equations. The outcome of this differential equation is solved and obtained the closed-form solution and energy equation in the form of hypergeometric functions.

The velocity, micro-rotation and temperature field are investigated versus a parametric variation. The physical domains of mass suction or injection and micropolar characteristics play an important role in specifying the presence, singleness and multiplanes of exact solutions. In addition, many nondimensional characteristics of the profiles of temperature, angular velocity and velocity profiles are graphically shown with substantial consequences. Furthermore, adding nanoparticles increases the heat transfer rate of the fluid used in manufacturing and industrial establishments. The current findings may be used for better oil recovery procedures, smart materials such as magnetorheological fluids, targeted medicine administration and increased heat transmission. Concerning environmental cleanup, nanomaterial fabrication and biomedical devices, demonstrate their potential influence in a variety of disciplines.

The originality of this paper is to analyze the impact of inclined MHD at an angle with the ternary nanofluid on a micropolar fluid over an expanding and contracting sheet with thermal radiation effect.

Understanding the mechanical and thermal behavior of materials is the goal of the branch of study known as fractional thermoelasticity, which blends fractional calculus with thermoelasticity. It accounts for the fact that heat transfer and deformation are non-local processes that depend on long-term memory. The sphere is free of external stresses and rotates around one of its radial axes at a constant rate. The coupled system equations are solved using the Laplace transform. The outcomes showed that the viscoelastic deformation and thermal stresses increased with the value of the fractional order coefficients.

The results obtained are considered good because they indicate that the approach or model under examination shows robust performance and produces accurate or reliable results that are consistent with the corresponding literature.

This study introduces a proposed viscoelastic photoelastic heat transfer model based on the Moore-Gibson-Thompson framework, accompanied by the incorporation of a new fractional derivative operator. In deriving this model, the recently proposed Caputo proportional fractional derivative was considered. This work also sheds light on how thermoelastic materials transfer light energy and how plasmas interact with viscoelasticity. The derived model was used to consider the behavior of a solid semiconductor sphere immersed in a magnetic field and subjected to a sudden change in temperature.

This study introduces a proposed viscoelastic photoelastic heat transfer model based on the Moore-Gibson-Thompson framework, accompanied by the incorporation of a new fractional derivative operator. In deriving this model, the recently proposed Caputo proportional fractional derivative was considered. This work also sheds light on how thermoelastic materials transfer light energy and how plasmas interact with viscoelasticity. The derived model was used to consider the behavior of a solid semiconductor sphere immersed in a magnetic field and subjected to a sudden change in temperature.

Enzymes play a pivotal role in orchestrating essential biochemical processes and influencing various cellular activities in tissue. This paper aims to provide the process of enzyme diffusion within the tissue matrix and enhance the nano system performance by means of the effectiveness of enzymatic functions. The diffusion phenomena are also documented, providing chemical insights into the complex processes governing enzyme movement.

A computational analysis is used to develop and simulate an optimal control model using numerical algorithms, systematically regulating enzyme concentrations within the tissue scaffold.

The accompanying videographic footages offer detailed insights into the dynamic complexity of the system, enriching the reader’s understanding. This comprehensive exploration not only contributes valuable knowledge to the field but also advances computational analysis in tissue engineering and biomimetic systems. The work is linked to biomolecular structures and dynamics, offering a detailed understanding of how these elements influence enzymatic functions, ultimately bridging the gap between theoretical insights and practical implications.

A computational predictive model for nanozyme that describes the reaction diffusion dynamics process with enzyme catalysts is yet not available in existing literature.

This study aims to model the important flow response quantities over a shrinking wedge with the help of response surface methodology (RSM) and an artificial neural network (ANN). An ANN simulation for optimal thermal transport of incompressible viscous fluid under the impact of the magnetic effect (MHD) over a shrinking wedge with sensitivity analysis and optimization with RSM has yet not been investigated. This effort is devoted to filling the gap in existing literature.

A statistical experimental design is a setup with RSM using a central composite design (CCD). This setup involves the combination of values of input parameters such as porosity, shrinking and magnetic effect. The responses of skin friction coefficient and Nusselt number are required against each parameter combination of the experimental design, which is computed by solving the simplified form of the governing equations using bvp4c (a built-in technique in MATLAB). An empirical model for Cf_x and Nu_x using RSM and ANN adopting the Levenberg–Marquardt algorithm based on trained neural networks (LMA-TNN) is attained. The empirical model for skin friction coefficient and Nusselt number using RSM has 99.96% and 99.99% coefficients of determination, respectively.

The values of these matrices show the goodness of fit for these quantities. The authors compared the results obtained from bvp4c, RSM and ANN and found them all to be in good agreement. A sensitivity analysis is performed, which shows that Cf_x as well as Nu_x are most affected by porosity. However, they are least affected by magnetic parameters.

This study aims to simulate ANN and sensitivity analysis for optimal thermal transport of magnetic viscous fluid over shrinking wedge.

The particle distribution in a fluid is mostly not homogeneous. The inhomogeneous dispersion of solid particles affects the velocity profile as well as the heat transfer of fluid. This study aims to highlight the effects of varying density of particles in a fluid. The fluid flows through a wavy curved passage under an applied magnetic field. Heat transfer is discussed with variable thermal conductivity.

The mathematical model of the problem consists of coupled differential equations, simplified using stream functions. The results of the time flow rate for fluid and solid granules have been derived numerically.

The fluid and dust particle velocity profiles are being presented graphically to analyze the effects of density of solid particles, magnetohydrodynamics, curvature and slip parameters. Heat transfer analysis is also performed for magnetic parameter, density of dust particles, variable thermal conductivity, slip parameter and curvature. As the number of particles in the fluid increases, heat conduction becomes slow through the fluid. Increase in temperature distribution is noticed as variable thermal conductivity parameter grows. The discussion of variable thermal conductivity is of great concern as many biological treatments and optimization of thermal energy storage system’s performance require precise measurement of a heat transfer fluid’s thermal conductivity.

This study of heat transfer with inhomogeneous distribution of the particles in a fluid has not yet been reported.