Computational Assessment of Microrotation and Buoyancy Effects on the Stagnation Point Flow of Carreau-Yasuda Hybrid Nanofluid with Chemical Reaction Past a Convectively Heated Riga Plate.

The present framework deliberated the mixed convection stagnation point flow of a micropolar Carreau-Yasuda hybrid nanoliquid through the influence of the Darcy-Forchheimer parameter in porous media toward a convectively heated Riga plate. In this investigation, blood is used as a base liquid and gold (Au) and copper (Cu) are the nanoparticles. The main novelty of the present investigation is to discuss the transmission of heat through the application of thermal radiation, viscous dissipation, and the heat source/sink on the flow of a micropolar Carreau-Yasuda hybrid nanoliquid. Further, the results of the chemical reaction are utilized for the computation of mass transport. Brownian motion and thermophoretic phenomena are discussed in the current investigation. The current problem is evaluated by using the connective and partial slip conditions and is formulated on the basis of the higher-order nonlinear PDEs which are converted into highly nonlinear ODEs by exploiting the similarity replacement. In the methodology section, the homotopic analysis scheme is employed on these resulting ODEs for the analytical solution. In the discussion section, the results of the different flow parameters on the velocity, microrotation, energy, and mass of the hybrid nanofluid are computed against various flow parameters in a graphical form. Some of the main conclusions related to the present investigation are that the velocity profile is lowered but the temperature is augmented for both nanoparticles volume fractions. It is notable that the skin friction coefficient is reduced due to the higher values of the Darcy-Forchheimer parameter. Further, the rising performance of the hybrid nanofluid Nusselt number is determined by the radiation parameter.

Introduction

In the past few years, researchers and scientists have been interested in studying non-Newtonian fluid flow problems due to the extensive variety of applications in different fields of industry and manufacturing. Applications of non-Newtonian fluid flow are oil recovery, food dispensation, movements of the biological fluids, clay mixtures, cosmetics, paper production, pharmaceuticals, nuclear and chemical industries, paints, molten polymers, geophysics, bioengineering, oil storage engineering, paper manufacturing, and many others. Mohamed et al.[1] used parallel plates for the analysis of non-Newtonian nanoliquid flow with the application of the Hall current through the porous media. In this work, it was shown that the energy profile of the nanofluid is increased with the increase of the heat source/sink, but it has the opposite behavior for the Prandtl number. Xia Li et al.[2] utilized the Prandtl and Cattaneo-Christov effective approach to study the periodic flow of the non-Newtonian Casson nanofluid with Darcy-Forchheimer and motile gyrotactic microorganism. From this analysis, it was detected that the nanofluid velocity declined due to the Casson liquid constant and inertial coefficient. Imtiaz et al.[3] addressed the non-Newtonian Jeffrey liquid flow under the curved stretched surface with the assistance of autocatalytic chemical reactions. They noted that the fluid concentration is enhanced due to the enhancement of the heterogeneous reaction. Gowda et al.[4] exploited the RK-45 scheme for the numerical estimation of the non-Newtonian nanoliquid flow with activation energy. It was detected that the heat transportation is enhanced due to the result of the porosity factor. Ramzan et al.[5] explored the non-Newtonian nanoliquid flow toward the thin needle with entropic and dipole characteristics with the implementation of the homotopic analysis technique. They proved that the nanoliquid entropic behavior is augmented with the enrichment of the Lewis and Eckert numbers. Soomro et al.[6] reviewed the non-Newtonian nanoliquid flow through the inclusion of thermophoretic and Brownian motion over the vertical stretched surface. They solved their problem numerically by exploiting the finite-difference Crank Nicolson scheme. Bilal and Urwa[7] explained the non-Newtonian liquid flow problem over the thin needle, and this model was evaluated under the prevalence of variable viscosity and activation energy. When they enhanced the value of the buoyancy ratio parameters the drag force increased. Prasannakumara[8] addressed the phenomena of heat transport over the non-Newtonian Maxwell nanoliquid flow by using a stretchy surface. With the use of this model, he showed good thermal performance for a Newtonian fluid as compared to a Maxwell liquid when he increased the ferromagnetic interaction and volume fraction parameters. Further, the flow problems related to the non-Newtonian fluids are studied and discussed in refs (9−11.)

