M. Fllh Njfdi , H. TleiRostmi , Kh. Hosseinzdeh , , D.D. Gnji

a Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran

b Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran

Keywords: Akbari-Ganji’s method Nanofluid Vertical channel Combined convection Polynomial boundary condition

ABSTRACT In this research, a vertical channel containing a laminar and fully developed nanofluid flow is inves- tigated. The channel surface’s boundary conditions for temperature and volume fraction functions are considered qth-order polynomials. The equations related to this problem have been extracted and then solved by the AGM and validated through the Runge-Kutta numerical method and another similar study. In the study, the effect of parameters, including Grashof number, Brownian motion parameter, etc., on the motion, velocity, temperature, and volume fraction of nanofluids have been analyzed. The results demonstrate that increasing the Gr number by 100% will increase the velocity profile function by 78% and decrease the temperature and fraction profiles by 20.87% and 120.75%. Moreover, rising the Brown- ian motion parameter in five different sizes (0.1, 0.2, 0.3, 0.4, and 0.5) causes lesser velocity, about 24.3% at first and 4.35% at the last level, and a maximum 52.86% increase for temperature and a 24.32% rise for ψo(hù)ccurs when N b rises from 0.1 to 0.2. For all N t values, at least 55.44%, 18.69%, for F (η), and Ω(η), and 20.23% rise for ψ(η) function is observed. Furthermore, enlarging the N r parameter from 0.25 to 0.1 leads F (η) to rise by 199.7%, fluid dimensionless temperature, and dimensional volume fraction to decrease by 18% and 92.3%. In the end, a greater value of q means a more powerful energy source, amplifying all velocity, temperature, and volume fraction functions. The main novelty of this research is the combined convection qth-order polynomials boundary condition applied to the channel walls. Moreover, The AMG semi-analytical method is used as a novel method to solve the governing equations.

Adding nanoparticles to the primary fluid forms nanofluids. This technique can be considered one of the most widely used meth- ods to increase heat transfer quality and enhance the system’s ef- ficiency [1–4] . One heat transfer method that is very common and popular in industries and nature is combined heat transfer. This kind of heat transfer happens when we have natural and forced heat transfer simultaneously [5–8] . When we have nanofluid and combined heat transfer together, a new field of study is created. Researchers have done many studies in the mentioned field, such as Mohammad et al. [9] studied a microchannel that was consid- ered a heat exchanger. They evaluated the flow and heat transfer and found that adding nanoparticles to the fluid can raise the heat transfer coefficient. Hajipour et al. [10] considered a vertical chan- nel full of porous media with the nanofluid flow. They assessed the nanofluid heat transfer for severalNbandNtwhen the chan- nel’s wall temperature was not changing. In another article, Ha- jipour et al. [11] numerically investigated a nanofluid flow in a ver- tical channel’s entrance area in the presence of natural and forced convection. In that article, the flow and temperature fields’ char- acteristics have been presented in terms of a mixed-convection parameter, Brinkman number, Darcy number, Lewis number, and other vital parameters. Torabi et al. [12] worked on a microporous channel to realize the entropy generation attitude. The achieve- ments indicated that when we have a robust temperature jump parameter, it is necessary to consider the interface’s entropy gen- eration rate in total entropy generation calculation. Nevertheless, when the temperature jump parameter is not strong enough, we are able to omit it. Seetharamu et al. [13] examined a micro-porous channel. The heat transfer for the channel was under the effect of heat flux and the channel’s wall temperature, which were in- constant. Their outcomes illustrated that the velocity slip coeffi- cient almost has no significant influence on the Nusselt numbers in every state. Ghadikolaei et al. [14] analyzed Magnetohydrody-

