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Nanofluid is a specially crafted fluid comprising a pure fluid with dispersed nanometer-sized particles. Incorporation these nanoparticles into pure fluid results in a fluid with improved thermal properties in comparison of pure fluid. The enhanced properties of nanofluids make them highly sought after, in diverse applications, consisting of coolant of devices, heat exchangers, and thermal solar systems. In this study hybrid nanofluid consisting of copper, alumina and titanium nanoparticles on a curved sheet has investigated with impact of chemical reactivity, magnetic field and Joule heating. The leading equations have converted to normal equations by using appropriate set of variables and has then evaluated by homotopy analysis method. The outcomes are shown through Figures and Tables and are discussed physically. It has revealed in this study that Cu-nanofluid flow has augmented velocity, temperature, and volume fraction distributions than those of Al 2 O 3 -nanofluid and TiO 2 -nanofluid. Also, the Cu-nanofluid flow has higher heat and mass transfer rates than those of Al 2 O 3 -nanofluid and TiO 2 -nanofluid., (© 2024. The Author(s).)

The fluid flow over an extending sheet has many applications in different fields which include, manufacturing processes, coating, thin film decomposition, heat and mass transfer, biomedical applications, aerospace engineering, environmental science, energy production. Keeping in mind these applications, the non-Newtonian hybrid nanofluid flow comprising of Cu and CuO nanoparticles over an extending sheet is analyzed in this work. Two different base fluids called kerosene oil and water have been incorporated. The sheet is considered to be thermally convective along with zero mass flux condition. The main equations of modeled problem have been transformed to dimensionless form by using similarity variables. The designed problem is evaluated computationally by using bvp4c Matlab function. Validation of the present results is also performed. The impacts of magnetic, Brownian motion, chemical reaction, suction and thermophoresis factors are analyzed and discussed in details. The outcomes of the present investigation declare that the kerosene oil-based hybrid nanofluid flow has greater velocity and concentration profiles than that of the water-based hybrid nanofluid flow. The water-based hybrid nanofluid has greater temperature distribution than that of kerosene oil-based hybrid nanofluid flow. The streamlines of the kerosene oil-based Newtonian and non-Newtonian hybrid nanofluid flows are more stretched than water-based Newtonian and non-Newtonian hybrid nanofluid flows., (© 2024. The Author(s).)

This article presents the two-dimensional flow of hybrid nanofluid comprising of gyrotactic microorganisms under the consequences of multiple slip conditions, magnetic field and thermal radiation across an elongating curved surface using porous media. The nanoparticles of TiO 2 and Fe 3 O 4 have dispersed in water for composition of hybrid nanofluid. Main equations of the problem are converted to ODEs by using an appropriate set of variables. Solution of the present model is determined with the help of bvp4c technique, which is explained in detail in the coming section. Validation of the current results is done versus the published work. The effects of various emerging factors on flow distributions have been considered and explained. Additionally, the slips conditions are incorporated to analyze various flow distributions. The present outcomes show that the rising magnetic factor lessens the velocity profile, whereas rises the temperature profile. The curvature factor has supported both temperature and velocity distributions. Growth in velocity, thermal, concentration, and microorganisms slip factors reduce the corresponding distributions. The greater impact of the embedded parameters is found on hybrid nanofluid flow when matched to nanofluid flow., (© 2023. The Author(s).)