Nanofluids play a significant role in the improvement of heat transportation of the base liquid. Because the nanoliquid is created by mixing nanoparticles with the base liquid, it has the power to improve the base fluid’s thermal performance. Nanofluid flow has an enormous variety of applications in various arenas of bioscience and engineering including geothermal power extraction, lubricants, cooling of motor vehicles, magnetic resonance imaging (MRI), purification of the biomolecules, and cooling of electronic devices and heat exchangers, nuclear reactor vehicles cooling, thermal management, and many others. Many scientists and researchers have expanded nanofluid research due to its numerous functions. Eswaramoorthi et al.[12] dissected the outcomes of nonlinear thermal radiation over magnetohydrodynamic (MHD) Cu–Ag/water-based nanoliquid flow under a heated plate with the inclusion of a heat transport mechanism and explained that the Nusselt number in Ag-nanoparticles is greater as compared to the Nusselt number in Cu-nanoparticles. Alshehri and Shah[13] discussed the hybrid nanoliquid flow toward an extending surface with the application of viscous dissipation and the Darcy-Forchheimer model. In this investigation, it was noted that the porosity of the fluid decreased the speed of the liquid particles. Waqas et al.[14] inspected the impacts of the motile microorganisms on the 3D Carreau–nanoliquid flow using the bvp4c-technique. Muhammad et al.[15] discovered the effects of the chemical reaction on the bioconvection three-dimensional Jeffrey nanofluid flow by using the stretching surface technique. They determined that the nanoliquid mass contour is decayed for the chemical reaction parameter. Nadeem et al.[16] used convective conditions to discuss the MHD Walter-B nanoliquid flow with Brownian and thermophoresis effects. Ramzan et al.[17] assessed the Burger nanoliquid flow by using a homotopic analysis scheme under a stretching cylinder and sheet and discussed that the escalating estimates of the thermal Biot number rise faster than the Burger nanoliquid Nusselt number. Bilal et al.[18] dissected the energy transport phenomenon over the mixed convection nanofluid flow with a magnetic field toward a revolving disk. Akbar et al.[19] elucidated the role of the Hall effect over the MHD Carreau–Yasuda nanoliquid flow surrounded by a channel through the porous media. From their concluding remarks, it can be determined that the mass rate transport is lower due to the enhancement of the Schmidt number. Li et al.[20] demonstrated the two-phase flow of a non-Newtonian nanoliquid through a permeable H-shaped cavity with mixed convection surrounded by a porous media. In this model, they also used four rotatable cylinders inside the enclosure. In this work, it is noted that the drag force is augmented with increasing Darcy number. Kavusi and Toghraie[21] numerically analyzed heat pipe performance under the applications of the two-dimensional nanofluid model with the help of the finite volume technique and found that the fluid pressure is enhanced with the enhancement of the thermal capacity of the fluid. Moraveji and Toghrai[22] computationally discussed the energy allocation rate and nanofluid flow features by using the different parameters of a vortex tube. In this article, it was seen that the mass transition rate is enhanced from the cold and hot cross-section area due to the increase in the length of the vortex tube. Ruhani et al.[23] explored statistically the rheological impact of the silica-ethylene glycol-nanoparticles on hybrid ferrofluid flow by manipulating the experimental data with a water-base fluid. In this problem, the fluid is Newtonian due to the linear relationship between shear stress and shear rate. Mostafazadeh et al.[24] scrutinized the consequence of laminar flow of a nanoliquid through a vertical channel with the use of single-and two-phase approaches and deliberated that the kinetic energy for the velocity field is enhanced due to the increment of the fluid temperature. Arasteh et al.[25] considered the role of local nonequilibrium conditions and the Darcy-Forchheimer model on the nanofluid flow. The strong numerical method recognized as the finite volume method was employed for the solution of their problem. More studies on the nanofluid flow problem can be found in refs (26−28).