https://doi.org/10.1016/j.taml.2022.100356

2095-0349/?2022TheAuthor(s).PublishedbyElsevierLtdonbehalfofTheChineseSocietyofTheoreticalandAppliedMechanics.Thisisanopenaccessarticleunderthe CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) namic Eyring-Powell flow in a channel. The channel’s surfaces were stretching, and the flow inside was unsteady. The most important results showed that were Lorentz force generated by the mag- netic field can decline the velocity and temperature. Najim et al. [15] studied the evaporation of a liquid on the channel’s wavy sur- face. The achievements indicated by rising the number of waves on the surface and enlarging their amplitude, heat, and mass transfer are improved. Shirkhani et al. [16] worked on the Newtonian fluid among parallel plates. The fluid was incompressible, and flow was unsteady and time-dependent. They tested the effect of Reynolds number and suction or injection parameters on the fluid veloc- ity. Jha et al. [17] assessed the flow in a vertical micro-concentric- annulus. The flow was steady, fully developed, and under natural and forced convection and heat generation or absorption. They re- alized that raising the heat generation is one way to increase the fluid’s velocity and temperature. However, it can decrease the heat transfer rate at the inner cylinder’s outer surface. Opposite conse- quences are noticed for heat absorption. Oni et al. [18] analyzed a vertical microchannel under the electrokinetic influence and in the presence of transient natural heat convection. Results showed that a suitable choice of flow parameters could improve skin friction and mass flux. Najim et al. [19] , in another paper, used a model to study the evaporation of nanofluids film a vertical channel. They examined how parameters like heat flux, inlet liquid mass flow, and volume fraction can affect evaporation. Bouhezza et al. [20] in- vestigated a nanofluid flow mixed convection in a vertical chan- nel. Results showed that nanofluid velocity and Nusselt number could be affected by nanoparticle volume fractions and the Brow- nian motion parameter. Moreover, the Cu-water nanofluid showed better heat transfer capability than Al2O3-water. Zangooee et al. [21] analyzed the MHD nanofluid flow between two disks when there was radiation from the disks. The outcomes showed that with an increase in the lower disk’s stretching rate values, the ra- dial and axial velocities boost near the lower disk, and the concen- tration and temperature rise. Also, it was proved that the Nusselt number enhances with stretching. Zhang and Yang [22] evaluated flow and natural convectionheat transfer in a vertical air channel. The results demonstrated that the flow transition, velocity promo- tion, and temperature increase, mainly occur in the near-wall ar- eas. Some other relevant studies are presented in Refs. [23–29] .

Based on the above, it can be said that using various materials, especially nanoparticles, to increase the efficiency of channels has been very successful, and so much research has been done in the field of channels, nanoparticles, and mixed convection. However, in this study, a laminar fully developed nanofluid flow in a verti- cal channel is studied. The nanofluid exchanges heat with its sur- roundings in the form of combined heat transfer, while the tem- perature and volume fraction distributions in the channel’s surface are considered polynomial functions of the channel length as the novelty. These forms of functions for temperature and volume frac- tion exist in many natural heat transfer phenomena and are criti- cal to investigate. The corresponding formulas for the problem are solved numerically and analytically by Runge-Kutta, and Akbari- Ganji’s method, respectively, and the effect of some critical param- eters such as Grashof number, Brownian motion parameter, ther- mophoresis parameter, buoyancy parameter, boundary conditions, and position in the channel were examined.

This study investigates the heat transfer and motion in a chan- nel containing nanofluid. Fluid flow has convection heat transfer via the channel’s surface, so the nanofluid temperature in any spot is affected by the surface’s temperature, which is not constant and increases along the channel. In a similar way, the volume fraction on the surface along they-direction changes with position, and it imposes an impression on the nanofluid volume fraction. Due to the effect of temperature and volume fraction from the surfaces, nanofluid velocity is also influenced. Fig. 1 shows the nanofluid in the vertical channel and functions for temperature and volume fraction on the channel’s surface.

Fig. 1. Nanofluid in the vertical channel.

