Entropy generation optimization of cilia regulated MHD ternary hybrid Jeffery nanofluid with Arrhenius activation energy and induced magnetic field

This study deals with the entropy generation analysis of synthetic cilia using a ternary hybrid nanofluid (Al–Cu–Fe2O3/Blood) flow through an inclined channel. The objective of the current study is to investigate the effects of entropy generation optimization, heat, and mass transfer on ternary hybrid nanofluid passing through an inclined channel in the proximity of the induced magnetic field. The novelty of the current study is present in studying the combined effect of viscous dissipation, thermophoresis, Brownian motion, exponential heat sink/source, porous medium, endothermic–exothermic chemical reactions, and activation energy in the proximity of induced magnetic field is examined. The governing partial differential equations (PDEs) are transformed into the ordinary differential equations (ODEs) using appropriate transformations. Applying the low Reynolds number and the long-wavelength approximation, resultant ODEs are numerically solved using shooting technique via BVP5C in MATLAB. The velocity, temperature, concentration, and induced magnetism profiles are visually discussed and graphically analyzed for various fluid flow parameters. Graphical analysis of physical interest quantities like mass transfer rate, heat transfer rate, entropy generation optimization, and skin friction coefficient are also graphically discussed. The entropy generation improves for enhancing values of Reynolds number, solutal Grashof number, heat sink/source parameter, Brinkman number, magnetic Prandtl number, and endothermic-exothermic reaction parameter while the reverse effect is noticed for chemical reaction and induced magnetic field parameter. The findings of this study can be applied to enhance heat transfer efficiency in biomedical devices, optimizing cooling systems, designing efficient energy conversion processes, and spanning from renewable energy technologies to aerospace propulsion systems.


List of symbols
Nanofluids are fluid mixtures having nanoparticles of 100 nm or fewer in the base fluid like oil or water.Nanoparticles of metals, oxides, or carbon-based substances have unique characteristics.Argonne National Laboratory scientist Choi 1 postulated nanofluids in 1995.Alnahdi et al. 2 investigate the utilization of a ternary Casson hybrid nanofluid in divergent/convergent channels for potential medication applications.This study's findings have applications in targeted drug delivery systems and thermal therapies.Alnahdi et al. 3 investigate blood-based ternary hybrid nanofluid flow through a punctured capillary for potential drug delivery applications.The study provides insights into enhancing drug delivery systems using nanofluids, which could optimize therapeutic efficacy.Nanofluids can be further classified based on the number of nanoparticles used to prepare them.Wang et al. 's 4 investigation into the effects of heat radiation and nanoparticle aggregation on nanofluid flow between a disc and cone gap.The research sheds light on the interplay between aggregation and radiation, which is vital for designing efficient heat transfer systems in many applications.In a ternary hybrid nanofluid flow with a Chemical reactions are integral to numerous applications, including industrial processes, energy production, and materials synthesis.They are crucial for manufacturing fuels, pharmaceuticals, and chemicals, as well as environmental remediation.The effects of chemical reaction, thermal radiation, and heat source on the axisymmetric MHD flow of a jeffrey nanofluid via a ciliated channel filled with a porous media were investigated by Shaheen et al. 31 .The research has implications in biomedical engineering, particularly in understanding fluid flow in biological systems and drug delivery mechanisms.Limitations may arise from the simplifications made in the model, which may not fully capture the complexity of real biological systems.Entropy production in exothermic/endothermic reactive magnetised nanofluid flow across porous curved space with variable permeability and porosity was theoretically analysed by Ullah et al. 32 .Nanofluid flow and heat transmission through a thin moving needle are investigated by Hussain et al. 33 , who consider the role of Arrhenius activation energy and chemical reaction.The phenomena of thermal energy transfer was recently investigated by Sharma et al. 34 , who looked at it in the context of hybrid nanofluid flow caused by a revolving Riga disc.The effects of heat radiation and chemical reaction on this procedure were the primary focus of the study.It's useful for comprehending how fluid flows and how heat is transferred in magnetohydrodynamic nanofluid systems.For the purpose of performing concurrent endothermic and exothermic processes, Chen 35 looked into the heat management of thermally coupled reactors.The study has potential applications in industrial processes where heat transfer optimization is crucial for efficient and sustainable operation.
A review of the above literature reveals that the nanofluid flows with the inclusion of various effects has been investigated by many researchers.Though, there is still a gap available in the literature on a cilia regulated ternary hybrid Jeffery nanofluid through a peristaltic channel with the consideration of various effects.The present article fills the gap with the consideration of ternary hybrid nano molecules on the fluid through a cilia regulated peristaltic channel with induced magnetic effect which never investigated before.Optimising entropy production, heat, and mass transfer in a ternary hybrid nanofluid as it flows down an inclined channel in the presence of an induced magnetic field and a heat source is also a key focus of this research.Cilia play important roles in human tissues and vital organs, such as cell migration and external fluid movement, therefore this study may benefit the medical field.The considered slanted geometry also gives crucial measurements for endoscopes and microscopes.For this reason, this investigation attempts to answer the following questions: 1. What are the significant effects of ternary hybrid nanoparticles controlled by the cilia on the Sherwood number, Nusselt number, temperature, and skin friction coefficient?2. What impact do Brinkman number, magnetic Prandtl number, endothermic-exothermic reaction parameter have on optimization of entropy generation? 3. How do the endothermic-exothermic chemical reactions influence the condensed and restrained skin friction factor, Nusselt number, and Sherwood number? 4. How the induced magnetic field affects the dynamics of a fluid with mixed convection hybrid nanofluid through inclined channel? 5. How do skin friction, Bejan number, and entropy formation physically relate to exponential heat source/ sink and activation energy?How fast do heat and mass transfer rates change?