Hybrid nanofluids are very useful in transmitter and biotechnology, ships, radioactive systems, electrical coolers, generators, automobile industry, air conditioners, heat converters, heat pumps, and solar energy, etc. Hybrid nanofluids provide a more satisfying outcome with respect to heat transport relative to nanoliquids and conventional liquids. As a result, scientists and researchers are very interested in exploring hybrid nanofluid flow difficulties. Xia et al.[29] determined the significance of the Joule heating effect on hybrid nanoliquid flow with entropy optimized dissipative. Here temperature graph is enlarged for the greater estimates of the radiation and magnetic parameters. Kumar et al.[30] deliberated the upshot of the nonuniform heat generation on the dusty hybrid nanoliquid flow toward the rotating disk through the porous media. Jamshed et al.[31] interpreted the Williamson hybrid nanoliquid flow with an engine oil-based liquid by employing the Cattaneo-Christov heat flux model. Anuar and Bachok[32] highlighted the unsteady micropolar hybrid nanoliquid flow at a stagnation point under the occurrence of thermal radiation. They attained numerical solutions to their problem. In this research work, it was distinguished that the hybrid nanoliquid flow is augmented for the higher estimation of the material parameter and unsteady parameter. Mabood et al.[33] documented the convective flow of a hybrid ferrofluid under a stretched sheet. In this problem, the consequence of nonlinear radiation and the irreversible evaluation of the hybrid nanoliquid are also computed. They observed that increasing the estimation of the Bejan number amplified the generation of the liquid entropy. Gowda et al.[34] demonstrated hybrid nanoliquid flow across an enlarging cylinder. In this investigation, it was detected that the fluid particles speed is decreased when the nanoparticulate quantity is enhanced. Haider et al.[35] calculated the occurrence of the Darcy-Forchheimer flow through an absorbent space. They noticed that the hybrid nanoliquid has a lower heat transport proportion as compared to the titanium dioxide nanoliquid. Ramzan et al.[36] presented the analytical solution of the MHD flow of mixed convection hybrid nanoliquid with slip conditions over a stretchable sheet. In this evaluation, they found that the escalating estimates of the suction parameter decrease the hybrid nanofluid temperature.

Chemical reactions have a variety of engineering and industrial applications. The crop damage owing to the freezing atmosphere, formation of the fog, polymers production, water emulsions, oil, dehydration processes, hydro-metallurgical industries, ceramics, food processing, and manufacturing of papers are some of the industrial and engineering applications of chemical reactions. Many researchers analyze the characteristics of chemical reactions after being inspired by their applicability. Khan et al.[37] evaluated analytically the flow of a micropolar nanoliquid through the impact of viscous dissipation with convective conditions over a rotating thin needle. In this observation, it was perceived that the rising estimates of the chemical reaction parameter decreased the mass allocation. Reddy and Lakshminarayana[38] studied the chemical reaction of the MHD three-dimensional Maxwell nanoliquid flow in the prevalence of an energy source and cross-diffusion under a stretchable sheet. Kumar et al.[39] described the result of the chemical reaction over the Casson nanoliquid flow by using a curved elongating sheet. In this inquiry, it was seen that the streamflow of the nanofluid decreases when the Casson fluid parameter is augmented. Gul et al.[40] explained the magnetohydrodynamic hybrid nanoliquid flow by using the behavior of the chemical reaction over a stretchable cylinder. Further, they noticed that the expanding estimates of the energy sink parameter increased the energy of the hybrid nanoliquid. Gopal et al.[41] established numerically the model of MHD nanoliquid flow with the manifestation of buoyancy forces through the porous media along a stretched sheet. In this article, it was predicted that the mass transport of the hybrid nanoliquid is decreased due to the intensifying chemical reaction effect. Lv et al.[42] described the consequence of the Hall effect over the nanoliquid flow as a result of a radiation effect through a rotating channel. They discussed that the energy frequency transport is higher for larger evaluations of the Prandtl number. Gowda et al.[43] described the second-grade nanoliquid flow with the existence of the chemical reaction and Brownian diffusivity. In this problem, they found that the augmented approximation of the porosity term decreased the nanoliquid Nusselt number. Rasheed et al.[44] used the convective conditions and internal heating source for the flow of a nanoliquid. It was seen that the expansion in the magnetic parameter led to improving the thermal profile of the nanoliquid.