The current article aims to examine the magnetohydrodynamics (MHD) impact on the flow of MgO-Ag/water-based hybrid nanoliquid with motile microorganisms and the fluid is allowed to flow over a Riga plate subject to slip effects and activation energy. Furthermore, the presence of a uniform heat source/sink is also addressed in the energy equation. In addition to this, the thermophoresis effect is highlighted in the concentration equation. From the present proposed model, we get a non-linear system of the governing equations. The obtained system of partial differential equations (PDEs) is converted to the dimensionless system of ordinary differential equations (ODEs) using the similarity transformation. The obtained high non-linear system of equations has been solved numerically, using the parametric continuation method (PCM). In the present analysis, the main motivation is to highlight the heat transfer rate of MgO-Ag/water-based hybrid nanofluid flow over a Riga plate. The second motivation of the present research is to highlight the impact of slip conditions on the velocity, energy, and mass profiles. From the graphical analysis, it is depicted that the slip conditions reduce the velocity, energy, and mass outlines. From the present analysis, we concluded that volume friction reduced the flow profile while increasing the temperature of the fluid flow over a Riga plate. All the parameters of the present research are highlighted in velocity temperature and concertation of the fluid. In addition to this in all the figures we have compared the hybrid nanofluid with mono nanofluid and the also the comparison between slip and no-slip conditions have carried out through graphs for velocity, temperature, and concentration., (© 2023. Springer Nature Limited.)

A stratified flow may be seen regularly in a number of significant industrial operations. For instance, the stratified flow regime is typically used by gas-condensate pipelines. Clearly, only a limited set of working situations for which this flow arrangement is stable allow for the achievement of the stratified two-phase flow zone. In this paper, the authors are considered the laminar, steady and incompressible magnetohydrodynamic flow of a non-Newtonian Casson fluid flow past a stratified extending sheet. The features of bio-convection, Brownian motion, thermal radiation thermophoresis, heat source, and chemically reactive activation energy have been employed. The set of equations administered flow of fluid is converted into ordinary differential equation by suitable variables. A semi-analytical investigation of the present analysis is performed with homotopy analysis method. Endorsement of the current results with previous results is also investigated. The outcomes showed that the velocity distribution of the fluid flow lessens with higher Casson and magnetic factors. The temperature profiles of fluid flow shrinkage as the Prandtl number and Casson factor increase and enlarges with higher values of thermal radiation, magnetic, and Brownian motion factors. It is found that the growing thermophoretic and Brownian motion factors reduce the rate of thermal flow of the Casson fluid flow. In contrast, the increasing thermal stratification parameter increases the thermal flow rate of fluid., (© 2023. The Author(s).)

Entropy creation by a blood-hybrid nanofluid flow with gold-tantalum nanoparticles in a tilted cylindrical artery with composite stenosis under the influence of Joule heating, body acceleration, and thermal radiation is the focus of this research. Using the Sisko fluid model, the non-Newtonian behaviour of blood is investigated. The finite difference (FD) approach is used to solve the equations of motion and entropy for a system subject to certain constraints. The optimal heat transfer rate with respect to radiation, Hartmann number, and nanoparticle volume fraction is calculated using a response surface technique and sensitivity analysis. The impacts of significant parameters such as Hartmann number, angle parameter, nanoparticle volume fraction, body acceleration amplitude, radiation, and Reynolds number on the velocity, temperature, entropy generation, flow rate, shear stress of wall, and heat transfer rate are exhibited via the graphs and tables. Present results disclose that the flow rate profile increase by improving the Womersley number and the opposite nature is noticed in nanoparticle volume fraction. The total entropy generation reduces by improving radiation. The Hartmann number expose a positive sensitivity for all level of nanoparticle volume fraction. The sensitivity analysis revealed that the radiation and nanoparticle volume fraction showed a negative sensitivity for all magnetic field levels. It is seen that the presence of hybrid nanoparticles in the bloodstream leads to a more substantial reduction in the axial velocity of blood compared to Sisko blood. An increase in the volume fraction results in a noticeable decrease in the volumetric flow rate in the axial direction, while higher values of infinite shear rate viscosity lead to a significant reduction in the magnitude of the blood flow pattern. The blood temperature exhibits a linear increase with respect to the volume fraction of hybrid nanoparticles. Specifically, utilizing a hybrid nanofluid with a volume fraction of 3% leads to a 2.01316% higher temperature compared to the base fluid (blood). Similarly, a 5% volume fraction corresponds to a temperature increase of 3.45093%., (© 2023. The Author(s).)