Mathematical formulation
Considered a laminar viscous, incompressible thermally and electrically conducting Al−Cu−Fe 2 O 3 −Blood ternary hybrid Jeffery nanofluid passing through an infinite two-dimensional inclined channel of thickness 2d.Cartesian coordinate system is considered in which the X-axis is oriented in the vertical direction, and Y-axis is oriented in the horizontal direction.Peristaltic flow is induced by traveling sinusoidal waves advancing with constant velocity c.Assume that the inner walls of the tube are covered with an unlimited number of constantly beating cilia, which will result in a symplectic metachronal wave that travels along the positive x-axis with an inclination angle η and a wave speed of c, as illustrated in Fig. 1.The medium between the cilia walls is porous having k 1 as the permeability constant.The flow is given a homogeneous magnetic field from outside.The magnetic field has a strength of B 0 and acts perpendicular to the flow direction.The channel is inclined at an angle of η with the vertical axis.Exponential heat source is considered with q 0 < 0 heat sink and q 0 > 0 heat source.Endothermic and exothermic chemical reactions with activation energy are considered in heat energy and concentration equations as per the Arrhenius law.The Cauchy stress tensor T, extra stress tensor S, for an incompressible Jeffrey fluid, are defined as 36 : where, P is the pressure, and I is the identity tensor.
Continuity equation: (5) www.nature.com/scientificreports/Induced magnetic field equation: where ρ thnf , σ thnf , µ thnf , k thnf and C p thnf are the ternary hybrid nanofluid's density, electrical conductivity, dynamic viscosity, thermal conductivity, and specific heat, in that order.The values of these thermophysical parameters for the existing ternary hybrid nanofluid are listed in Table 1 and are mathematically expressed as 3 : The following transformations are used to convert the considered fluid flow problem from fixed frame to wave frame: Introducing the following dimensional parameters: where, p is pressure, θ is dimensionless temperature, is dimensionless concentration, is the temperature ratio, and is concentration ratio.The rate constant of endothermic and exothermic chemical reactions with activation energy, R c is assumed to be dependent on the absolute temperature T * and is provided by the modi- fied Arrhenius law 44,45 : Thus, Eqs.(5, 6, 7, 8, 9) simplify to the dimensionless form as follows: Table 1.Thermophysical properties of ternary hybrid nanofluid.where A 1−4 represent the ratios of ternary hybrid nanofluid thermophysical properties to that of base fluid and defined as follows: The non-dimensional parameters used in Eqs.(13, 14, 15, 16, 17) are described as follows: Introducing the stream function ψ and ϕ for velocity fields and induced magnetic fields, respectively: Hence, the following form represent the stress components: By using the long wavelength ( → ∞ ⇒ δ → 0) and low Reynolds number (Re → 0) approximations in the above-mentioned stress components and dimensionless governing Eqs.(13, 14, 15, 16, 17) become: ( 13) From Eq. ( 18), the pressure gradient is eliminated via cross-difference.The resulting equation is obtained as follows: With the following boundary conditions 46 : Engineering quantities of interests.This study also analyses the relevant physical parameters including shear stress, mass transfer rate, and heat transfer rate.The mathematical formulas for the Nusselt number, skin friction coefficient, and Sherwood number are 47,48 : where, q w is the heat flux, τ u is the axial stress, and m w is the mass flux.They are defined as: The above expressions' non-dimensional forms are as follows: Entropy generation.Now, we will discuss the entropy generation analysis for the considered problem.It is noted that the impacts of heat transfer, mass transfer, thermal radiation, fluid friction, Joule heating, homogeneous and heterogeneous chemical reaction are the main causes of entropy in the medium.Considering the second rule of thermodynamics, the following is the expression for entropy generation 49 : Using Eqs.(11) and (12) in Eq. ( 28), the characteristic entropy, lubrication approximations, and the formula for the total entropy generation number yield the following: where N T is the thermal irreversibility, N V is the viscous irreversibility, N F the fluid friction irreversibility, N J is the Joule heating irreversibility, and N M is the mass transfer irreversibility.The ratio between irreversibility caused by heat transfer and total irreversibility caused is given by Bejan number, Be.It is defined in mathematical form: (19) (23)

