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The ternary hybrid nanofluids have potential in different technological arenas such as biomedical engineering, solar energy, atomic reactors, the automotive industry, and heat pipes. Given these facts, along with the recent advancements in nanotechnology and their extensive applications, this research focuses on the MoS₂-Fe₃O₄-ZrO₂/CH₃OH ternary nanofluid flow through bidirectional stretching sheets. We have transformed the coupled nonlinear partial differential equations for the advanced model into nondimensional ordinary differential equations using similarity transformations, and then semi-analytically apply the homotopy analysis methodology (HAM). We have displayed the physical features of potential factors graphically alongside the flowing factors based on velocity and temperature. We presented a physical evaluation in tabular format for the rate of heat transmission and compared the results with existing work to ensure their validity. These meaningful outcomes indicate that the axial fluid velocity is compressed by the magnetic interaction, inertial drag, porosity and stretchable ratio, while it is augmented by the Powell-Eyring factor and the changed Hartmann value. The effect of increasing transverse speed boosts inertial drag.

Novel nanomaterial applications claim distinct uses in thermal engineering, cooling processes, heat transfer devices, and automobile industries, among others. Motivated research uses modified heat and mass flux theories to present thermal observations for the unsteady flow of magnetized Maxwell nanofluid, confined by porous bidirectionally stretched surfaces. The heat transfer model’s extension is based on Joule heating and heat source effects. The Cattaneo–Christov theories govern the expansion of mass and heat transfer. We analyze thermal problems under zero-mass diffusion constraints. The use of proper variables simplifies mathematical modeling into a dimensionless form. The Homotopy Analysis Method (HAM) solves the dimensionless system. The paper highlights the convergence criteria for the HAM procedure. Graphics underline the problem’s physical perspective. We observe that the Deborah number enhances heat and mass transfer. The temperature profile decreases when the parameter becomes unstable.

The theme of the current effort is to theoretically analyze the entropy generation and heat transfer aspects of Casson nanofluid flow triggered by rotating porous disc with the presence of magnetic dipole, nonlinear thermal radiation, viscous dissipation and Joule heating. The modeling of the nanofluid can be described with the combination of Brownian motion and thermophoresis by incorporating the passive control boundaries, and the governing PDEs are transformed into a set of highly nonlinear ODEs. The resulting equations are then solved analytically using HAM technique. The present results are compared with previously published results, which are in excellent agreement. The effect of pertinent nondimensional parameters on the entropy generation, hydrodynamic, heat and mass transport aspects is discussed via graphical illustrations. Both radial and tangential velocities are affected by accelerating the values of Hartmann number and porosity parameter. The temperature profile is upsurged by improving the radiation and thermal ratio parameter. Increasing the Casson parameter and Brinkman number leads to improved entropy generation rate. Moreover, skin friction, heat and mass transfer rates are examined with the help of the tables. It is believed that this study can be utilized as coolants by numerous automotive and engineering industries, namely the electronic devices, electrical motor, spin coating, fabrication of spacecraft, thermal insulation, nuclear reactors, etc.

Due to its remarkable thermodynamic development in various engineering sciences, the dispersion of nanostructures in classical base liquids is receiving greater attention from researchers and scientists. In light of the aforementioned inspirations, the suggested research is associated with using a water-based ternary-hybrid nanofluid that contains three unique nanostructures, silicon dioxide, titanium dioxide and aluminum oxide to optimize the heating process in science and engineering. In this situation, the influence of a chemically reactive magneto-hydrodynamics (MHD) Darcy–Forchheimer ternary hybrid composites transport involving nonlinear radiation across an exponentially permeable stretched surface is highlighted with the model’s physical initial and boundary constraints. The simulation of the heat expression also includes the impacts of nonlinear thermal radiations, energy supply, energy dissipation and magnetic force. Furthermore, the interpretation of Buongiorno’s theory involves spontaneous and thermophoresis diffusions. Entropy analysis is explained using the second law of thermodynamics. We converted the model framework for fluid movement, energy and mass transfer (together with ternary hybrid nanofluid characteristics) into self-similar nondimensional differential equations, that were then further analytically evaluated by using the homotopy asymptotic method (HAM) technique via Mathematica software. The acquired results are illustrated numerically and graphically to study the behavior of fluid flow, heat, concentration distribution, drag force, rate of heat and mass transfer for various emergent components to gain scientific comprehension. With the growing values of magnetic, porosity and Forchheimer number, the nature of the velocity profile goes down.