In the light of the above-cited literature, the current problem is framed for the evaluation of the heat and mass transport phenomena on the mixed convection flow of Carreau–Yasuda hybrid nanofluid toward the convectively heated Riga plate through the porous medium. The hybrid nanofluid is formed by mixing the gold (Au) and copper (Cu) nanoparticulates in the blood-base liquid. The roles of the Darcy-Forchheimer flow, viscous dissipation, chemical reaction, and thermal radiation are computed in the present scrutiny. The micropolar Carreau–Yasuda hybrid nanofluid is the non-Newtonian fluid. So, the daily life examples of such fluids are corn starch, silly putty, some brakes pads in cars, quicksand, and corn flour, etc. A simulation of the higher-order ODEs is performed with the implementation of the homotopic analysis scheme. Effects of the several flow constraints over the velocity, microrotation, mass, and energy of the hybrid nanofluid are computed and discoursed in a graphical form. The Nusselt number and skin friction are scrutinized in contour graphs against numerous flow parameters. At the end of the present investigation, we will be able to find out the answers to the following questions:

How do the gold and copper nanoparticles affect the mixed convection micropolar Carreau–Yasuda hybrid nanofluid?

How does the Lorentz force affect the velocity of the micropolar Carreau–Yasuda hybrid nanoliquid by using the Hartmann number?

What is the behavior of the velocity of the micropolar Carreau–Yasuda hybrid nanofluid versus different flow parameters?

How does the microrotation profile of the micropolar Carreau–Yasuda hybrid nanofluid behave against the microrotation parameter and microrotation slip parameter?

What is the behavior of the micropolar Carreau–Yasuda hybrid nanofluid temperature via discrete flow parameters?

What is the role of the radiation parameter on the Nusselt number?

How does the skin friction coefficient behave against the Darcy-Forchheimer parameter?

Problem Formulation

Consider the 2D, incompressible and steady flow of a mixed convection micropolar Carreau–Yasuda hybrid nanofluid along with Darcy-Forchheimer behavior in a porous medium under the convectively heated Riga plate. Also, assume that the fluid is pseudoplastic. In this study, blood is used as a base liquid and gold (Au) and copper (Cu) are used as the nanomaterials. By combining the electrodes and magnets the Riga plate is designed for the present problem. Effects of thermal radiations, heat source, and viscous dissipations are employed for the evaluation of heat transport under convective and partial slip conditions. Additionally, the roles of Brownian motion and thermophoretic are evaluated. The phenomena of mass transport are considered by using the chemical reaction. The velocity of the stretching sheet is u(x) = ax and the ambient velocity of the liquid is u∞(x) = bx, but the origin is fixed. Here T is the convective liquid temperature but the ambient temperature of the liquid is T∞. For the formulation of the current problem, the x-axis is taken upright while the y-axis is horizontal over the Riga plate in the Cartesian coordinate system. Figure panels a and b are designed for the physical illustration of the current model.

Figure 1

(a) Physical view of the Riga plate. (b) Schematic illustration of the problem.

The equation for a Carreau–Yasuda fluid is[45]Here μ0 and μ∞ are the zero and infinite shear rate viscosity. The parameters for the Carreau–Yasuda liquid are Γ and d. A1 is the Rivlin-Ericksen tensor and τ is the extra stress tensor, γ̇ is and A1 = [(gradv) + gradv]. When μ∞ = 0 eq becomes

