# Heat and Mass Transfer of the Darcy-Forchheimer Casson Hybrid Nanofluid Flow due to an Extending Curved Surface

Authors: Gohar; Tahir Saeed Khan; Ndolane Sene; Abir Mouldi; Ameni Brahmia
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

The current paper describes a Darcy-Forchheimer flow of Casson hybrid nanofluid through an incessantly expanding curved surface. Darcy-Forchheimer influence expresses the viscous fluid flow in the porous medium. Carbon nanotubes (CNTs) with a cylindrical form and iron-oxide are utilized to make hybrid nanofluids. Using Karman’s scaling, the principal equations are rearranged to nondimensional ordinary differential equations. The “Homotopy analysis method” is used to further build up the analytic arrangement of modeled equations. The impact of flow variables on the velocity and temperature profiles has been tabulated and explained. The flow velocity is raised when both the curvature and volume fraction parameters are elevated. The temperature and velocity profiles exhibit the opposite tendency when the Forchheimer number is increased, since the fluid velocity decreases while the energy profile grows. The addition of CNTs and iron nanocomposites improves the thermophysical characteristics of the base fluid significantly. The obtained consequences show that hybrid nanofluids are more efficient to improve the heat transfer rate. Using CNTs and nanomaterials in the base fluid to control the coolant level in industrial equipment is a wonderful idea.