In the pump of different machines, the vacuum pump oil (VPO) is used as a lubricant. The heat rate transport mechanism is a significant requirement for all industries and engineering. The applications of VPO in discrete fields of industries and engineering fields are uranium enrichment, electron microscopy, radio pharmacy, ophthalmic coating, radiosurgery, production of most types of electric lamps, mass spectrometers, freeze-drying, and, etc. Therefore, in the present study, the nanoparticles are mixed up into the VPO base liquid for the augmentation of energy transportation. Further, the MHD flow of a couple stress nanoliquid with the applications of Hall current toward the rotating disk is discussed. The Darcy-Forchheimer along with porous medium is examined. The prevalence of viscous dissipation, thermal radiation, and Joule heating impacts are also considered. With the aid of Cattaneo-Christov heat-mass flux theory, the mechanism for energy and mass transport is deliberated. The idea of the motile gyrotactic microorganisms is incorporated. The existing problem is expressed as higher-order PDEs, which are then transformed into higher-order ODEs by employing the appropriate similarity transformations. For the analytical simulation of the modeled system of equations, the HAM scheme is utilized. The behavior of the flow profiles of the nanoliquid against various flow parameters has discoursed through the graphs. The outcomes from this analysis determined that the increment in a couple-stress liquid parameter reduced the fluid velocity. It is obtained that, the expansion in thermal and solutal relaxation time parameters decayed the nanofluid temperature and concentration. Further, it is examined that a higher magnetic field amplified the skin friction coefficients of the nanoliquid. Heat transport is increased through the rising of the radiation parameter., (© 2022. The Author(s).)

The research of fluid containing nanoparticles for the heat transport characteristics is very famous because of its variety of real-life applications in various thermal systems. Although the thermal efficiency of the nanofluid was effective but still the nano scientists were trying to introduce some new advance class of fluid. Therefore, an advance class of fluid is developed by the dispersion of two different nano sized particles in the conventional base fluid known as "Hybrid nanofluid" which is more effective compared to simple nanofluids in many engineering and industrial applications. Therefore, motivated from the hybrid type of nanofluids in the current research we have taken two-dimensional laminar and steady flow of second grade fluid passing through porous plate. The engine oil base fluid is widely used fluid in the engineering and industrial problems. Keeping these applications in mind the engine oil is considered and two different nanoparticles Copper and aluminum oxide are added in ordered to get the required thermal characteristics. In addition to this the thermal radiation, chemical reaction, activation energy, Brownian motion and thermophoresis are also addressed during the current research. The present proposed higher-order PDE's is transformed to the non-linear system of ODE's. For the solution of the proposed high non-linear model HAM method is employed. As the hybrid nanofluid are highlighted on the second-grade fluid flow over a horizontal porous flat plate. During the present analysis and experimental study, it has been proved that the performance of hybrid nanofluid is efficient in many situations compared to nanofluid and regular fluid. For physical interpretation all the flow parameters are discussed through graphs. The impact of volume fraction is also addressed through graphs. Moreover, the comparative analysis between hybrid and nanofluid is carried out and found that hybrid nanofluid performed well as compared to nanofluid and regular fluid. The engineering quantities obtained from the present research have been presented in tables., (© 2022. The Author(s).)

Hybrid nanofluids play a significant role in the advancement of thermal characteristics of pure fluids both at experimental and industrial levels. This work explores the mixed convective MHD micropolar hybrid nanofluid flow past a flat surface. The hybrid nanofluid flow is composed of alumina and silver nanoparticles whereas water is used as a base fluid. The plate has placed vertical in a permeable medium with suction and injection effects. Furthermore, viscous dissipation, thermal radiation and Joule heating effects are taken into consideration. Specific similarity variables have been used to convert the set of modeled equations to dimension-free form and then has solved by homotopy analysis method (HAM). It has revealed in this investigation that, fluid motion upsurge with growth in magnetic field effects and mixed convection parameter and decline with higher values of micropolar factor. Micro-rotational velocity of fluid is upsurge with higher values of micropolar factor. Thermal flow behavior is augmenting for expended values of magnetic effects, radiation factor, Eckert number and strength of heat source. The intensification in magnetic strength and mixed convection factors has declined the skin friction and has upsurge with higher values of micropolar parameter. The Nusselt number has increased with the intensification in magnetic effects, radiation factor and Eckert number., (© 2022. The Author(s).)