The present investigation computes the heat transport phenomenon of the magnetohydrodynamic (MHD) flow of CuO-Ag/H₂O hybrid nanofluid over a spinning disc. The authors are confident that there is very less analysis covering the fluid flow containing silver and copper oxide nanoparticles over a rotating disk. Therefore, the authors are interested to consider the water-based nanoliquid flow over a spinning disk. Furthermore, the velocity slip and thermal convective conditions are taken into consideration. The formulation of the problem is made in the form of PDEs and is then converted into the nonlinear ODEs by employing suitable similarity transformations. The homotopic analysis approach is applied for the semi-analytical solution of these resulting equations. The convergence of homotopic approach has also revealed with the help of figure. The performance of the hybrid nanofluid flow velocities and temperature has been shown in a graphical form against distinct flow parameters. Also, the numerical results of skin friction coefficient and Nusselt number have been calculated in a tabular form. The outcomes of the current problem show that the increase in the skin friction of the water-based copper oxide nanofluid is greater than the water-based silver nanofluid at 4% of the nanoparticle volume fraction. Also, the skin friction of the hybrid nanofluid is increased by 8% compared to the silver nanofluid at 4% of the nanoparticle volume fraction. Furthermore, the heat transfer rate of the water-based copper oxide nanofluid is greater than the water-based silver nanofluid at 4% of the nanoparticle volume fraction. Also, the heat transfer rate of the hybrid nanofluid is 52% greater than that of silver nanofluid at 4% of the nanoparticle volume fraction. It is found that the Nusselt number of the hybrid nanofluid is highly affected by the embedded parameters as compared to nanofluids.

In this research work, we will carry out analytical investigation of MgO–CuO∖H₂O, hybrid nanofluid magnetohydrodynamics (MHD) stagnation point flow with the influence of viscous dissipation for enhancement of heat transfer ratio. The flow system takes into account the impact viscous dissipation and transport dependence on the shape factor. Furthermore, velocity and temperature at the stretching surface are also considered in this study. To convert a collection of PDEs to nonlinear ODEs, we applied appropriate transformations. We utilize the Homotopy analysis method (HAM) to solve this set of equations. A physical description is used to simulate and evaluate the structures of flow features such as velocity, skin friction, Nusselt number and temperature profiles in response to changes in developing factors, the effects of different factors on temperature and velocities are shown in the form of graphs. It is hoped that this theoretical approach would contribute positively to improve the heat transformation ratio to satisfy the demands of the manufacturing and engineering sectors.

This study examines the analytical study of magnetic hydrodynamic stagnation point flow with the impact variable viscosity on a movable surface along with the impact of thermal radiation. The problem is modeled with the help of momentum and energy conservation laws in the form of NLPDEs. The novelty of this study is the combined impact of variable viscosity and thermal radiation with the analytical method. Aluminum oxide nanoparticles and water are used as base fluids in this research work. The authors applied appropriate transformations to convert a collection of dimension forms of NLPDEs to dimensionless forms of NODEs. The transformed NODEs are solved with the help of an approximate analytical method known as the HAM. The effects of different parameters, including electric field, magnetic field, stagnation point flow, thermal radiation PN, and EN on energy and momentum profiles intended, and the results are planned with the help of graphs.

This study examines the effects of viscous dissipation, thermal radiation, nanofluid over a stretched surface, and viscous dissipation on a two-dimensional couple stress blood base for the enhancement of heat transfer rate. Gold and multiwall carbon nanotubes are two forms of nanoparticles that are taken into consideration, with blood serving as the base fluid. The NLPDE controls the considering problem. The NLPDE was converted to NODEs using the mentioned similarity transformation. The analytical method known as HAM was used to analyze the transform NODE analytically. Graphs are used to illustrate the effects of many parameters, such as magnetic factors, nanoparticle volume friction, velocity power index, PN, thermal radiation factors, and EN, which are derived from TE and VE. The current research work highlights how important it is to include viscous dissipation in nanofluid dynamics. The results show complex interactions among stretching, thermal properties, and micro-scale effects. The results may have an impact on the development and enhancement of biomedical devices and treatments that use nanofluidic systems, especially those that deal with blood.