Fig. 2. AGM and 4th-order Runge-Kutta method for F (η) with ξ= 1 , q = 2 , N b = 0.1 , N t = 0.1 , N r = 0.1 , Gr = 0 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

As evident in Fig. 1 , temperature (θ) and volume fraction (?) functions are consideredq-th-degree polynomial with just two sentences –one is an independent variable, and another is a con- stant number.CandDare constant in the formula, and tempera- ture and volume fraction in origin are consideredθi,?i.

Fig. 3. AGM and 4th-order Runge-Kutta method for Ω(η) with ξ= 1 , q = 2 , N b = 0.1 , N t = 0.1 , N r = 0.1 , Gr = 0 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

Fig. 4. AGM and 4th-order Runge-Kutta method for ψ(η) with ξ= 1 , q = 2 , N b = 0.1 , N t = 0.1 , N r = 0.1 , Gr = 0 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

Fig. 5. Gr number influence on F (η) with ξ= 1 , q = 2 , N b = 0.1 , N t = 0.1 , N r = 0.1 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

To formulate the problem in the first phase, we use the equa- tion of conservation of total mass in two dimensions:

whereuandvare velocity inxandydirections. In the next phase, we use the famous relationship of momentum conservation in they-direction (direction of motion is only in they-direction):

In the presented equation above,pis pressure,μis dynamic viscosity,ρbandρnrepresent base fluid and nanoparticle density,βis volumetric expansion coefficient, andgis equal to 9.81 as gravitational acceleration. In the third phase, we use energy con-

servation correlation:

In Eq. (3) ,αis thermal diffusivity,cbandcnare base fluid and nanoparticle specific heat capacity,DBMis Brownian diffusion co- efficient, andKthis the thermophoretic diffusion coefficient. In the last phase for nanoparticle volume fraction, we have:

We assume that the nanofluid flow is fully developed, so the boundary conditions for the problem are:

whereWis the channel’s width. To eliminate the geometrical ef- fect, we introduce new dimensionless parameters:

Moreover, we form dimensionless functions for pressure (Δ), velocity (F), temperature (Ω), and volume fraction (ψ) in the nanofluid:

In the presented formula,Vis the average velocity,ReandPrare Reynolds and Prandtl numbers, with the following definition:

Parameters includingGras Grashof number,Zas pressure pa- rameter,Leas the Lewis number,Nbas Brownian motion parame- ter,Ntas the thermophores is parameter, andNras buoyancy ratio, are needed to construct the final differential equations of the prob- lem:

Consequently, by using terms in Eqs. (6) –(9) and neglecting the pressure gradient, the Eqs. (1) –(4) transform into three coupled nonlinear ordinary differential equations:

New boundary conditions for the Eqs. (10) –(12) considering the boundary conditions in Eq. (5) ,are as follows:

Thus, the final equations and boundary conditions for the verti- cal channel with polynomial boundary conditions for temperature and volume fraction are formed. Now by solving the primary equa- tions in different conditions (for solving,ξis equal to 1, andqis equal to 2), we are able to understand the effect of each critical parameter on the velocity, temperature, and volume fraction in the nanofluid.

In order to solve nonlinear ordinary or partial differential equa- tions, Akbari-Ganji’s Method (AGM) is one of the most practical methods. This method was invented by Mohammadreza Akbari and Davood Domairry Ganji [30–32] . For solving a set of equations by the AGM, we follow the procedure below. Suppose a nonlinear differential equation (G) is as follows:

With boundary conditions of

And we assume the answer is a series (this series can either be a summation of polynomials, a set of sin-cos, or exponential terms)

Based on the seriesh(σ)and its derivatives inσ=0, we have:

and whenσ=a:

Substituting Eqs. (17) and (18) in Eq. (14) , we can obtain:

With regard to selectingnin Eq. (16) (obviously, we cannot con- sider it as infinity, and when the number of unknowns is more than the number of available equations), and in order to make a set of equations that consist ofn+1equations andn+1unknowns, we differentiate Eq. (14) :

Applying the boundary conditions on the derivatives ofG, we have:

