DARCY-FORCHHEIMER TANGENT HYPERBOLIC

Journal of Porous Media, 26(5):1–14 (2023) DARCY-FORCHHEIMER TANGENT HYPERBOLIC NANOFLUID FLOW THROUGH A VERTICAL CONE WITH NON-UNIFORM HEAT GENERATION Husna A. Khan,1 Ghazala Nazeer,1 & Sabir Ali Shehzad2,∗ 1 Department of Mathematics, Government Sadiq College Women University, Bahawalpur, Pakistan 2 Department of Mathematics, COMSATS University Islamabad, Sahiwal 57000, Pakistan *Address all correspondence to: Sabir Ali Shehzad, Department of Mathematics, COMSATS University Islamabad, Sahiwal 57000, Pakistan, E-mail: [email protected] Original Manuscript Submitted: 7/26/2022; Final Draft Received: 8/28/2022 The convective flows through different geometries have numerous applications in high-speed aerodynamics, nuclear cooling systems, fiber technology, and polymer engineering. In the present paper, we investigate the non-linear, mixed convective, boundary-driven, tangent hyperbolic nanofluid flow through a cone. The flow takes place under nonuniform heat sink/source. Darcy-Forchheimer effects have also been taken into account in mathematical modeling and analysis. The Buongiorno model is implemented to examine the effects of thermophoresis and Brownian motion parameters. The governing equations are constructed through the laws of conservation. The modeled flow problem is converted into a set of ordinary differential equations with the help of proposed similarity transformations. To interpret the modified system of equations, the homotopy analysis method (HAM) is applied. The roles of versatile parameters of interest are analyzed and sketched for better understanding. The velocity profile increases by increasing the Darcy number, and converse behavior is found by giving rise to the Forchheimer inertial drag parameter. The rise in temperature profile occurs by increasing a non-uniform heat source variable. The concentration profile enhances when the value of the thermophoresis parameter increases, and shows inverse behavior for the Brownian motion parameter. In the Buongiorno model, nanoparticle concentration has an inverse relation with the Brownian motion parameter. So, the concentration profile declines for greater Brownian motion parameter. To understand the behavior of flow through a cone, the values of Nusselt number and Sherwood numbers are examined. KEY WORDS: Darcy-Forchheimer flow, tangent hyperbolic fluid, nanofluid, vertical cone, heat generation 1. INTRODUCTION In literature, numerous studies are available on energy and mass species analysis by considering the Newtonian and non-Newtonian materials. The non-Newtonian materials have a spatial importance in many engineering and industrial developments. Non-Newtonian fluids require a great deal of mathematical modeling. In such fluids, the relation between the rate of strain and shear stress is complicated as compared to viscous fluids. Physiological liquids (bile, synovial), cosmetic products, coal-oil slurries, shampoos, and paints are some examples of such fluids. A single nonNewtonian model can never predict each of the properties of non-Newtonian materials. Therefore, multiple models for non-Newtonian materials have been proposed and addressed in the past. One of the models for these materials is the tangent hyperbolic (TH) fluid (Pop and Ingham, 2001). This rheological model has some technical and physical stability advantages over other non-Newtonian models. Peristaltic hyperbolic tangent fluid movement through an asymmetric tunnel with energy transport was presented by Nadeem and Akram (2011). This model, under convective type boundary conditions, was investigated by Akbar (2014). 