By considering the above-mentioned assumptions on the flow behavior, the governing equations of the presented model are[45−48]The components of velocity along the directions of x-axis and y-axis are u and v. The ambient velocity is u∞, the hybrid nanofluid dynamics viscosity is μ, the vortex viscosity is k1*, N* is the microrotation parameter, the hybrid nanoliquid viscosity is v, n is the power index, the parameters for the Carreau–Yasuda hybrid nanoliquid are Γ and d, the gravitational acceleration is g, C∞ is the ambient concentration of the liquid, β is the thermal expansion factor, j0 is the current density, the magnetization of the magnet is M0, ρ is the base liquid density, ρ is the density of the particles, T and T∞ are the temperature and ambient temperature of the liquid correspondingly, the inertia is in which the drag coefficient is C, is the spin gradient, where the length scale parameter is expressed by , the thermal conductivity of the hybrid nanoliquid is k, at a constant pressure the specific heat is C, σ* is the Stefan–Boltzmann constant and the coefficient of absorption is k*, τ is the ratio between the nanoparticles heat capacity and base liquid capacity, D and D are the Brownian and thermal diffusion factors, Q0 is the heat source/sink, and k2 is the chemical reaction rate constant.

The boundary conditionsIn the above-mentioned boundary conditions, u is the velocity of the stretching surface, a and b are constants, N* is the microrotation parameter, the microrotation slip parameter is m0, the thermal conductivity is k, and the ambient velocity is u∞.

The thermophysical properties of the hybrid nanofluid are defined asHere hybrid nanoliquid thermal conductivity is k, the base liquid thermal conductivity is k, ϕ1 is the nanoparticle volume fraction of the first nanoparticle, the nanoparticle volume fraction of the second nanoparticle is ϕ2, ρ is the hybrid nanofluid density, the base liquid density is ρ, the density of the first and second nanoparticles is ρs1 and ρs2, μ is the dynamic viscosity of the hybrid nanofluid, μ is the base fluid dynamic viscosity, (C) is the specific heat constant of the hybrid nanofluid, the specific heat constant of the base fluid is (C), (ρC) and (ρC) are the specific heat constants of the first and second nanoparticles, and σ is the electrical conductivity of the base fluid.

Table lists the physical properties of the base liquid and nanoparticles.

Table 1

Thermophysical Characteristics of the Base Liquid and Nanoparticles[49]

property	blood	Au (gold)	Cu (copper)
Cp (J/(kg·K))	3594	129	385
ρ (kg/m3)	1063	19320	8933
k (W/(m·K))	0.492	314	401
βT × 10–5 (1/K)	0.18	1.4	1.67
σ (S/m)	6.67 × 10–1	4.10 × 107	59.6 × 106

Similarity transformation is defined as

By using the above similarity variables defined in Eq. , it is obtained thatand the converted boundary conditions areAfter the simplification of the present problem some important nondimensional parameters are listed here:

Weissenberg number:

Stretching parameter:

Darcy-Forchheimer parameter:

Buoyancy ratio parameter:

Mixed convection parameter:

Hartmann number:

Width parameter:

Thermal radiation:

Heat generation parameter:

Brownian motion parameter:

Thermophoresis:

Eckert number:

Schmidt number:

Chemical reaction constraint:

Microrotation parameter:

Prandtl number:

Porosity parameter:

In the current examination the physical quantities are discussed aswith τ and qAfter the simplification, the dimensionless form of eq is is the Reynolds number.

Solution of the Problem

In the physical situation, sometimes the mathematically modeled problem is not solved easily especially in the case of highly nonlinear differential equations. For the solution of such a nonlinear system of equations, the scientists and mathematicians have developed a different type of techniques. By using the HAM technique,[50−55] the various physical models related to fluid mechanics have been studied by different researchers. The advantage of the homotopy analysis method over the other methods are as follows:

Homotopy analysis method is a generalized method because it is valid for both strongly and weakly nonlinear problems.

This scheme is independent of small and large constraints.

Any sort of nonlinear PDEs without discretization and linearization can be solved by using this technique.

This method is linear and does not require any base function.

Series and convergent solutions of the system are determined by using this method.

The present problem is simulated by using the HAM scheme. For this, the initial guess is determined asand the linear operators arewith propertieswhere J(i = 1, 2, 3, ···, 9) are the constants.