Finally,n+1equations andn+1unknownsare formed in a solv- able manner, and it is possible to calculate the coefficients and gain the final answer ash(σ). With the same attitude, a set of ODEs can be solved. For our problem, we consider the following equations:

and, we assume theF,Ω, andψfunctions are series with polyno- mial sentences, as described below:

By replacing the series ofF,Ω, andψin Eqs. (23) –(25) , we have:

Concerning applying the boundary conditions in Eq. (13) , a set of equations and unknowns is formed. Solving that set of equations gives us the unknown coefficients:

By substituting the unknowns in Eq. (32) in Eqs. (26) –(28) , we have the final answers forF,Ω, andψ:

Note that these coefficients are solved for specific conditions (ξ=1,q=2,Gr= 0,Nt= 0.1,Nb= 0.1,Nr= 0.1,Pr= 1,Re= 1,Z= 0,Le= 10) and are different for other conditions of the problem.

To ensure the accuracy of the AGM semi-analytical method used to solve the Eqs. (10) –(13) is acceptable; results are compared with the numerical4th-order Runge-Kutta method. Due to the vali- dation, results for dimensionless velocity, temperature, and volume fraction of the nanofluid are depicted in Figs. 2–4 . By comparing the outcomes of the numeric method and AGM, we can see that our results are very close to each other, and the error obtained in most cases was less than 0.6% for all three functions, which proves the correctness of the applied method. Moreover, to prove that AGM can solve these problems and equations correctly, this method has been used to solve the main equations in Fakour et al. [33] for a relevant topic. Table 1 shows the results. Again it can be understood that AGM is a proper method to solve these kinds of problems.

Table 1 Comparison between the results of AGM and Fakour et al. [33] .

After verifying the AGM, we examined the effect of essential parameters such asGr,Nb,Nt,Nr,q, andξon the dimensionless velocity, temperature, and volume fraction functions in the main equations.

The velocity diagram for different Grashof numbers (1, 2, 3 and 4) is shown in Fig. 5 . From the figure, it can be seen that the nanofluid velocity at the channel’s surfaces is equal to zero due to friction and fluid adhesion but is extreme in the middle (x=W) in a symmetric form (it should be mentioned that nanofluid velocity and dimensionless velocity function have a direct relationship with each other based on Eq. (7)).The Grashof number indicates the ra- tio of buoyancy to viscosity, so by increasing the Grashof number, the velocity will also increase. Nevertheless, this increase makes an inconstant rate of change in the dimensionless velocity in ev- ery step. In other words, increasing theGrfor one unit from 1 to 4 raises theF(η)for 78%, 22%, and 13%. Figure 6 shows the nanofluid dimensionless temperature diagram. Nanofluid in the vicinity of the surface receives heat through conduction, and then in the sub- sequent layers, the heat transfer changes from conduction to con- vection, which is much stronger. So, nanofluid dimensionless tem- perature, which indicates the difference between surface temper- ature and fluid temperature (Eq. (7)), is equal to zero near the channel’s surface and reaches its maximum possible value in the channel’s center. At each stage of increasing the Grashof number, convection heat transfer intensifies, and unlike the velocity profile, we have a nearly 20.87% constant decline on average. The dimen- sionless volume fraction graph is illustrated in Figure 7 . This func- tion designates the variance between surface volume fraction and nanofluid volume fraction (Eq. (7)). Like the temperature function, we have the maximum volume fraction in the middle of the chan- nel. Increasing theGrnumber similarly reduces the volume frac- tion function by about 120.75%, 48.41%, and 26.16% at each level.