1091–028X/23/$35.00 © 2023 by Begell House, Inc. www.begellhouse.com 1 2 Khan, Nazeer, & Shehzad A few decades ago, Choi and Eastman (1995) in Argonne National Laboratory (ANL) introduced the new concepts of fluid known as nanofluid—a phase fluid composed of nanoparticles having length scales ranges of 1 to 100 nm. The conductivity of standard fluid is weaker and can be exceptionally enhanced by introducing nanoparticles. Nanofluids have several applications in various fields such as nano-drug delivery, smart fluids, cancer therapeutics, nuclear reactors, nanofluid detergents, electronics, and automotive applications. The nanofluid improves the thermal conductance and convective energy transportation coefficient of the base-materials. Thermal properties are inadequate in oils, water, and ethylene glycol. Multiple techniques have been developed in order to boost the thermal conductance of such fluids. Introduction of nano-sized material particles to the liquids is one of these techniques. The nanofluids are further classified into the Buongiorno nanofluid model and Tiwari-Das nanofluid model. The Buongiorno model is used to examine the impacts of thermophoresis and Brownian motion parameters on mass and heat transfer. To determine the impact of nanoparticles, volume-fraction, and energy transportation properties, Vajravelu et al. (2011) used a combination of temperature-dependent heat absorption/production and thermal buoyancy factors. Later on, Reddy et al. (2019) disclosed the solar effects in non-Newtonian nanomaterial flow under transverse Lorentz force. The peristaltic nanofluid behavior of Jeffrey-six constant liquid through a non-uniform vertical tube is reported by Imran et al. (2020). Hassan et al. (2021) reported the experimental data based modeling of homogeneous hybrid nanomaterial flow. The activation energy mechanism in a bio-convected nanofluid flowing through the porous cavity is explored by Balla et al. (2022). Das and Kumbhakar (2022) evaluated the magnetized bio-convected non-Newtonian nanofluid flowing over a slippery surface activation energy and Joule dissipation. Mixed convection flows have become the center of interest for researchers and scientists due to their versatile technological and industrial appliances in nature, including solar receivers, electronic devices, heat exchangers, nuclear reactors for emergency shutdown, and flows through the ocean and atmosphere. The mixed convected boundarydriven flow of a nanofluid over a stretching plate was represented by Malik et al. (2015). Gaffar et al. (2015) examined the non-linear, steady, convective flow and energy transfer using the TH fluid theory. The nonlinear thermal radiation effect of TH nanofluid flow under Robin’s conditions was addressed by Hayat et al. (2016a). Hayat et al. (2016b) showed a two-dimensional mixed convective flow with thermal radiation effects on a TH fluid. Bhatti et al. (2021) addressed the convection effects in a thermally-transported flow of Williamson hydro-magnetized nanofluid. They also executed the entropy analysis of the considered problem. Darcy’s law is prominent amongst the most standup models in examining the flow in porous media. This law has provided the base for the modeling of physical problems related to heat convection and porous media. The flow in a porous media is tremendously precious in grain stockpiling, framework of ground water, development of water in repositories, oil assets, raw petroleum generation, and fermentation processes. Darcy’s law, on the other hand, is inadequate for turbulent flow circumstances because inertial properties in porous media are not taken into account. These movements occur in large capacity gas and distillation petroleum reservoirs near the wellbore. The DarcyForchheimer model of drag force is the most often used method in porous medium for modeling the high velocity transportation. This concept is constructed from the geometrical structures of the porous media. Vafai and Tien (1981) explored the effect of Forchheimer inertial on the convective porous media flows. Patil et al. (2021) addressed the time-dependent, exponentially-stretched flow of viscous fluid through the Darcy-Forchheimer porous medium. Two-dimensional convective flows through conical geometries have gained much consideration by the scientists and investigators of this field due to their conspicuous usages