Zeroth Order Deformation Problem

In the present problem, the zero-order deformations areHere ε is the embedding, and h, h, hθ, and hϕ are the nonzero auxiliary factors. N, N, Nθ, and Nφ are the nonlinear operator and discoursed asFor ε = 0 and ε = 1 eqs –25 becomeThe Taylor series expansion is applied to eqs –37 and it is obtained thatBy putting ε = 1, in eqs –41, the convergence of the series is achieved as

mth Order Deformation Problem

The mth order form of the problem isThe Rm(ζ), Rm(ζ), Rθm(ζ), and Rϕm(ζ) are defined asThe general solution of the present problem is attained with the use of the particular solution

Results and Discussion

The mixed convection stagnation point flow of the micropolar Carreau–Yasuda hybrid nanoliquid toward the convectively heated Riga plate is elaborated. For the analytical explanation of the model the HAM method is used on eqs –14 along with boundary conditions eq . The physical behavior of the velocity, microrotation, mass, and energy of the hybrid nanofluid are displayed graphically. The computation for the skin friction coefficient and Nusselt number of the hybrid nanoliquid are performed in contour graphs.

Velocity Profile

The influence of the Hartmann number Ha, Darcy-Forchheimer factor Fr, microrotation parameter K, mixed convection parameter λ, power index number n, buoyancy ratio parameter N, and Weissenberg number We versus velocity contour are observed in Figures –10. Figure is captured to evaluate the variation of the hybrid nanofluid velocity under the effect of the Ha. In this figure, it is noticed that the hybrid nanoliquid velocity is amplified when the Hartmann number Ha is enhanced. The external electrical field increases due to the increase of the Hartmann number Ha which consequently increases the hybrid nanoliquid velocity. The boundary layer thickness rises when Ha rises. Also, the Lorentz force is produced when the Hartmann number Ha increases. Figure determines the significance of the Darcy-Forchheimer Fr on hybrid nanofluid velocity. The hybrid nanofluid velocity is reduces against the expanding values of the Darcy-Forchheimer parameter Fr. The Forchheimer concept is applied for the modeling of the Darcy-Forchheimer parameter, and the Fr has a nonlinear relationship against the flow of the liquid. Further, the thickness of the motion of the fluid particles is enhanced when the Fr is enhanced. When the applications of the Forchheimer theory are discussed then the retardation force is visualized. The liquid motion is slowed due to the upsurge of the porosity of the surface because the Fr is associated with the porosity of the surface. So, the liquid velocity is lower when the Darcy-Forchheimer parameter is higher.

Figure 2

Results of velocity outline versus Ha.

Figure 10

Results of velocity outline versus ϕ2.

Figure 3

Results of velocity outline versus Fr.

Figure depicts the fluctuation of the velocity outline via discrete values of the microrotation parameter K. It is viewed that the velocity profile is heightened due to the escalating estimates of the microrotation constant K. It is spotted that with the escalation of the microrotation parameter K, the boundary layer thickness diminishes. The outcome of λ on the velocity profile is debated in Figure . In this inquiry, it can be seen that the greater estimates of the mixed convection parameter λ lead to boosting the hybrid nanofluid velocity profile. The role of the power index number n over the velocity profile is discussed in Figure . In this figure, the reduction in the hybrid nanoliquid velocity versus larger values of the power index number n is examined. Figure establishes the result of the buoyancy ratio constraint Nr over the velocity profile. In this scrutiny, it is detected that the velocity curve is raised through the higher approximation of the buoyancy ratio parameter Nr.

Figure 4

Results of velocity outline versus K.

Figure 5

Results of velocity outline versus λ.

Figure 6

Results of velocity outline versus n.

Figure 7

Results of velocity outline versus Nr.

Figure determines the deviation of the velocity outline with the increase of the Weissenberg number We. It is predicted that the velocity curve is elevated for rising values of We. The consequence of the nanoparticles volume fraction ϕ1 on the velocity field is elaborated in Figure . In this inquiry, it is noticed that when the nanoparticles volume fraction ϕ1 is higher, the velocity of the hybrid nanofluid decreases. Figure explores the variation of the hybrid nanofluid velocity versus escalating values of ϕ2. The decreasing behavior in the velocity profile is perceived to expand the estimation of the nanoparticles volume fraction ϕ2. The reason behind this is that when the nanomaterial quantities are enhanced then the number of the nanoparticles in the base fluid increase, which consequently diminishes the hybrid nanofluid velocity. The boundary layer thickness also increases by the mixing of the nanoparticulates in the base liquid which opposes the fluid motion.