The swift advancement of heat transfer technologies can be attributed to the growing need for effective heating and cooling systems in various sectors, including the automotive, chemical, and aerospace industries. This work aims to examine the impact of radiation on the behavior of Casson hybrid nanoparticles (Al₂O₃-CuO) mixed convective flow in three distinct scenarios. The physical properties of copper oxide (CuO) and aluminum oxide (Al₂O₃) nanoparticles are utilized when mixed with CMC-water as the solvent. This paper aims to analyze the influence of mixed convective flow on the thermal integrity of hybrid nanoparticles when subjected to a wedge, cone, and plate. The analysis of chemical reactions and the existence of a permeable substance is also incorporated. The partial differential systems are appropriately transformed into a system of ordinary differential equations (ODEs). In addition, the calculation of this system of ODEs is carried out using the analytical technique known as the homotopy analysis approach (HAM). The study examines potential resolutions for flow issues in three distinct configurations: wedge, cone, and plate. A comprehensive examination and record of the impacts of various physical characteristics is carried out. The concepts of wall friction, Nusselt number, and Sherwood number, among others, are explained through the utilization of graphical representations. The porosity and Casson fluid characteristics cause a decrease in the performance of the velocity profile. Hybrid nanofluids have superior heat transfer efficiency compared to conventional nanofluids.

In this paper, quadratic Riccati differential equation of fractional order has been solved by employing the optimal homotopy asymptotic method (Optimal HAM) with application to random processes, optimal control and diffusion problems. Optimal HAM uses simple computations with quite acceptable approximate solutions which have close agreement with exact solutions as compared to other techniques. To illustrate the efficiency and reliability of the method, some examples are provided and the results are discussed with tables.

In this paper, an innovative form of nanofluids is identified as tri-hybrid nanofluid, which is synthesized by dispersing three or more varieties of nanomaterials in the considered base fluid. So, in this study, we comparatively examined SiO₂/H₂O nanofluid, TiO₂+Al₂O₃/H₂O hybrid nanofluid and SiO${}_{2+}$TiO₂+Al₂O₃/H₂O ternary hybrid nanofluid. Stretching of the flat surface enables us to develop the nanofluids flow. Additional considerations include the impacts of MHD, viscid dissipation, nonlinear thermal convection and radiation, joule heating and the presence of a heat source. For transforming PDEs (continuity, motion, heat equation and boundary constraints) into ODEs, an appropriate transformation procedure is used. HAM technique is used to solve these nonlinear coupled ODEs. Graphs are used to evaluate and examine the effect of numerous describing variables on nano, hybrid and tri-hybrid nanofluids speed and heat distribution. Furthermore, the computed values of engineering-relevant parameters ($C_f$ and Nu) are tabulated and analyzed. The velocity of nanofluids acquires enhancing tendency for nonlinear thermal and mix convection parameter, but reverse upshot is assured due to nanoparticle volume fraction, Weissenberg number and magnetic parameters. Thermal field gets intensified in nature for magnetic and Eckert number, heat generation, thermal radiation and nanoparticles volume fractions. The ternary hybrid nanofluid has the most efficient behavior according to the comparative examination of ternary, hybrid and nanofluids.

This paper explores the bioconvective Maxwell fluid flow over a horizontal stretching sheet. The Maxwell fluid flow is considered in the presence of gyrotactic microorganisms. The velocity slips and convection conditions are used in this investigation. Additionally, the Cattaneo–Christov heat and mass flux model, Brownian motion, thermophoresis, and activation energy are employed in the flow problem. The model formulation has been transferred to a dimension-free format using similarity variables and solved by the homotopy analysis approach. Figures have been sketched to depict the HAM convergence. The consequences of this study are that the velocity of Maxwell fluid flow reduces for higher Hartmann number, buoyancy ratio factor, and bioconvective Rayleigh number, whereas the increasing behavior in velocity profile is seen against Deborah number. The thermal characteristics of the Maxwell fluid flow diminish with developing values of the thermal relaxation factor and Prandtl number, while augmenting with the increasing Brownian motion, thermal and concentration Biot numbers and thermophoresis factor. The rate of thermal transmission of the Maxwell fluid flow enhances with the increasing Prandtl number, and mixed convective factor, while diminishing with the increasing buoyancy ratio factor, thermophoresis factor and Brownian motion factor.

This paper discusses the problem of fingero-imbibition phenomenon with magnetic field effect. The problem arises in two-phase flow through homogeneous porous medium during secondary oil recovery process. The mathematical formulation gives us one-dimensional nonlinear partial differential equation. Homotopy analysis method is adopted to solve the governing equation. The solution is discussed numerically and graphically with the help of Mathematica BVPh package.