Numerical solution
The model's governing Eqs.(19, 20, 21, 22, 23) are a set of coupled, highly nonlinear ordinary differential equations.The gunshot method is used to resolve this system of differential equations.By turning a boundary value problem into an initial value problem of a first order differential equation, the shooting method can solve a BVP.It entails searching for IVP solutions that also satisfy the boundary conditions of the BVP for a variety of beginning conditions.A practical shooting approach is used to solve the governing differential Eqs.(19, 20, 21, 22, 23)  and the accompanying boundary conditions (24).Here are the substitutions that are used: These substitutions result in an initial value problem consisting of nine first-order differential equations: (30)   Parameter

Results validation
Figure 2a,b show a comparative study of velocity u and temperature θ .The present model is reduced to an already published work of Alla et al. 36 .Figure 2a,b illustrate the velocity and temperature profiles for a combination of dimensionless parameters, respectively.It can be concluded from both figures that the present study is in accordance with the earlier published work.Table 2 shows the range of the values of parameter and corresponding references of parameters.
The numerical results for Eq. ( 32) with the boundary conditions (33), are obtained using MATLAB's BVP5C programme.

Results and discussions
A combined effect of external and induced magnetic field, viscous dissipation, highly porous medium, thermophoresis, Brownian motion, exponential heat source/sink, activation energy, and endothermic-exothermic chemical reactions are scrutinized in this section.For a given set of flow parameters, various graphs of velocity, temperature, concentration, and induced magnetism profiles are drawn.
Velocity profiles.Figure 3a-d illustrate the impact of the solutal Grashof number, channel inclination, Forchheimer parameter, and induced magnetic field number on the axial velocity of the fluid.The link between the solutal Grashof number and the fluid's velocity is depicted in Fig. 3a.It is noted that, for enhanced values of solutal Grashof number, the axial velocity first increase and then decrease.The change happens at almost y = 0.1.The ratio of a species' buoyancy to viscous force is physically represented by the solutal Grashof number.The values in the figure are all greater than 1, meaning that the buoyancy force's effect is larger than the viscous force.Figure 3b shows how increasing the channel inclination affects the fluid velocity.The figure depicts that for raised values of inclination, the axial velocity increases for the first half and then decreases for the second half.When the inclination increases, the magnetic field's effect dominates that of gravity.As the magnetic field acts perpendicular to the channel, the velocity increases for the fallen magnetic strength.For the second half, gravitational acceleration dominated the magnetic field, and on increasing the inclination, the gravitational acceleration dropped the speed.Figure 3c depicts the effect of the Forchheimer number on the axial velocity.It can be seen in the figure that for raised values of the Forchheimer number, the velocity decreases.Increasing the Forchheimer number decreases fluid velocity since the inertia factor is directly correlated with the medium's porosity and drag coefficient.As a result, as the Forchheimer number rises, so does the drag coefficient and the porosity of the medium.A greater Forchheimer number is associated with lower velocities since the liquid's resistive force increases with decreasing speed.Figure 3d explains the importance of the induced magnetic field on fluid velocity.The figure signifies that for enhanced values of induced magnetic field number, the velocity first increases and then decreases.This change happens at y = 0. Induced magnetism happens in nanofluids when an external magnetic field causes nanoparticles suspended in a base fluid to take on magnetic characteristics.The nanoparticles' magnetic moments can align with the magnetic field, resulting in a net magnetic moment in the fluid, which causes this effect.3e-h signifies the effect of the heat source number, Brownian motion parameter, endothermic-exothermic chemical reaction number, and the thermophoresis number on the fluid's temperature profiles.The effect of the heat source parameter on the temperature profile is seen in Fig. 3e.The figure shows that the temperature increases when the heat source strength increases.From the physical point of view, these results make sense, as enhanced heat source strength implies more heat generation from the region's surface.Figure 3f illustrates how the Brownian motion parameter affects temperature.The figure demonstrates that the temperature profiles also increase for increased values of Brownian motion parameters.The fluid's swiftly moving atoms and molecules, have a greater influence on the Brownian motion of the suspended particles in the base fluid.The relationship between Brownian motion and particle size and the fact that these particles (32)