Figure 8

Results of velocity outline versus We.

Figure 9

Results of velocity outline versus ϕ1.

Microrotation profile

The fallout of the microrotation parameter K and microrotation slip parameter m0 on the microrotation profile of the hybrid nanoliquid is discussed in Figures and 12. Figure established the results of the microrotation constraint K on the hybrid nanoliquid microrotation profile. It is perceived that the hybrid nanoliquid microrotation profile shows decrement performance for intensifying values of the microrotation factor K. The significance of the microrotation slip factor m0 on the microrotation profile is constructed in Figure . This figure explained that the expanding estimation of the microrotation slip parameter m0 augmented the microrotation profile of hybrid nanofluid.

Figure 11

Results of hybrid nanofluid microrotation profile due to K.

Figure 12

Results of hybrid nanofluid microrotation profile due to m0.

Temperature Profile

The roles of thermal radiation parameter R, Bi, Ec, thermophoresis parameter Nt, and heat generation parameter Q on the hybrid nanofluid temperature are demonstrated in Figures –17. Figure signifies the temperature for larger values of R. It can be seen that the rising estimates of R increase the temperature. Figure displays the behavior of the energy curve under the effect of thermal Biot number Bi. In this figure, it is evident that the escalating values of the Bi produce an increment in the hybrid nanoliquid temperature. Further, the boundary layer thickness and energy rate increase when Bi rises; therefore, the hybrid nanofluid temperature profile increases. The analysis of Ec on the temperature contour is displayed in Figure . It is depicted that the temperature increases for greater values of the Eckert number Ec. The drag force between the particles of the liquid is amplified due to the intensification of the Eckert number which consequently increases the heat transport rate and alternatively the temperature of the fluid is enhanced. Further, the relationship between the kinetic energy and the heat enthalpy variation is called the Eckert number. So, the kinetic energy of the hybrid nanoliquid increases with the increase of Ec. The temperature is discussed as the average kinetic energy. Therefore, the hybrid nanofluid temperature rises. Figure exhibits the results of the Nt on the hybrid nanofluid temperature. It can be seen that the temperature declines for rising values of Nt.

Figure 13

Results of energy outline versus R.

Figure 17

Results of energy outline versus Q.

Figure 14

Results of energy outline versus Bi.

Figure 15

Results of energy outline versus Ec.

Figure 16

Results of energy outline versus Nt.

Figure is graphed to evaluate the impact of Q over the temperature profile. In this analysis, it is revealed that the hybrid nanoliquid temperature is augmented for rising values of Q. Figure is sketched for the assessment of the hybrid nanoliquid temperature with respect to the growing estimation of the nanoparticles volume fraction ϕ1. In this figure, it is apparent that the intensifying values of the ϕ1 led to an improvement of the temperature profile. The role of the nanoparticles volume fraction ϕ2 on the energy contour is shown in Figure . This figure shows that the hybrid nanoliquid temperature graph is augmented due to the augmentation of ϕ2. The hybrid nanofluid thermal conductivity rises when the nanoparticles are added in the base liquid. So, the thermal efficiency of the hybrid nanofluid is also enhanced, thus the temperature increases due to the increase of nanoparticles volume fraction.

Figure 18

Results of energy outline versus ϕ1.

Figure 19

Results of energy outline versus ϕ2.