in modern industry and engineering applications. The subject of two-dimensional free convective flows over a vertical cone has attracted a number of modern researchers to survive with the industrial demands. Roy (1974) proposed the natural convection flow via a perpendicular cone considering large values of Prandtl number. Free convection flow across the cone and a wedge was described by Ramanaiah and Malarvizhi (1992). Hossain and Paul (2001) investigated thermal convective flows with buoyancy and pressure effects from a vertical permeable circular cone at the surface of a non-uniform temperature. The flow of nanofluid above a vertical cone implanted in a porous media with mixed boundary layer is described by Gorla et al. (2011). Due to combined convective effects, thermal and mass gradients cause mass and heat transfer phenomena through the vertical cone in the vicinity of boundary of the cone. Examples of industrial applications of heat and mass transfer phenomena include turbines, automobiles, power generators, and metallurgical products. The vibrational curves contain mass transfer and an in-depth treatment of heat in fluid (Ingham and Pop, 1998; Nield and Bejan, 2006). Journal of Porous Media Darcy-Forchheimer Tangent Hyperbolic Nanofluid Flow 3 Heat transfer enrichment due to small sized suspended particles was first time reported by Masuda et al. (1993). In the condition of convective thermal radiation and heat conduction, Hsiao (2016) investigated the energy conversion problem of viscous fluids on nonlinear stretched boundaries. Many geophysical, geothermal, industrial, and engineering submissions such as nuclear-powered devices are interested in magnetohydrodynamic (MHD) flows and heat transfer. Vajravelu et al. (1992) examined an MHD convection at wedge and cone with different exterior temperatures and interior heat immersion. Cheng (2009) explored the convected thermal and mass transportation of fluid by involving the wall concentration and temperature over a vertical cone. Chamkha and Rashad (2014) investigated the mass transfer and thermal heat by diverse convective MHD flow over a spinning erect cone in natural fluid. By internal heat absorption/generation, heat energy produces inside the body. There exist two models to determine the effect of heat source/sink on the body. In the first model, heat source/sink depends upon uniform temperature, while it depends upon temperature and space in the second model. The change in the decomposition of particles and temperature distribution is caused by heat generation effects. Few of the burning applications of heart source/sink include reactor safety analysis, combustion analysis, metal waste, electronic devices, and semiconductors. In the fields of mechanical, chemical, and civil engineering, nanofluid flows with nonlinear heat source and permeable media have a wide range of uses, including different catalysis to speed up chemical reactions and heat transfer, fast thermal aerodynamics, thermonuclear cooling systems, and sprig revelation process. Vasu et al. (2017) demonstrated the collective effects of heat sink/source on thermophoresis-based, temporary free convection of mass transport in a viscoelastic fluid past a vertical sheet. Saif et al. (2020) examined the heat-mass transport analysis of magnetized Jeffrey nanomaterial flowing over the curved moving sheet. In view of existing literature, the foremost motivation of this research is to interpret the non-linear mixed convective Darcy-Forchheimer TH nanofluid flow through cone. The flow is considered with the involvement of magnetic force and non-linear heat production and absorption. To our knowledge, no such study exists in literature. The homotopy analysis technique (Liao, 2012; Sheikholeslami et al., 2012; Arqub and El-Ajou, 2013; Farooq et al., 2015) has been used to solve the modeled problem. The nature of various emerging constraints on interesting physical quantities have been analyzed and presented graphically. 