Temperature profiles. Figure
Vol  www.nature.com/scientificreports/absorb energy from the surroundings to overcome the activation energy barrier.The surrounding temperature decreases, and the fluid temperature increases.The importance of thermophoresis numbers on the temperature profiles is shown in Fig. 3h.The figure signifies that the temperature profiles increase by elevating the thermophoretic effect.It has been demonstrated that temperature profiles are enhanced when the thermophoretic parameter has higher values.This occurs due to the thermophoretic force that particle adjacent to a hot surface produce.By increasing the thickness of the temperature boundary layer, this force encourages particle disintegration outside of the fluid regime.
Concentration profiles.Figure 3i-l represents the impact on the fluid concentration for fitting constant, chemical reaction number, activation energy, and temperature ratio.Figure 3i depicts the consequence of smaller and larger fitting constants on the temperature.For larger fitting constant, the concentration profiles improve in the positive direction.The temperature fitting constant is used to adjust the thermal conductivity and viscosity of the nanofluid to account for these differences.Figure 3j shows the aftermath of increased chemical reaction parameters on the concentration profiles.The figure shows that for enhanced values of chemical reaction parameters, the concentration profiles of fluid move in the upper direction.Increases in the concentration of the ternary hybrid nanofluid and the thickness of the boundary layer result from an increase in the chemical reaction parameter close to the surface.It is thought to increase the likelihood of collisions between fluid particle atoms.Figure 3k indicates the concentration profile behavioral change for the activation energy parameter.An increased behavior is seen in the concentration profiles for escalated activation energy because the activation energy is regarded as a strong potential barrier separating the potential energy minima.Because activation energy supplies the energy to initiate the chemical reaction, increasing concentration.Because of amplification in activation energy, which supplies significant energy to process the reaction and increase the concentration field, the concentration profile grows.Figure 3l is used to demonstrate the temperature ratio changes in the concentration.The concentration parameter for the expanding temperature ratio exhibits an incline behavior.This phenomenon occurs because the parameter describing the thermal state of a fluid, the temperature ratio, increases as the temperature rises.It describes the fluid's thermal condition; as this parameter rises, so does the temperature.www.nature.com/scientificreports/ the induced magnetism profiles increase by increasing the Reynolds number.This is because viscosity and the Reynolds number are inversely correlated.As the Reynolds number rises, viscosity falls, making the fluid less thick and increasing velocity.Conversely, viscosity rises as the Reynolds number falls, making the fluid thicker and decreasing velocity.For enhanced velocity, the fast-moving nanoparticles induce much greater magnetism.Figure 3n signifies the changes in magnetic Prandtl number on the induced magnetic field.The figure shows how the induced magnetism increases on increasing the magnetic Prandtl number.Physically, the magnetic Prandtl number is proportional to the heat transfer rate.For larger values of magnetic Prandtl number, the temperature and Brownian motion of nanoparticles increase, which eventually increases the velocity.Again, for more velocity, more magnetism is induced in the fluid.Figure 3o depicts the effect of enhanced Darcy number.It shows that the induced magnetic field rises for a larger Darcy number.This is due to the fact that the medium becomes more permeable at higher Darcy values, which in turn creates more room for nanofluid recirculation and a greater velocity.Once more, greater velocity creates more magnetism.Figure 3p shows the significance of the larger thermal Grahsof number on the induced magnetism profile of the fluid.On enhancing the thermal Grashof number, the fluid developed more induced magnetism.Physically, the thermal Grashof number indicates the ratio of buoyancy force to the viscous force.The viscous forces become lower for incremented values of the thermal Grashof number.These lower viscous forces increase the fluid velocity, eventually increasing the fluid's induced magnetism.part of the channel.On making Fr 10, a smaller orange-coloured bolus is formed at the top.The annular region of the lower part also decreased.Small bubbles can be seen in the middle portion of the flow.Figure 4b shows the velocity contour behaviour for increasing the solutal Grashof number.By increasing Gc from 1 to 4, the rightward waves become sharper.The bubbles become narrower from the right side.There is also a formation of orange bolus in the topmost annulus.For the Gc value of 8, the waves become more sharer at the right side.The formation of orange bolus in the upper two annuli enhanced.The next Fig.4c, demonstrates how the changing induced magnetic field impacts the velocity.The figure shows that the innermost yellow bubbles become smaller when the induced magnetism parameter increases from 2 to 8. In the middle annulus, it got disappeared.For the Mf value of 14, the inner bolus again starts emerging.One can see an orange-coloured bubble in the middle of the upper bolus.The purpose of Fig. 4d is to show how increasing the Reynolds number changes the flow velocity.On making Re as 2.5, it can be seen that the size of the inner bolus becomes smaller.There is also a formation of 2.8 valued bolus in the uppermost annulus.For the Re value of 5, the formation of boluses again starts, but now of higher value (in the negative direction). Figure 4e represents the thermal Grashof number vs the velocity contour.It shows that when Gr is changed to 4 from 1, waves are leftward squeezing.The wavelength decreases, and the amplitude increases drastically.For the Gr value of 8, there is again a significant jump in amplitude is seen.There is also a formation of 2.8 valued bolus in the uppermost annulus.