Concentration Profile

Figures –22 visualize the change in the concentration of the hybrid nanoliquid via Sc, Nt, and kr. The physical description of the hybrid nanofluid concentration for intensifying values of Sc is scrutinized in Figure . In this inquiry, it is eminent that the hybrid nanoliquid concentration is weakened for rising values of Sc. The molecular diffusivity decreases when the Schmidt number increases. So, the hybrid nanofluid concentration is enhanced with the expanding of Sc. Figure epitomizes the impact of the Nt on the mass outline. In this graph, it can be seen that the concentration field is decayed due to enhancement in the values of the Nt. Figure addresses the results of kr on the concentration. The decline behavior in the concentration is noticed for the expanding estimation of kr. It is perceived that the collision between the particles of the fluid is enhanced due to the expansion of the chemical reaction parameter kr near the surface which consequently reduces the concentration of the hybrid nanofluid and boundary layer thickness.

Figure 20

Results of mass outline versus Sc.

Figure 22

Results of mass outline versus kr.

Figure 21

Results of mass outline versus Nt.

Skin Friction and Nusselt Number

The deviation of C and Nu versus different flow parameters in two-dimensional contour graphs are deliberated. Figure examined the role of the Darcy-Forchheimer Fr factor on skin friction. In this figure, it is identified that C is decayed due to the intensifying Fr values. Figure displayed the performance of Nusselt number Nu for higher values of the radiation parameter R. The increasing behavior in Nu of the hybrid nanofluid is detected for expanding values of the radiation parameter R.

Figure 23

2D view of skin friction coefficient against Fr.

Figure 24

2D view of Nusselt number against R.

Conclusion

In the current analysis, the two-dimensional mixed convection flow of Ag–Cu/blood-based micropolar Carreau–Yasuda fluid with Darcy-Forchheimer parameter in a porous medium by using the convectively heated Riga plate is examined. The main objective of this problem is to simulate the influence of the thermal radiation, viscous dissipation, and heat source/sink on the Ag–Cu/blood-based micropolar Carreau–Yasuda hybrid nanofluid. The phenomenon of the heat and mass transport is analyzed under the convective and partial slip conditions. The HAM technique is operated for the analytical exploration of the problem. The mathematical framework of the present problem is discussed in terms of the system of the of PDEs. With the assistance of suitable similarity transformations, these higher-order nonlinear PDEs are converted into highly nonlinear ODEs. The physical significance of the different flow parameters such as Hartmann number, Darcy-Forchheimer factor, microrotation, mixed convection, power index number, buoyancy ratio, Weissenberg number, microrotation slip, thermal radiation, thermal Biot number, Eckert number, thermophoresis parameter, heat generation parameter, Schmidt number, and chemical reaction on the velocity, microrotation, mass and energy profiles of the hybrid nanofluid are computed. The major concluding remarks of the present work are

Increment performance of the hybrid nanofluid velocity is examined for Hartmann number, microrotation factor, mixed convection constraint, buoyancy ratio parameter, and Weissenberg number.

The Darcy-Forchheimer parameter, power index number, and nanoparticle concentrations diminished the hybrid nanoliquid velocity.

The microrotation profile of the hybrid nanoliquid increases for microrotation slip parameter but decreases for microrotation parameter.

An enhancing behavior of the hybrid nanoliquid temperature is perceived due to the increasing radiation term, Eckert number, thermal Biot number, heat generation factor, and nanoparticle concentrations.

The thermophoresis parameter declined the energy profile.

The hybrid nanofluid concentration is weakened for Schmidt number and chemical reaction effect.

It is noticed that the skin friction coefficient of the hybrid nanofluid is diminished with the escalating of the Darcy-Forchheimer parameter.

Nusselt number of the hybrid nanofluid is higher for the radiation parameter.

Future work

The present problem can be extended in the future as follows:

The present problem can be studied in the form of three-dimensional flow.

This problem can be extended for different boundary conditions.

This model can be extended for different base fluids such as kerosene oil, honey, engine oil, and so forth.

In this problem the mixed convection micropolar Carreau–Yasuda hybrid nanofluid flow is analyzed, but in the future it can be studied for any non-Newtonian fluid such as Maxwell fluid, Jeffrey fluid, Williamson fluid, or Casson fluid, etc.

Behaviors of bioconvection, Cattaneo-Christov energy, and mass flux, entropy generation, and activation energy can be analyzed.

For the simulation, different numerical and analytical techniques can be used.