Scientists and researchers are much interested in studying graphene and silver nanoparticles for the enhancement of heat transport due to their extensive variety of applications in different areas of industrial and engineering such as drug delivery, medical devices, ultra-light, excellent electrical conductivity, strong medical strength, health care, consumer, food, etc. Therefore, in the existing investigation, the MHD flow of a mixed convective hybrid nanoliquid with graphene and silver nanoparticles past a rotating disk is considered. Water and ethylene glycol (50:50) is used as a base liquid in the existing model. The mechanism for heat transport is computed with the existence of thermal radiation and thermal convective condition. Homogeneous and heterogeneous chemical reactions are assumed in the flow behavior. The mathematical formulation of the proposed problem is based on the nonlinear PDEs which are then transformed to nonlinear ODEs by manipulating the appropriate similarity transformation. The simulation of the existing problem has been performed with the help of the homotopy analysis technique. The outcomes of the different flow parameters on the velocities, temperature, concentration, skin friction coefficient, and Nusselt number of the hybrid nanofluid are attained via graphs and tables. Some significant results from the existing problem demonstrate that the rate of heat transport is greater for the thermal Biot number and nanoparticles volume fraction. Further, it is noticed that the velocity of the liquid particles becomes lower for a higher magnetic field parameter., (© 2022. The Author(s).)

This report presents the three-dimensional electromagnetohydrodynamic flow of a zinc-oxide-water nanofluid past a bidirectional Riga plate with velocity slips and thermal and mass convection conditions. The Cattaneo-Christov heat and mas flux model, thermal radiation, chemical reaction and activation energy are considered to analyze the flow problem. The volume fraction of the ZnO nanoparticles is taken 6% in this analysis. An appropriate set of similarity variables is used to transform the partial differential equations into ordinary differential equations. During this process, some parameters are found and influences of these factors on the flow profiles are shown and discussed in detail. A numerical technique called NDSolve is considered for the solution of the nanofluid flow problem. The results showed that higher solid volume fraction and slip parameter have reduced velocities profiles and the increasing solid volume fraction and thermal Biot number have increased the temperature profile. Additionally, the concentration Biot number has increased the concentration profile. The modified Hartmann number has significantly increased the velocity profile. Dual impacts in velocity profiles along primary and secondary direction has been observed due to stretching ratio parameter. A comparison of current results has been carried with a fine agreement amongst current and established results., (© 2022. The Author(s).)

The Jeffrey fluid model is capable of accurately characterizing the stress relaxation behavior of non-Newtonian fluids, which a normal viscous fluid model is unable to perform. The primary objective of this paper is to provide a comprehensive investigation into the effects of MHD and thermal radiation on the 3D Jeffery fluid flow over a permeable irregular stretching surface. The consequences of the Darcy effect, variable thickness and chemical reaction are also considered. The phenomena have been modeled as a nonlinear system of PDEs. Using similarity substitution, the modeled equations are reduced to a dimensionless system of ODEs. The parametric continuation method (PCM) is used to determine the numerical solution to the obtained sets of nonlinear differential equations. The impact of physical parameters on temperature, velocity and mass profiles are presented through Figures and Tables. It has been noticed that the energy profile magnifies with the increment of porosity term, thermal radiation and heat source term, while diminishing with the flourishing upshot of power index and Deborah number. Furthermore, the porosity term and wall thickness parameter enhance the skin friction., (© 2022. The Author(s).)