Velocity contour plots.
Velocity surface plots.Figure 5 shows the velocity surface plots for Darcy, solutal Grashof, and magnetic field number.Figure 5a shows how changing Darcy number affects the fluid velocity.On raising the Darcy number, the velocity decreases.Physically, as the medium porosity increases, the fluid experiences more resistance for its motion.For the increased value of Darcy number, there is an upward shift of the plots, meaning that the velocity increased.An increasing effect is seen on the left and right sides of the flow, meaning the boundary nanoparticles also boost their velocity.Figure 5b shows the effect of the solutal Grashof number.It says that on raising the values, the velocity increases.Physically, the Grashof number is inversely proportional to the viscous forces.When these forces decrease, the velocity increases.The plot shows a better velocity increase for the leftward nanoparticles.There is more elevation in the leftward waves, and the rightward waves also suffer a lower shift.The next Fig.5c, shows the effect of magnetic field number.When the magnetic field number is increased, the Lorentz force is developed.This force decreases the fluid velocity.The figure shows that for increased M values, the plots shift downward, meaning a decrease in the velocity.The right side sees a good velocity drop.However, the left wave experiences a lower drop than the other side.
Temperature surface plots.Figure 6 shows the effect on temperature behaviour for the heat source, Brinkman, and endothermic-exothermic chemical reaction parameters.Figure 6a shows the increasing change of heat source number on temperature.As the heat source number increases, more heat energy is released to the fluid and the fluid's temperature increases.For enhanced values of heat source, the temperature plots experience an upward shift, representing an increase in temperature.For a Q value of 5, there is a good jump in temperature at later y-values.Figure 6b expresses how the increasing Brinkman number affects the temperature.For elevated Brinkman number, the kinematic collisions rise, and the entropy rises.This result can be seen for lower y-values with rising temperature curves.However, for higher y-values, the effect does not dominate in the presence of other factors, and a temperature decrease is seen.Figure 6c shows the change of endothermic-exothermic reaction parameter on temperature.As the values for this parameter increase, the temperature also increases.The plots show an upward shift of curves for enhanced Kr2.Induced magnetism surface plots.Figure 8 reveals the impact on induced magnetism for increasing magnetic, magnetic Prandtl, and Reynolds numbers.Figure 8a reveals that for enhanced values of the magnetic number, the Lorentz forces start developing on the fluid.In response to these forces, lesser magnetism is induced for higher y-values.However, for lesser y-values, there is a good magnetism development in the fluid.The middle portion of the channel also sees a jump at the peaks of the curves.Figure 8b represents the enhanced behavior for magnetic Prandtl number and induced magnetism in the fluid.The magnetic Prandtl number is inversely proportional to the velocity of the fluid.For larger Pr m , the induced magnetism decreases for the right side of the channel.For the left waves of the channel, the induced magnetic field almost remains the same, or a little increment is seen.Furthermore, Fig. 8c is made for the Reynolds number and induced magnetism.The figure depicts that when the Reynolds number is increased, there is a higher shift of magnetism for the middle part of the channel, as the Reynolds number is inversely proportional to the viscous forces.As these forces decrease, the velocity increases, increasing the inducing effect.The Y-axis's higher part shows a slight decrease in the induced magnetism effect.
Correlation matrices.The correlation coefficient expresses the degree of association between two or more variables.It indicates the strength of the relationship between two variables, as well as the direction of the relationship (positive or negative).Typically, correlation is depicted by a coefficient ranging from − 1 to 1.A perfect positive correlation is represented by a correlation coefficient of 1, whereas a perfect negative correlation is represented by a correlation coefficient of 1.If the coefficient is 0, there is no association between the two variables.
The formula for the correlation coefficient (Pearson's r ) between two variables p and q can be written as: where n is the size of sample used in the study, p i and q i are the values of p and q for the ith observation, p and q are the sample means of p and q, respectively.Figure 9a,b show the correlation matrix for M f = 1 and M f = 5.When the induced magnetic field number rises, the positive correlation between the velocity and the temperature increases.The negative correlation between the velocity and concentration decreases as M f increase.The velocity and induced magnetism correlation become negative from positive.The temperature and concentration positive correlation decreases.The negative correlation between temperature and induced magnetism decreases.The concentration and induced magnetic field increase in the positive direction for the third decimal place.Figure 9c,d represent the correlation matrix for Fr = 1 and Fr = 5.On increasing the Forchheimer number, the positive correlation between velocity and temperature decreases.The negative correlation of velocity and concentration decreases in the negative direction.The r = n i=1 p i − p q i − q n i=1 p i − p     When the concentration of aluminium nanoparticles is increased, it is seen that the skin friction coefficient and mass transfer rate rise, however the opposite impact is seen for the rate of heat transfer coefficient.The skin friction coefficient, the mass transfer coefficient, and the heat transfer rate all rise with increased concentrations of copper nanoparticles, but the trend for the latter two reverses.The same effect is seen for iron oxide concentration.