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## Body

## 1. Introduction

The flow over an extending surface has received much importance due to its significant role in several sectors of industry and engineering, such as condensation process, spinning of fiber and continuous casting of fiber, plastic sheet extraction, paper production, and many others. Crane [1] was the first to study the flow over an expanding planar surface. Many researchers have since followed the concept of the crane [2–5], expanding the sheets to investigate various aspects of this form of flow. Sajid et al. [6] addressed boundary layer flow and micropolar fluid, concluding that the curvature effect leads to a reduction in boundary layer size. Gul et al. [7] have investigated the flow of the boundary layer on the stretching surface using the Fractional Order Derivatives Scheme. Imtiaz et al. [8] demonstrated the fluid flow under the upshots of the magnetic field over an extending curved surface. It has been noticed that with the action of curvature coefficient, the energy profile is enhanced. Rosca et al. [9] have analyzed the flow caused by contracting and expanding sheets. Saeed et al. [10] offered a complete investigation of the Darcy hybrid nanoliquid flow through a curved surface that is exponentially expanding. The outcomes signify using SWCNTs, MWCNTs, and Fe3O4 nanomaterials for the increase in the nusselt number. Ali et al. [11] analyzed the hydrological importance of wave propagation of hybrid nanofluid over a warmed extended curved surface with the impacts of a magnetic field using bvp4c. The suspension of carbon nanotubes in a magnetite nanoliquid promotes local surface drag but reduces local heat flow. Kumar et al. [12] have reported the radiation impact on the Casson fluid across the exponentially curved sheet. Hayat et al. [13] explored ferroliquid flow with the mass and heat transition across a curved stretching surface. Hussain et al. [14] reported the findings of their investigation on hybrid nanofluid flow across a curved sheet. The outcomes of the survey revealed that the energy transference efficiency in hybrid nanocrystals is higher than that in nanofluids for large frequencies of the curvature index. Qian et al. [15] and Khan evaluated that how heat transmission and radiation were affected by the conducting flow over a curved extending surface. Their study was found to be in good accord with a previously published finding.The heat transmission in carbon nanofluids has gotten a lot of interest from researchers in a variety of fields in the last few years. CNTs are carbon nanotubes with a fundamental chemical structure and a carbon atom formation wrapped in a cylindrical shape. CNTs have superior chemical, thermophysical, and mechanical characteristics, making them ideal for usage as a particulate in a base fluid. They offer various advantages over other nanomaterials due to their tiny size, structure, configuration, dimension, and hardness. Haq et al. [16] evaluated the computational findings for conductive fluid using carbon nanomaterials along an extensive surface. Ahmadian et al. [17] addressed a 3D model of an unsustainable hybrid nanofluid flow with fluid and momentum transmission caused by surface accelerating displacement. The use of hybrid nanoparticles is thought to have enhanced the carrier fluid’s thermal properties substantially. Because of the C–C link, CNTs are more effective than other forms of nanoparticles in the carrier fluid. CNTs nanofluid may be further functionalized to get the desired result, which may be used in a range of applications through noncovalent and covalent manipulation [18]. Saeed et al. [19] have considered the nanofluid containing CNTs and iron oxide nanomaterials using the flow of fluid over a curved surface. Gul et al. [20, 21] studied the flow of nanofluids to enhance heat transfer and thermal applications. Alghamdi et al. [22] have observed the flow of hybrid nanofluid through a blood artery for medications. Using the bvp4c tool, Li et al. [23], and Ding et al. [24] used (MWCNTs) in the base liquid to evaluate heat transmission. Akbar et al. [25] described the influence of a magnetic field on the flow of CNT nanofluids through a moving permeable channel. Gul et al. [26] and Bilal et al. [27] used an inclined extending cylinder to examine the Darcy-Forchhemier hybrid nanoliquid flow. They examined the carbon nanotubes (CNTs) and iron oxide Fe3O4 as two distinct nanomaterials. Ahmed et al. [28] represented temperature propagation in a wavy-wall impermeable enclosure through nanofluids. It was discovered that increasing the waviness of the sheet boosts both the heat transmission rate. Yarmand et al. [29] investigated how graphene nanoplatelets/platinum hybrid nanofluids with diverse properties may improve heat transfer rates. Sajid et al. [30] examined the thermophysical characteristics of hybrid and single-form nanotubes using numerical methods. They determined that the size, type, concentration, and temperature fluctuation of nanoparticles had a significant impact on the thermophysical characteristics of nanofluid. Kumar et al. [31] examined the solar radiation impact on the flow of ferromagnetic hybrid nanofluid. Gowda et al. [32] studied the flow of nanofluid over the stretched and curved surface using (KKL) relation. Kumar et al. [33] have used the concept of the magnetic dipole for the flow of nanofluid over a cylinder. Zeeshan et al. [34] have studied the couple stress nanofluid flow using the paraboloid model.The curved surface for the fluid flow has many applications in the mechanical and automotive industry. Sanni et al. [35], Jawad et al. [36], and Saeed et al. [37] have studied the fluid flow on a curved surface using various kinds of nanofluids for the heat transfer enhancement. Hayat et al. [9, 38], Rosca, and Pop [39] have explained the homogeneous-heterogeneous reaction phenomena using the curved surface for the flow pattern. Okechi et al. [40], Asghar et al. [41], and Hayat et al. [42] have used the nonewtonian fluid flow over the curved surface with various extensions considering Darcy-Forchheimer flow medium. The related work to the proposed model can be seen in the References [9, 35, 43–45].The inertia effect is taken into account by incorporating a squared component to the momentum equation, called Forchheimer’s modification [46]. Muskat [47] used the term “Forchheimer factor” to describe this new concept. It is critical to include non-Darcy consequences in convective transport analysis to properly represent real-world challenges. The novelty of the model has been presented as
(i)
For the hybrid nanofluid flow, heat and mass transmission is examined simultaneously(ii)
TheCNTs+Fe3O4/H2O hybrid nanoliquid flow across a stretching surface with the mass and heat transition has been addressed(iii)
The non-Newtonian Casson hybrid nanofluid has been used as another extension in the existing literature(iv)
Heat obsorption\omisson has also been considered in the flow regime(v)
This study intends to evaluate and simulate the Darcy-Forchheimer water-based hybrid nanoliquid flow induced by a curved surface that extends(vi)
The second priority is to modify the Saba et al. [48] and Xue [49] model for hybrid nanofluid flow(vii)
The proposed model has been solved by the homotopy analysis method