2. MATHEMATICAL FORMULATION A tangent hyperbolic fluid flow in an axisymmetric laminar, stable, and incompressible natural convection past a vertical cone with a nonlinear thermal source is accounted. A hydro-magnetic field normal to the mass flux and surface in a saturated porous media is considered. Nonlinear thermal radiation and nonlinear mixed conduction are used to study viscous flow in porous media. Convective circumstance executes the geometries of the surface boundary. Brownian motion and thermophoretic features are taken into consideration. The physical model is expressed in Fig. 1. The surface’s stretching velocity, calculated as uw = (xν)/L2 vw = −(2ν)/L, is the denoted suction/injection ′ ′ velocity. The temperature T ′ = T∞ + axr1 θ (η), surface temperature is taken as Tw′ = T∞ + axr1 where a is the ′ ′ constant and r1 is the thermal factor. Near the surface, the concentration is denoted by Cw = C∞ + axr2 where r2 is the nanofluid concentration parameter. The governing equations with Boussinesq’s approximation are (Mumtaz et al., 2020): ∂(ur) ∂(vr) + = 0, ∂x ∂y u ∂u ∂u ∂2u √ ∂u ∂ 2 u σβ2∗ +v = ν (1 − n̂) 2 + 2Γn̂ν − u + gβ1 cos α (T − T∞ ) 2 ∂x ∂y ∂y ∂y ρ ∂y ν b 2 2 + gβ1 cos α (T − T∞ ) + gβ∗ cos α (C − C∞ ) + β∗ cos α (C − C∞ ) − u − u2 , K K [ ] ( ) 2 ∂T ∂T ∂C ∂T ∂2T DT ∂T 1 ′′′ u +v = α 2 + τ DB + + q , ∂x ∂y ∂y ∂y T∞ ∂y ρCp ∂y Volume 26, Issue 5, 2023 (1) (2) (3) 4 Khan, Nazeer, & Shehzad FIG. 1: Physical configuration of the flow problem u ∂C ∂2C DT ∂ 2 T ∂C +v = DB 2 + , ∂x ∂y T∞ ∂y 2 ∂y (4) with boundary conditions (Abdullah et al., 2021): u = uw , u → 0, v = −vw , T → T∞ , ∂T ∂C DT ∂T = h (Tw − T ), DB + =0 ∂y ∂y T∞ ∂y C → C∞ as y → ∞, −k at y = 0 (5) where the velocity components u and v are measured x and y directions. The kinematic viscosity is denoted by ν, density is taken as ρ, the electrical conductivity is denoted by σn̂ is the material power law index, β∗ is the magnetic field strength, gravitational acceleration is g, volumetric thermal expansion coefficient is β, T is the temperature, concentration is C, k denotes thermal conduction, φ is the semi-vertical cone angle, b is Forchheimer geometric constant, Γ is positive time constant, α is the thermal diffusivity, DT represent thermophoresis, DB is the Brownian diffusion coefficients, T∞ is ambient temperature, and τ is extra stress tensor. The heat source/sink is defined as a non-uniform term q ′′′ (Gireesha et al., 2018). q ′′′ = kρU 0 [A1 (Tw − T∞ )f ′ + A2 (T − T∞ )]. µ (6) A2 and A1 are temperature and space-dependent coefficients. They are connected to inner heat if A1 > 0 and A2 > 0. If A1 < 0 and A2 < 0, it is the internal heat sinks. Appropriate transformations for cone are defined as (Gireesha et al., 2018; Abdullah et al., 2021): u= xν ′ f (η), L2 v=− 2ν f (η), L η= y , L θ (η) = T − T∞ , Tw − T∞ φ (η) = C − C∞ . Cw − C∞ (7) By using Eq. (6), Eqs. (2)–(5) reduce to the following dimensionless forms: 1 ′ Fs δ ′ 2 f − f + Wen∗ f ′′ f ′′′ − M f ′ Da Da + Gr(θ cos α + θ2 λ cos α + Gc φ cos α + Gc φ2 λ1 cos α) = 0, (1 − n∗ ) f ′′′ + 2f f ′′ − f ′ − 2 (8) Journal of Porous Media Darcy-Forchheimer Tangent Hyperbolic Nanofluid Flow 5 θ′′ + Pr [2f θ′ − r1 f ′ θ + θ′ (N bφ′ + N tθ′ )] + (A1 f ′ + A2 θ), φ′′ + Sc [2f φ′ − r2 f ′ φ] + N t ′′ θ . Nb (9) (10) Boundary conditions: f (0) = S, f ′ (0) = 1, f ′ (∞) = 0 θ′ (0) = γ (θ (0) − 1), θ (∞) = 0 (11) Nt ′ ′ φ (0) = − θ (0), φ (∞) = 0, Nb √ where Da = K/L2 is Darcy number, We = ( 2νxΓ)/L3 is Weissenberg number, M = (σβ2∗ L2 )/(ρν) is magnetic parameter, Fs = b/L is Forchheimer inertial parameter, Pr = ν/α is Prandtl number, Sc = ν/DB is Schmidt number, Gc = [gL4 β∗ (Cw − C∞ )]/(xν2 ) is concentration Grashof number, Gr = [gL4 β1 (Tw − T∞ )]/(xν2 ) is ′ thermal Grashof number, N t = [τDT (Tw − T∞ )]/(νT∞ ) is Brownian motion, and N b = [τDB (Cw − C∞ )]/ν is thermophoresis parameter. Nusselt number and Sherwood number are addressed in dimensionless form as: Nux = xqw , k(T w − T∞ ) qw = − k Shx = ∂T ∂y qm = − DB xqm , DB (Cw − C∞ ) , (12) (13) y=0 ∂C ∂y , (14) y=0 where qw expresses the heat flux and qm shows the mass flux: Nux = −θ′ (0) , Shx = −ϕ′ (0). (15) 2.1 HAM Convergence HAM is a semi-analytic method which is used to report the solution of linear and non-linear expressions. This technique contains the convergence parameters which are known as supporting parameters. For convergent solution appropriate selection of such parameters is very important. For the convergence of HAM, the supporting parameters hf , hφ , hθ play an important role. The suitable range of hf , hφ , hθ are −0.35 ≤ hf ≤ 0.1, −0.35 ≤ hφ ≤ 0.25, and −0.15 ≤ hθ ≤ 0.35 [see Figs. 2(a)–2(c)]. 3. RESULTS AND DISCUSSION To investigate the physical nature of numerous emerging constraints on temperature, concentration, and velocity profiles, Figs. 3–8 are plotted. The proper values of fixed parameters are taken as follows for the latest analysis: n∗ = 0.5, Da = 0.1, W = 0.1, δ = 0.5, Fs = 1.1, S = 0.5, M = 1.5, Gc = 0.5, r1 = 0.5, A1 = 0.2, Pr = 1.0, N t = 0.3, N b = 0.3, A2 = 0.5, Sc = 0.5, A2 = 0.4, r = 0.5, and α = 60◦ . Table 1 interprets the Nusselt number values against dissimilar N t, Pr, A1 , and A2 . The Nusselt number is reduced for augmenting A1 and A2 values. The Sherwood number values against N t, N b, Sc, and Da are expressed in Table 2. The Sherwood number is improved against the higher N t and Sc values. The effects of suction constraints, concentration, thermal Grashoff, and Weissenberg numbers on velocity are presented in Figs. 3(a)–3(d). The higher wall suction reduced the liquid velocity as exposed in Fig. 3(a). The nanofluid is inspected by the geometrical wall due to the intense wall suction, which results in a decrease in the substantial flow decrease. So, as the wall suction increases, the velocity distribution decreases. Figure 3(b) depicts the influence of the thermal Grashof constraint on the velocity. As the thermal Grashof number is augmented, the viscosity of nanofluid Volume 26, Issue 5, 2023 6 Khan, Nazeer, & Shehzad (a) (b) (c) FIG. 2: }-curve for HAM solutions (a) (b) FIG. 3. Journal of Porous Media Darcy-Forchheimer Tangent Hyperbolic Nanofluid Flow 7 (c) (d) FIG. 3: (a) Impact of S on f ′ (η) for cone when n∗ = 0.5, Da = 0.1, We = 0.1, δ = 0.5, Fs = 1.1, Gc = 0.5, λ = 0.5, M = 0.7, λ1 = 0.2, Gr = 0.5, and α = 45◦ ; (b) impact of Gr on f ′ (η) for cone when n∗ = 0.5, Da = 0.1, We = 0.1, δ = 0.5, Fs = 1.1, S = 0.5, Gc = 0.5, λ = 0.5, M = 0.7, λ1 = 0.2, and α = 45◦ ; (c) impact of Gc on f ′ (η) for cone when n∗ = 0.5, Da = 0.1, We = 0.1, δ = 0.5, Fs = 1.1, S = 0.5, λ = 0.5, M = 0.7, λ1 = 0.2, Gr = 0.5, and α = 45◦ ; (d) impact of We on f ′ (η) for cone when n∗ = 0.5, Da = 0.1, δ = 0.5, Fs = 1.1, S = 0.5, Gc = 0.5, λ = 0.5, M = 0.7, λ1 = 0.2, Gr = 0.5, and α = 45◦ (a) (b) (c) (d) FIG. 4. Volume 26, Issue 5, 2023 8 Khan, Nazeer, & Shehzad (e) (f) FIG. 4: (a) Impact of α on f ′ (η) for cone when n∗ = 0.5, Da = 0.1, We = 0.1, δ = 0.5, Fs = 1.1, S = 0.5, Gc = 0.5, λ = 0.5, M = 0.7, λ1 = 0.2, and Gr = 0.5; (b) impact of Da on f ′ (η) for cone when n∗ = 0.5, We = 0.1, δ = 0.5, Fs = 1.1, S = 0.5, Gc = 0.5, λ = 0.5, M = 0.7, λ1 = 0.2, Gr = 0.5, and α = 45◦ ; (c) impact of δ on f ′ (η) for cone when n∗ = 0.5, Da = 0.1, We = 0.1, Fs = 1.1, S = 0.5, Gc = 0.5, λ = 0.5, M = 0.7, λ1 = 0.2, Gr = 0.5, and α = 45◦ ; (d) impact of Fs on f ′ (η) for cone when n∗ = 0.5, Da = 0.1, We = 0.1, δ = 0.5, S = 0.5, Gc = 0.5, λ = 0.5, M = 0.7, λ1 = 0.2, Gr = 0.5, and α = 45◦ ; (e) impact of M on f ′ (η) for cone when n∗ = 0.5, Da = 0.1, We = 0.1, δ = 0.5, Fs = 1.1, S = 0.5, Gc = 0.5, λ = 0.5, λ1 = 0.2, Gr = 0.5, and α = 45◦ ; (f) impact of n∗ on f ′ (η) for cone when Da = 0.1, We = 0.1, δ = 0.5, Fs = 1.1, S = 0.5, Gc = 0.5, λ = 0.5, M = 0.7, λ1 = 0.2, Gr = 0.5, and α = 45◦ decreases, and as a result, the velocity is enhanced. Figure 3(c) reports the nature of the concentration Grashof on the velocity. The increasing values of concentration Grashoff number intensify the velocity profile. The influence of the Weissenberg