In biological systems, the MHD boundary layer bioconvection flow through permeable surface has several applications, including electronic gadgets, heating systems, building thermal insulation, geological systems, renewable energy, electromagnetism and nuclear waste. The bioconvection caused by the hydromagnetic flow of a special form of water-based nanoliquid including motile microorganisms and nanoparticles across a porous upright moving surface is investigated in this report. The combination of motile microbes and nanoparticles causes nanofluid bioconvection is studied under the cumulative impact of magnetic fields and buoyancy forces. The Brownian motion, thermophoresis effects, heat absorption/generation, chemical reaction and Darcy Forchhemier impact are also unified into the nonlinear model of differential equations. The modeled boundary value problem is numerically computed by employing a suitable similarity operation and the parametric continuation procedure. The parametric study of the flow physical parameters is evaluated versus the velocity, energy, volume fraction of nanoparticles, motile microorganisms' density, skin friction, Sherwood number and Nusselt number. It has been observed that the velocity profile reduces with the effect of porosity parameter k 1 , inertial parameter k 2 , Hartmann number and buoyancy ratio. While the energy transition profile significantly enhances with the flourishing values of Eckert number Ec, heat absorption/generation Q and Hartmann number respectively., (© 2022. The Author(s).)

The energy transmission through micropolar fluid have a broad range implementation in the field of electronics, textiles, spacecraft, power generation and nuclear power plants. Thermal radiation's influence on an incompressible thermo-convective flow of micropolar fluid across a permeable extensible sheet with energy and mass transition is reported in the present study. The governing equations consist of Navier-Stokes equation, micro rotation, temperature and concentration equations have been modeled in the form of the system of Partial Differential Equations. The system of basic equations is reduced into a nonlinear system of coupled ODE's by using a similarity framework. The numerical solution of the problem has been obtained via PCM (Parametric Continuation Method). The findings are compared to a MATLAB built-in package called bvp4c to ensure that the scheme is valid. It has been perceived that both the results are in best agreement with each other. The effects of associated parameters on the dimensionless velocity, micro-rotation, energy and mass profiles are discussed and depicted graphically. It has been detected that the permeability parameter gives rise in micro-rotation profile., (© 2022. The Author(s).)

Bioconvection has recently been the subject of dispute in a number of biotechnological fields that depend on fluids and their physical properties. When mixed nanofluids are subjected to heat and mass transmission, the process of bioconvection occurs. This attempt conveys the theoretical analysis of two-dimensional electrically conducting and magnetically susceptible binary fluid containing nanoparticles and gyrotactic microorganisms past a stratified stretching surface. Furthermore binary chemical reaction, thermal radiation, and activation energy are taken into assumptions. The analytical solution based on HAM has been performed. The convergence of HAM is presented with the help of figures. The present study is compared with previously published results and has established an excessive agreement which validate the present study. It is perceived that the presence and absence of an electric field influences the variations in fluid velocities due to presence of magnetic field. The micropolar constant heightens the velocity and microrotation of the fluid flow. The buoyancy parameter and bioconvection Rayleigh number diminish the velocity function while these parameters show dual impact on microrotation function. The skin friction and couple stress escalates with the increasing buoyancy ratio parameter and bioconvection Rayleigh number., (© 2021. The Author(s).)

Fractional calculus characterizes a function at those points, where classical calculus failed. In the current study, we explored the fractional behavior of the stagnation point flow of hybrid nano liquid consisting of TiO 2  and Ag nanoparticles across a stretching sheet. Silver Ag and Titanium dioxide TiO 2  nanocomposites are one of the most significant and fascinating nanocomposites perform an important role in nanobiotechnology, especially in nanomedicine and for cancer cell therapy since these metal nanoparticles are thought to improve photocatalytic operation. The fluid movement over a stretching layer is subjected to electric and magnetic fields. The problem has been formulated in the form of the system of PDEs, which are reduced to the system of fractional-order ODEs by implementing the fractional similarity framework. The obtained fractional order differential equations are further solved via fractional code FDE-12 based on Caputo derivative. It has been perceived that the drifting velocity generated by the electric field E significantly improves the velocity and heat transition rate of blood. The fractional model is more generalized and applicable than the classical one., (© 2021. The Author(s).)