## 2. Mathematical Formulation

The Darcy-Forchheimer flow considers CNTs andFe3O4 nanomaterials across an expanding curved sheet. The viscous fluid flow has been expressed in the permeable space by the Darcy-Forchheimer effect. The flow is assumed across the stretching sheet, with radius R, as depicted in Figure 1. The term r,s is taken as the space coordinate and u,v is the velocity component. Here, Uws=aes/L is the exponential stretching velocity, Tw is the curved surface, and T∞ is the ambient temperature. Keeping in view, the above superposition, the energy, and momentum equations along with their boundary conditions are expressed as [9, 36–39, 46]
(1)∂∂rr+Rv+R∂u∂s=0,(2)u2r+R=1ρhnf∂p∂r,(3)v∂u∂r+Rr+Ru∂u∂s+uvr+R=−1ρhnfRr+R∂p∂s+νhnf1+1β∂2u∂r2−ur+R2+1r+R∂u∂r−νhnfK∗u−1ρhnfFu2,(4)v∂T∂r+u∂T∂sRr+R=αhnf∂2T∂r2+1r+R∂T∂r+QρcphnfT−T∞,(5)v∂C∂r+u∂C∂sRr+R=Dhnf∂2C∂r2+1r+R∂C∂r.Figure 1
Physical sketch.The basic flow conditions are(6)v=0,u=Uws=aes/L,T=T∞+T0eAs/2L=Tw,C=C∞+C0eAs/2L=Cw,atr=0,u⟶0,∂u∂r,T⟶T∞,C⟶C∞,atr⟶∞.Here,K∗ and F=Cb/sK∗1/2, are the permeability and inertia factors.
(7)υhnf=μhnfρhnf,μhnfμf=1−ϕ1−5/21−ϕ2−5/2,ρhnfρf=1−ϕ21−1−ρMsρfϕ1+ρCNTρfϕ2,khnfkbf=2ϕ2kCNTkCNT−kbf+1−ϕ2−lnkCNT+kbf2kbf2ϕ2kbfkCNT−kbf+1−ϕ2−lnkCNT+kbf2kbf−1,kbfkf=kMS+m−1kf−ϕ1kf−kMS−1kMS+m−1kf−m−1ϕ1kf−kMS,ρCphnfρCpf=ρCpCNTρCpfϕ2+1−ϕ21−1−ρCpMsρCpfϕ1.Thekhnf is the thermal conductivity, ϕ1 and ϕ2 are the volume friction parameters, CpMS is the specific heat capacity, ρMS and ρCNT are specified densities of Fe3O4 and CNTs, and Sc is the Schmidt number, respectively.The transformation variables are [50]
(8)η=aes/L2υfL1/2r,v=−Rr+Raνfes/L2Lfη+ηf′η,u=Uw=aes/Lf′η,p=ρfa2e2s/LHη,T=T∞+T0eAs2LΘη,C=C∞+C0eAs2LΦη.Thus, by using Eq. (8), Eqs. (2)–(7) yield
(9)H′=ρhnfρf1η+Kf′2,(10)1+1βf‴+1η+Kf″−1η+K2f′−2λf′−1−ϕ12.51−ϕ22.5ρhnfρfη+2Kη+K2Kf′2−Kη+Kff″−Kη+K2+2Frf′2=1−ϕ12.51−ϕ22.5Kη+K4H+ηH,(11)khnfkfΘ″+1η+KΘ′+ρCphnfρCpfPrKη+KfΘ′−Af′Θ+δΘ=0,(12)1−ϕ11−ϕ2Φ″+1η+KΦ′+ScKη+KfΦ′=0.By eliminatingH from Eqs. (9) and (10), we get
(13)1+1βf′v+2η+Kf‴−1η+K2f″+1η+K3f′−2λf″+1η+Kf′+ρhnfρfKη+K2ff″+Kη+Kff‴−Kη+K3ff′3Kη+K2f′2−3Kη+Kf′f″−2Fr2f′f″+1η+Kf′2=0.The transform conditions are(14)f=0,f′=1,Φ=1,Θ=1atη=0,f′⟶0,f″⟶0,Φ⟶0,Θ⟶0atη⟶∞,whereFr, λ, and k are the Forchheimer, porosity, and curvature parameters, respectively, which can be rebound as
(15)Fr=CbK∗1/2,Pr=υfαf,δ=2QLUwρcp,k=aes/L2υfL,λ=υfLK∗Uw,Sc=υfDf.The local Nusselt number, Sherwood Number, and Skin friction are expressed as(16)LSRe2−12Nux=−khnfkbfΘ′0,LSRe2−12Shx=−Φ′0,Re2Cfx=11−ϕ12.51−ϕ22.51+1βf″0,where local Reynolds number is(17)Rex=u0x2vl.