Fig. 6. Gr number influence on Ω(η) with ξ= 1 , q = 2 , N b = 0.1 , N t = 0.1 , N r = 0.1 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

Fig. 7. Gr number influence on ψ(η) with ξ= 1 , q = 2 , N b = 0.1 , N t = 0.1 , N r = 0.1 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

Fig. 8. N b influence on F (η) with ξ= 1 , q = 2 , N t = 0.1 , N r = 0.1 , Gr = 0 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

Fig. 9. N b influence on Ω(η) with ξ= 1 , q = 2 , N t = 0.1 , N r = 0.1 , Gr = 0 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

Fig. 10. N b influence on ψ(η) with ξ= 1 , q = 2 , N t = 0.1 , N r = 0.1 , Gr = 0 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

Next, we examined howNb(Brownian motion parameter) af- fects the three functionsF(η),Ω(η), andψ(η). This parameter is a scale of nanoparticles’ random motion in a nanofluid due to colli- sion with other nanoparticles, which in this research is examined in five different sizes (0.1,0.2,0.3,0.4, and 0.5). As we are able to un- derstand from Fig. 8 , because of the growth in the number of colli- sions with increasingNbsize, the nanofluid one-direction motion is disturbed; as a result, we have a lower velocity profile. This reduc- tion is 24.3% at first and 4.35% at the last level. From another per- spective, when we raise the Brownian motion parameter, dimen- sionless temperature increases by 52.86% in the maximum state (Fig. 9) because by increasing collisions, the nanoparticles’ kinetic energy transforms into internal energy and is the reason forthe in- crease in the temperature profile. Furthermore, Fig. 10 displays the growth in the volume fraction function as an effect ofNbrise. This phenomenon is happening because more movement of the parti- cles means more required space or volume, so the random motion of nanoparticles opposes the fraction, and we have a 24.32% in- crease inψwhenNbrises from 0.1 to 0.2.

Fig. 11. N t influence on F (η) with ξ= 1 , q = 2 , N b = 0.1 , N r = 0.1 , Gr = 0 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

Fig. 12. Nt influence on Ω( η) with ξ= 1 , q = 2 , N b = 0.1 , N r = 0.1 , Gr = 0 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

Fig. 13. N t influence on ψ(η) with ξ= 1 , q = 2 , N b = 0.1 , N r = 0.1 , Gr = 0 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

Fig. 14. N r influence on F (η) with ξ= 1 , q = 2 , N b = 0.1 , N t = 0.1 , Gr = 0 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

Fig. 15. N r influence on Ω(η) with ξ= 1 , q = 2 , N b = 0.1 , N t = 0.1 , Gr = 0 , Pr = 1 , Re = 1 , Z = 0 , L e = 10.

Our investigation demonstrated that, by enlarging theNt, the nanofluid temperature function increases, leading to less dimen- sionless volume fraction and more dimensionless velocity. TheNtis a parameter that can show the amount of heat received by nanoparticles from their atmosphere; by increasing this parame- ter, particles’ tendency to absorb surrounding heat will increase. Consequently, we have an enhancement in nanoparticle tempera- ture and a decrease in surrounding temperature, which can explain the rise of nanoparticles velocity and degradation of volume frac- tion. For all thermophoresis parameters in the plots (Figs. 11–13), at least 55.44% and 18.69% rise forF(η), andΩ(η), and 20.23% de- crease forψ(η)functions is observed.

Figures 14–16 show how the buoyancy ratio parameter (Nr) in five different values (0.05, 0.1, 0.15, 0.2, and 0.25) affects the three functions of velocity, temperature, and volume fraction. According to the concept of theNrparameter, it can be comprehended that increasing the size of this parameter will strengthen the buoyancy force. Therefore, the upward movement becomes more accessible for the nanofluid and will move quicker. This phenomenon is well noticeable in Fig. 14 . Based on the data, increasing theNrparam- eter from 0.15 to 0.25 (66% increase inNrvalue) causesF(η)to rise by 199.7%.On the other hand, high velocity can reduce heat ex- change opportunities between the channel’s surface and nanofluid. Thus, as shown in Fig. 15 , increasing theNrparameter in the same interval decreases the fluid’s dimensionless temperature by 18%. Moreover, Fig. 16 shows that growth of the buoyancy ratio param- eter has a direct relationship with dimensional volume fraction; increasing it induces a lower volume fraction by 92.3%.