The current study provides a detailed analysis of steady two-dimensional incompressible and electrically conducting magnetohydrodynamic flow of a couple stress hybrid nanofluid under the influence of Darcy-Forchheimer, viscous dissipation, joule heating, heat generation, chemical reaction, and variable viscosity. The system of partial differential equations of the current model (equation of motion, energy, and concentration) is converted into a system of ordinary differential equations by adopting the suitable similarity practice. Analytically, homotopy analysis method (HAM) is employed to solve the obtained set of equations. The impact of permeability, couple-stress and magnetic parameters on axial velocity, mean critical reflux condition and mean velocity on the channel walls are discussed in details. Computational effects show that the axial mean velocity at the boundary has an inverse relation with couple stress parameter while the permeability parameter has a direct relation with the magnetic parameter and vice versa. The enhancement in the temperature distribution evaluates the pH values and electric conductivity. Therefore, the [Formula: see text] hybrid nanofluids are used in this study for medication purpose., (© 2021. The Author(s).)

To enhance the surface properties of solids the mechanism of thin films is frequently used. Penetration, degradation, stiffness, illumination, diffusion, absorption, and electric performance are all characteristics of a bulk substance medium that a thin film can improve. In nanotechnology, thin film processing can be extremely useful. Therefore, the time-dependent nonlinearly convective stream of thin film nanoliquid over an inclined stretchable sheet with magnetic effect is investigated in current work. The features of mass and heat transport processes are explained using important factors like thermophoresis and Brownian movement. Nonlinear partial differential equations are obtained to model the time-dependent liquid film flow over an inclined surface, which are then turned into couple ordinary differential equations utilizing appropriate alterations. The results of the computation of the model problem are collected using an analytical approach Homotopy Analysis Method and presented the final finding numerically and graphically. During the flow assessment, the impact of individual flow factors such as magnetic, Brownian, and thermophoresis parameters on regular profiles (temperature, velocity, and concentration) are analyzed and found to be quite remarkable. Furthermore, the consequence of M and N t factors on the velocity, concentration and thermal distribution leads to diminishing conduct. On the other hand, the thermal profile of the liquid film rises in response to the thermophoresis factor. The % wise variation in the skin friction, Nusselt number and Sherwood number versus physical parameters has been obtained and discussed., (© 2021. The Author(s).)

This investigation describes the hydromagnetic flow of gravity-driven couple stress hybrid nanofluid past a heated plate. The carbon nanotubes (CNTs) are used to characterize the hybrid nanofluid. The heated plate is placed vertically with an application of homogenous-heterogeneous reactions to the assumed flow system. The homogeneous reaction governs by isothermal cubic autocatalytic kinetics while the heterogeneous reaction governs by the first order kinetics. For current study the couple stress hybrid nanofluid is presumed to be conducted electrically with impact of non-uniform magnetic effects. An appropriate set of dimensionless quantities has employed to governing equations and then has solved by homotopy analysis method. The influence of emerging parameters encountered in this work has discussed in detail with the help of graphs. In this study it has examined that, flow of fluid reduces with upsurge in magnetic parameter and volumetric concentrations, whereas thermal and concentration characteristics augment with increase in volumetric concentrations. Moreover, growth in Prandtl number leads to a reduction in thermal characteristics and growth in Schmidt number result a reduction in concentration profile. The impact of various emerging parameters has also studied numerically upon physical quantities. It has established that, with augmentation in values of buoyancy parameter there is a growth in the values of skin friction. A comparison has also carried out between current and established results with a fine agreement in both results., (© 2021. The Author(s).)

This work investigates numerically the solution of Darcy-Forchheimer flow for hybrid nanofluid by employing the slip conditions. Basically, the fluid flow is produced by a swirling disk and is exposed to thermal stratification along with non-linear thermal radiation for controlling the heat transfer of the flow system. In this investigation, the nanoparticles of titanium dioxide and aluminum oxide have been suspended in water as base fluid. Moreover, the Darcy-Forchheimer expression is used to characterize the porous spaces with variable porosity and permeability. The resulting expressions of motion, energy and mass transfer in dimensionless form have been solved by HAM (Homotopy analysis method). In addition, the influence of different emerging factors upon flow system has been disputed both theoretically in graphical form and numerically in the tabular form. During this effort, it has been recognized that velocities profiles augment with growing values of mixed convection parameter while thermal characteristics enhance with augmenting values of radiation parameters. According to the findings, heat is transmitted more quickly in hybrid nanofluid than in traditional nanofluid. Furthermore, it is estimated that the velocities of fluid [Formula: see text] are decayed for high values of [Formula: see text] and [Formula: see text] factors., (© 2021. The Author(s).)