Conclusion
This work investigates the ternary hybrid nanofluid made of copper (Cu), aluminium (Al), and iron oxide (Fe 2 O 3 ) with blood as the base fluid for the combined effect of viscous dissipation, gravity, external and induced magnetic field, highly porous medium, thermophoresis, Brownian motion, exponential heat source/sink, activation energy, and endothermic-exothermic chemical reactions are investigated.Velocity, temperature, concentration, and induced magnetism profiles, skin friction coefficient, entropy production, Nusselt number, Sherwood number, Bejan number, and mass transfer rates are discussed visually with appropriate arguments.The study's key findings include the following: • The fluid temperature increases for the enhanced values of heat source, thermophoresis, Brownian motion, and endothermic-exothermic chemical reaction parameter.• By raising the fitting constant, concentration ratio, chemical reaction parameter, and activation energy param- eter, the concentration rises.• The induced magnetism goes up on boosting the Reynolds, magnetic Prandtl, Darcy, and thermal Grashof numbers.
• The fluid velocity increases by decreasing the Forchheimer number.It first increased and then decreased for enhanced solutal Grashof number, channel inclination, and induced magnetic field number.• The entropy generation improves for increasing values of Reynolds, solutal Grashof, heat source/sink, Brink- man, magnetic Prandtl, endothermic-exothermic reaction parameter while reverse effect is noticed for chemical reaction and induced magnetic field parameter.• Bejan number increases by reducing the Reynolds number, Brinkman number, chemical reaction parameter, induced magnetic field parameter and enhancing the solutal Grashof, heat sink/source parameter, magnetic Prandtl number, endothermic-exothermic reaction parameter.• Skin friction coefficient, the Nusselt number, and the Sherwood number rise as the concentrations of Al, Cu, and Fe 2 O 3 rise.
The study finds applications in enhancing the design and efficiency of biomedical devices and drug delivery systems, optimize heat transfer processes in microfluidic systems, improve the performance of magnetohydrodynamic power generators, optimize energy conversion in renewable energy systems, and develop efficient cooling systems for electronic devices.Additionally, the findings can also be relevant in the development of advanced materials with tailored thermal properties, optimization of microscale chemical reactions, and understanding biological transport phenomena in living organisms.The research opens possibilities for numerous applications in the areas of biomedical engineering, energy systems, microfluidics, and materials science.
Validating the results of present study would require conducting experiments that closely mimic the conditions and parameters studied in the theoretical analysis.Experimental setups could involve the use of specially designed channels or flow systems that incorporate ciliated walls, magnetic field induction, and the ternary hybrid Jeffery nanofluid.Measurements of fluid velocity, temperature, and concentration gradients, as well as entropy generation rates, would need to be taken and compared with the predicted values from the theoretical model.These experiments would provide valuable empirical evidence to support the validity and applicability of the theoretical findings.
Further research avenues include exploring more biologically accurate models of cilia behavior, investigating multi-scale variations in nanoparticle distribution, employing advanced numerical techniques for enhanced accuracy, conducting experimental validation in controlled medical scenarios, and collaborating across disciplines to develop practical medical applications, such as optimized drug delivery and artificial organ design.