## 3. Problem Solution

For analytical findings, the HAM approach has been utilized to solve the modeled equations, which was firstly introduced by Liao [51–53]. The initial guesses for velocity f0 and temperature Θ0 are given as
(18)f0η=e−η−e−2η,Θ0η=e−η,Φ0η=e−η.The linear terms are(19)Lff=f′vandLΘΘ=Θ″.The expanded form ofLf,LΘandLΦ is
(20)Lfχ1+χ2η+χ3η2+χ4η3=0,LΘχ5+χ6η=0,LΦχ7+χ8η=0.


### 3.1. OHAM Convergence

The converging of the OHAM approach was achieved employing Liao’s concept [51–61].
(21)εmf=1l+1∑j=1lNf∑k=1mfηη=jδη2,(22)εmΘ=1l+1∑j=1lNΘ∑k=1mfη,∑k=1mΘη,η=jδη2,(23)εmΦ=1l+1∑j=1lNΦ∑k=1mΦηη=jδη2.The sum of residual error isεmt=εmf+εmΘ+εmΦ.


## 3.1. OHAM Convergence

The converging of the OHAM approach was achieved employing Liao’s concept [51–61].
(21)εmf=1l+1∑j=1lNf∑k=1mfηη=jδη2,(22)εmΘ=1l+1∑j=1lNΘ∑k=1mfη,∑k=1mΘη,η=jδη2,(23)εmΦ=1l+1∑j=1lNΦ∑k=1mΦηη=jδη2.The sum of residual error isεmt=εmf+εmΘ+εmΦ.