Fig. 17. 3D graph of F under the change of ηand q , with ξ= 1 , N b = 0.1 , N t = 0.1 , N r = 0.1 , Gr = 1.5 , Pr = 1.5 , Re = 1 , Z = 0 , L e = 10.

Fig. 18. 3D graph of Ωunder the change of ηand q , with ξ= 1 , N b = 0.1 , N t = 0.1 , N r = 0.1 , Gr = 1.5 , Pr = 1.5 , Re = 1 , Z = 0 , L e = 10.

Fig. 19. 3D graph of ψu(yù)nder the change of ηand q , with ξ= 1 , N b = 0.1 , N t = 0.1 , N r = 0.1 , Gr = 1.5 , Pr = 1.5 , Re = 1 , Z = 0 , L e = 10.

Fig. 20. 3D graph of F under the change of ηand ξ, with q = 2, N b = 0.1 , N t = 0.1 , N r = 0.1 , Gr = 1.5 , Pr = 1.5 , Re = 1 , Z = 0 , L e = 10.

Figures 17–19 show each of the three principal functions’ be- havior due to the change in power ofq(power used in correla- tions for temperature and volume fraction in Fig. 1) at different points in the channel. According to the proposed graphs, increas- ingqraises the absolute value of all threeF(η),Ω(η), andψ(η)functions. The parameterqis directly related to the temperature and volume fraction distribution, and increasing the value of this parameter means a more robust energy source. Finally, Figs. 20–22 show the nanofluid hydrothermal specifications at different points in the channel length and width. An increase inξequals fluid en- croach to the upstream points in the channel. Based on the plots, as nanofluid advances and moves away from the origin, veloc- ity and temperature functions increase while the volume fraction function declines.

Fig. 21. 3D graph of Ωunder the change of ηand ξ, with q = 2 , N b = 0.1 , N t = 0.1 , N r = 0.1 , Gr = 1.5 , Pr = 1.5 , Re = 1 , Z = 0 , L e = 10.

Fig. 22. 3D graph of ψu(yù)nder the change of ηand ξ, with q = 2 , N b = 0.1 , N t = 0.1 , N r = 0.1 , Gr = 1.5 , Pr = 1.5, Re = 1, Z = 0 , L e = 10.

In this paper, we investigate a nanofluid content channel af- fected by heat transfer. The formulation for this problem was solved through the semi-analytical AGM and validated through the numerical technique and comparison to other methods. The sum- mary of significant outcomes for this study is as follows:

By rising the Grashof number for one unit from 1 to 4, at each stage, dimensionless velocity (F) grows by 78%, and dimensionless volume fraction (ψ) decreases by 120.75% at their maximum state, while dimensionless temperature (Ω) constantly lessens by 20.87% on average. Increasing the Brownian motion parameter (Nb) in five different sizes (0.1, 0.2, 0.3, 0.4 and 0.5)causes lesser velocity. This reduction is around 24.3% at first and 4.35% at the last level. More- over, the utmost 52.86% increase for temperature function and a 24.32% increase forψo(hù)ccurs whenNbrises from 0.1 to 0.2. For allNtvalues, at least 55.44%, 18.69% increase forF(η) andΩ(η), and 20.23% decline forψ(η) functions is detected. Enlarging theNrpa- rameter from 0.25 to 0.1 leadsF(η) to rise by 199.7%, fluid dimen- sionless temperature and dimensional volume fraction decline by 18% and 92.3%, respectively. A greater value of “q”means a more powerful energy source, amplifying all velocity, temperature, and volume fraction functions.

Declaration of Competing Interest

The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors certify that they have no affiliations with or in- volvement in any organization or entity with any financial inter- est (such as honoraria; educational grants; participation in speak- ers’ bureaus; membership, employment, consultancies, stock own- ership, or other equity interest; and expert testimony or patent- licensing arrangements) or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript. The manuscript is original, and it has not been submitted to another journal.