Limitations
The study considers some limitations and assumptions: 1.The investigation assumes synthetic cilia, thereby simplifying the behavior of real biological cilia.This simplification may overlook the intricate dynamics and interactions that natural cilia exhibit within physiological contexts.
2. The analysis focuses on a specific ternary hybrid nanofluid (Al-Cu-Fe2O3/Blood) with uniform nanoparticle distribution.Consequently, the study neglects potential variations in nanoparticle concentration that might exist in real-world applications.3. The utilization of the long-wavelength and low Reynolds number approximations could introduce limitations in predicting accurate outcomes, particularly when dealing with scenarios characterized by higher flow rates or substantial variations.4. Exploration confines itself to a single channel geometry, restricting the generalizability of findings to more complex and diverse anatomical structures.

Figure 1 .
Figure 1.Geometrical description of the cilia problem.

Figure 3 .
Figure 3. (a) Velocity VS Thermal Grashof number.(b) Velocity VS channel inclination.(c) Velocity VS Forchheimer number.(d) Velocity VS induced magnetic field number.(e) Temperature VS Heat source number.(f) Temperature VS Brownian motion number.(g) Temperature VS endo-exo reaction number.(h) Temperature VS thermophoresis number.(i) Concentration VS fitting constant.(j) Concentration VS chemical reaction number.(k) Concentration VS activation energy number.(l) Concentration VS temperature ratio.(m) Induced magnetism VS Reynolds number.(n) Induced magnetism VS magnetic Prandtl number.(o) Induced magnetism VS Darcy number.(p) Induced magnetism VS thermal Grashof number.

Figure 10
shows the entropy generation and Bejan number contour plots for various flow parameters.Figure 10a,b show the entropy generation and Bejan number plots for the Reynolds and solutal Grashof numbers.The figure shows that the entropy generation increases by enlarging

Figure 10 .
Figure 10.(a) EG VS Re VS Gc.(b) Be VS Re VS Gc.(c) EG VS Q VS Br.(d) Be VS Q VS Br.(e) EG VS Kr 1 VS Kr 2 .(f) Be VS Kr 1 VS Kr 2 .(g) EG VS M f VS Pr m .(h) Be VS M f VS Pr m .

Table 2 .
Ranges of the values of flow parameters.

Table 3 .
Physical significance quantities for nanoparticle concentrations.The skin friction coefficient is a dimensionless quantity that characterizes the frictional forces exerted by a fluid on a solid surface.It is an important parameter in the study of nanofluids and has important practical applications in various industries, including transportation, electronics, oil recovery, and biomedical applications.The mass transfer rate (Sherwood number) is defined as the rate at which the mass of a particular species is transported across a surface in a given direction due to the concentration gradient of that species.It has significant applications in various engineering fields, such as heat transfer, energy systems, chemical engineering, environmental engineering, and biotechnology.Heat transfer rate in nanofluids refers to the rate at which heat energy is transferred through a nanofluid.The concept is used in electronics cooling applications, solar thermal systems, nuclear reactors, and heat exchangers.Table3displays the relationship between skin friction coefficient, heat transfer rate (Nusselt number), and mass transfer rate (Sherwood number) and nanoparticle concentration.