## 4. Results and Discussion

The goal of this portion is forward to see how the temperature and velocity profiles function under the effect of the predicted factors. The flow configuration is shown in Figure1. The OHAM technique’s progress has been computed and is depicted in Figure 2(a). Figure 2(b) displays the influence of velocity field versus M. The Lorentz force augments the resistance against the fluid motion and as a result, the velocity reduces with the greater value of the magnetic parameter. The augmentation in the Casson parameter declines the velocity profile. The Casson parameter at the infinity tends to the Newtonian fluid. The larger value of the heat absorption and omission parameter improves the temperature distribution as shown in Figure 2(c). The greater value of the parameter δ improves the temperature distribution. Figure 3 illustrates the effects of the (porosity term) and k on velocity and temperature. This conclusion can be drawn that the velocity f′η decrease, while the temperature field is increased versus rising values of porosity parameter λ as illustrated in Figures 3(a) and 3(b). Practically, the kinetic viscosity and length of the extending surface are enhanced with the action of the porosity parameter; therefore, such a phenomenon has been observed. On the other hand, the action of curvature parameter k enhances the velocity field and declines the temperature propagation as illustrated in Figures 3(c) and 3(d).Figure 2
(a)–(c) HAM solution, Casson parameter, and heat absorption\omission parameter influence.
(a)(b)(c)Figure 3
(a, b) Porosity parameterλ impact on the velocity f′η and temperature profile θη. (c, d) Curvature parameter k effect on velocity f′η and temperature profile, respectively.
(a)(b)(c)(d)Figures4(a)–4(d) are sketched to illustrate the consequences of volume friction coefficients ϕCNT and ϕFe3O4 on velocity and energy propagation. The specific heat capacity of H2O is greater than much iron and carbon nanoparticles. The addition of nanoparticles in the water reduces its heat-absorbing capacity, which results in an excessive amount of heat in the fluid. These factors cause the enhancement of fluid velocity and thermal energy transition.Figure 4
(a)–(d) Volume friction parametersϕ1 and ϕ2 impact on the velocity f′η and temperature profiles θη, respectively.
(a)(b)(c)(d)Figures5(a) and 5(b) revealed the influence of Forchhemier number Fr on velocity and temperature profiles, respectively. The increment in the Forchhemier term reduces the fluid velocity and enhances the thermal energy profile. Because the permeability of fluid reduces by the action of the Forchhemier term, therefore, such a phenomenon has been observed. The energy profile declines with the effect of temperature exponent coefficient A and Prandtl number Pr as displayed in Figures 5(c) and 5(d). The thermal diffusivity of fluid rises with the increasing credit of Prandtl number, which results in declination of fluid temperature θη as shown in Figure 5(d). The thickness of the boundary layer improved with the increasing value of Pr and consequently, the temperature profile reduces. Figures 6(a) and 6(b) illustrate to elaborate the consequences of curvature parameter k and Schmidt number Sc on mass transport Φη profile. The mass transition rate reduces with the influence of Schmidt number while enhancing with the positive effects of curvature term, because the fluid mean viscosity becomes thick as the number of carbon and iron oxide particulates continues to increase.Figure 5
(a, b) Forchheimer parameter Fr impact on the velocityf′η and temperature profiles θη, (c) temperature exponent coefficient, and (d) Prandtl number Pr effects on temperature.
(a)(b)(c)(d)Figure 6
(a) Schmidt number Sc, (b) curvature parameter, and (c, d) volume friction parameters effects on mass transfer profile, respectively.
(a)(b)(c)(d)The surface drag forceRe/2Cfx for carbon nanoliquid and Fe3O4 is declared via Figures 7(a) and 7(b). It is been evidenced that as the curvature and the volumetric parameters are increased, the skin friction drops. Figures 7(c) and 7(d) demonstrate the numerical results for the Nusselt number L/SRe/2−1/2Nux. It has been discovered that the heat conversion rate accelerated as the number of carbon nanomaterials in the conventional fluids and the Prandtl number expanded. Figure 7(e) indicates that the Sherwood number L/SRe/2−1/2Shx is an enhancing factor of the Schmidt number. Table 1 displays the thermophysical properties of solid substrates and basic fluids. The OHAM technique’s consolidation has been computed up to the 30th iteration and is reported in Table 2. Table 3 offers a comparative analysis of the current study to the existing literature.Figure 7
(a) Skin frictionRe/2Cfx for Fr and k, (b) skin friction Re/2Cfx for λ and ϕ2, (c) Nusselt number L/SRe/2−1/2Nux for λ and ϕ2, (d) Nusselt number L/SRe/2−1/2Nux for λ and Pr, and (e) Sherwood number L/SRe/2−1/2Shx for k and Sc.
(a)(b)(c)(d)(e)Table 1
The numerical properties of nanomaterials and base fluid [27].
ρkg/m3Cpj/kgKkW/mKPure water997.141790.613Fe3O452006706SWCNTs26004256600MWCNTs1600796300Table 2
The total residual errors, whenFr=k=0.6,ϕ1=0.02,ϕ2=0.2,λ=0.2,Pr=6.3, and A=0.4..
mεmtSWCNTsεmtMWCNTsεmtFe3O451.8168×10−41.9479×10−41.4257×10−4131.1223×10−51.2354×10−51.18312×10−5231.3599×10−60.4698×10−60.4489×10−6303.2578×10−74.3689×10−74.1464×10−7Table 3
The comparative analysis with the published work, whenϕ1=ϕ2,Fr=k=0.6,λ=0.2,Pr=6.3, and A=0.4..
Hayat et al. [35]Presentf″00.7350.7352130−Θ′0-1.375-1.3752410−Φ′0………….-1.3620189


## 5. Conclusion

We addressed the Darcy-Forchheimer flow of Casson hybrid nanoliquid induced by an extended curved surface in this problem. The momentum and energy equations are included in the flow model, which is set up as a system of partial differential equations. The “Homotopy analysis method” is used to further build up the analytic arrangement of modeled equations. This mathematical model attempts to highlight the dominance of nanofluid in heat and mass transmission in advanced technologies and industries. The following are the core findings:(i)
The velocity and temperature fields both show positive behaviors against the increasing values ofϕ1 and ϕ2 (volume fraction parameters) of CNTs andFe3O4The accumulative values of the Casson parameter decline the hybrid nanofluid motion.
(ii)
The employment of CNT andFe3O4 nanomaterials in the base fluid, to regulate the coolant level in industrial equipment, is quite beneficial(iii)
The thermal energy profile shows a reducing trend versus larger values of temperature exponent coefficientA(iv)
High fluid velocity is achieved by increasing the value ofk (curvature parameter), while the fluid temperature drops(v)
The temperature and velocity profiles exhibit the opposite tendency when the Forchheimer number is elevated since the fluid velocity decreases, whereas the temperature profile improves(vi)
The temperature distribution increases for the larger values of the absorption parameter(vii)
The comparison of the recent work with the published work authenticates the obtained results



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