In this investigation, heat transportation together with irreversibility analysis for the flow of couple stress hybrid nanofluid past over a stretching surface is considered. The innovative characteristics and aims of this work are to note that the transportation heat couple stress model involves EMHD, viscous dissipation, Joule heating, and heat absorption, and omission. The hybrid nanofluid is prepared due to the suspension of the solid nanoparticles of the SWCNTs and MWCNTs in pure human blood. This mathematical model is an appropriate model for biological advantages including testing of human blood for drug deliveries to various parts of the human body. Particularly, the Prandtl number used for the blood is 21 and very large as compared to the other base fluids. Necessary modifications are used to translate the defining partial differential equations and boundary conditions into a layout that can be computed. To obtain mathematical approximations for the resulting scheme of nonlinear differential equations, the innovative homotopy analysis method (HAM) is used. The explanation for velocity, energy, and entropy are exposed and the flow against various influential factors ([Formula: see text]) is discussed graphically. The numerical values are calculated and summarized for dimensionless [Formula: see text] In addition, the current study is compared for various values of [Formula: see text] to that published literature and an impressive agreement in terms of finding is reported. It has also been noticed that the [Formula: see text] and [Formula: see text] factors retard the hybrid nanofluid flow, while the temperature of fluid becomes upsurges by the rise in these factors. 11.95% enhancement in the heat transfer rate has been attained using the hybrid nanofluids.

The current study focuses on the laminar flow of copper and copper oxide ([Formula: see text] and [Formula: see text]) hybrid nanoliquid, considering blood as a carrier fluid in a rectangular domain between two permeable channels. This study may manipulate for the purpose such as the drug delivery process, flow dynamic mechanism of the micro-circulatory system. In the proposed model, MHD and heat source/sink on the flow pattern have been studied. Furthermore, the sides of each channel are permeable, allowing the nanoliquid to escape, filter, squeezing and dilating with a fixed velocity. Appropriate transformations are incorporated to convert the governing partial differential equations and the boundary conditions suitable for computation. The elegant homotopy analysis method (HAM) is used to obtain analytic approximations for the resulting system of nonlinear differential equations. The features of flow characteristics such as velocity, and temperature profiles in response to the variations of the emerging parameters are simulated and examined with a physical explanation. The magnetic field plays a vital role in the blood flow and therefore the existing literature has been extending with the addition of magnetic field. Among the many outputs of the study, it is found that the pressure distribution decline with the accumulated values of the magnetic parameter at the center of the flow regime. The augmentation in the temperature distribution estimates the pH values and electric conductivity. Therefore, the [Formula: see text] hybrid nanofluids are used in this study for medication purposes. The magnetic field has an important role in the blood flow and therefore the extending study has been extending using the magnetic field. The heat emission/absorption term is added to the energy equation to maintain the homogeneous temperature for the blood flow. We expect that this work will provide efficient outputs for medical purposes such as drug delivery.

In this work, the thermal analysis for bio-convective hybrid nanofluid flowing upon a thin horizontally moving needle is carried out. The chemical reaction and viscous dissipation has also considered for flow system in the presence of microorganism. The hybrid nanoparticles comprising of Copper [Formula: see text] and Alumina [Formula: see text] are considered for current flow problem. Mathematically the flow problem is formulated by employing the famous Buongiorno's model that will also investigate the consequences of thermophoretic forces and Brownian motion upon flow system. Group of similar variables is used to transform the model equations into dimensionless form and have then solved analytically by homotopy analysis method (HAM). It has established in this work that, flow of fluid declines due to increase in bioconvection Rayleigh number, buoyancy ratio and volume fractions of nanoparticles. Thermal flow grows due to rise in Eckert number, Brownian, thermophoresis parameters and volume fraction of nanoparticles. Concentration profiles increase due to growth in Brownian motion parameter and reduces due to increase in thermophoresis parameter and Lewis number. Motile microorganism profile declines due to augmentation in Peclet and bioconvection Lewis numbers. Moreover, the percentage enhancement in the drag force and rate of heat transfer using conventional nanofluid and hybrid nanofluid are observed and discussed. The hybrid nanofluid increases the skin friction and heat transfer rate more rapidly and efficiently as compared to other traditional fluids. A comparison of the present study with the existing literature is also conducted with a closed agreement between both results for variations in thickness of the needle.

In the current work, the unsteady thermal flow of Maxwell power-law nanofluid with Welan gum solution on a stretching surface has been considered. The flow is also exposed to Joule heating and magnetic effects. The Marangoni convection equation is also proposed for current investigation in light of the constitutive equations for the Maxwell power law model. For non-dimensionalization, a group of similar variables has been employed to obtain a set of ordinary differential equations. This set of dimensionless equations is then solved with the help of the homotopy analysis method (HAM). It has been established in this work that, the effects of momentum relaxation time upon the thickness of the film is quite obvious in comparison to heat relaxation time. It is also noticed in this work that improvement in the Marangoni convection process leads to a decline in the thickness of the fluid's film.

In this new world of fluid technologies, hybrid nanofluid has become a productive subject of research among scientists for its potential thermal features and abilities, which provides an excellent result as compared to nanofluids in growing the rate of heat transport. Our purpose here is to introduce the substantial influences of magnetic field on 2D, time-dependent and stagnation point inviscid flow of couple stress hybrid nanofluid around a rotating sphere with base fluid is pure blood, [Formula: see text] as the nanoparticles. To translate the governing system of partial differential equations and the boundary conditions relevant for computation, some suitable transformations are implemented. To obtain the analytical estimations for the corresponding system of differential expression, the innovative Optimal Homotopy Analysis Method is used. The characteristics of hybrid nanofluid flow patterns, including temperature, velocity and concentration profiles are simulated and analyzed in detail due to the variation in the evolving variables. Detailed research is also performed to investigate the influences of relevant constraints on the rates, momentum and heat transport for both [Formula: see text] and [Formula: see text]. One of the many outcomes of this analysis, it is observed that increasing the magnetic factor will decelerate the hybrid nanofluid flow velocity and improve the temperature profile. It may also be demonstrated that by increasing the Brownian motion factor, significant improvement can be made in the concentration field of hybrid nanofluid. The increase in the nanoparticle volume fraction from 0.01 to 0.02 in the case of the hybrid nanofluid enhances the thermal conductivity from 5.8 to 11.947% and for the same value of the nanoparticle volume fraction in the case of nanofluid enhance the thermal conductivity from 2.576 to 5.197%.

The magnetohydrodynamic hybrid second-grade nanofluid flow towards a stretching/shrinking sheet with thermal radiation is inspected in current work. Main concern of current investigation is to consider hybrid [Formula: see text] nanofluid which is perceived by hanging two dissimilar kinds of nanoparticles known as alumina and copper within the base fluid. The fluid motion is produced by non-linear stretching/shrinking sheet. The modeled equations which comprise of energy, motion and continuity equations are changed into dimensionless form using group of similar variables. To determine the solution of transformed problem, the Homotopy Analysis technique is used. The findings of this work revealed that the magnetic parameter improves the heat transfer rate. This work also ensures that there are non-unique solutions of modeled problem for shrinking case and a unique solution for stretching case. Higher values of [Formula: see text] results in declining of flow field. Rise in [Formula: see text] agrees to a decline in velocity distributions. Higher values of second order fluid parameter reduces the viscosity of fluid and accordingly velocity increases. Velocity profile is also a decreasing function of volume friction.