number on velocity profile is shown in Fig. 3(d). For higher Weissenberg number, a reducing effect on velocity distribution is seen. Physically, the increasing Weissenberg number specifies that the nanofluid flow particles will take more time to achieve their original state. The velocity distribution is reduced by increasing fluid viscosity. In fact, higher Weissenberg values give more resistance to nanofluid flow, leading to an increase in heat and nanofluid flow concentration. As a result, velocity appears to be decreasing, while temperature and concentration distributions appear to be increasing. Figures 4(a)–4(f) display the effects of semi-vertical angle, Darcy number, Forchheimer inertial parameter, magnetic parameter, and index of material power-law on the velocity. Figure 4(a) illustrates the effect of the cone’s semi-vertical angle on the liquid velocity. In the boundary layer regime, a rise in α significantly decreases the velocity. The velocity against Darcy number with the new radial coordinates closer to the top edge is shown in Fig. 4(b). An increment in the Darcy number enhanced the permeability of the porous medium, which prompted the flow to accelerate, thus the velocity increases. The velocity against δ values is shown in Fig. 4(c). As δ rises, the velocity tends to decrease. The effect of the Forchheimer inertial parameter on the velocity profile is shown in Fig. 4(d). When the Forchheimer inertial drag parameter is raised, the flow decelerates, which causes a decline in the velocity profile. The porous medium flow becomes progressively turbulent when the Forchheimer parameter is enhanced. The magnetic parameter’s, M influence on the velocity is represented in Fig. 4(e). A resistive force acted against the flow direction when a magnetic force is implemented normal to direction of the flow. As the magnetic field continues to intensify, this force slows down the liquid movement along the cone and causes a decrease in velocity. The characteristics of the power law index on the velocity are executed in Fig. 4(f). It has been observed that as n∗ increases, velocity and corresponding layer thickness decrease. The effects of modification of the space dependent parameter, mixed convected constraint, temperature dependent factor, and Prandtl number on temperature are exposed in Figs. 5(a)–5(d). The temperature field by space, as well as the temperature dependent heat source/sink variables, is executed in Figs. 5(a) and 5(b). The temperature is augmented by improving the space and temperature-dependent constraints. As a consequence of this, heat source produces more heat which causes the temperature to rise and corresponding layer density. The behavior of mixed convected constraint on the temperature is evaluated in Fig. 5(c). When this parameter rises, the layer thickness falls due to the higher Journal of Porous Media Darcy-Forchheimer Tangent Hyperbolic Nanofluid Flow 9 (a) (b) (c) (d) FIG. 5: (a) Impact of A1 on θ(η) for cone when Pr = 0.9, r1 = 1.0, N t = 0.4, N b = 0.2, A2 = 1.0, and γ = 0.9; (b) impact of A2 on θ(η) for cone when Pr = 0.9, r1 = 1.0, N t = 0.4, N b = 0.2, A1 = 1.5, and γ = 0.9; (c) effect of r1 on θ(η) for cone when Pr = 0.9, N t = 0.4, N b = 0.2, A1 = 1.5, A2 = 1.0, and γ = 0.9; (d) effect of Pr on θ(η) for cone when r1 = 1.0, N t = 0.4, N b = 0.2, A1 = 1.5, A2 = 1.0, and γ = 0.9 (a) (b) FIG. 6. Volume 26, Issue 5, 2023 10 Khan, Nazeer, & Shehzad (c) (d) FIG. 6: (a) Impact of r2 on φ(η) for cone when N t = 0.4, N b = 0.2, and Sc = 0.5; (b) impact of N b on φ(η) for cone when N t = 0.4, Sc = 0.5, and r2 = 1.0; (c) impact of N t on φ(η) for cone when N b = 0.4, Sc = 0.5, and r2 = 1.0; (d) impact of Sc on φ(η) for cone when N t = 0.4 and r2 = 1.0 (a) (b) ′ FIG. 7: Influence of (a) We on f (η) and (b) N b on ϕ (η) (a) (b) FIG. 8: Influence of (a) N t and (b) N b on θ (η) Journal of Porous Media Darcy-Forchheimer Tangent Hyperbolic Nanofluid Flow 11 TABLE 1: Values of Nusselt number at the surface Nt 0.1 0.2 0.3 — 0.5 — — 0.5 — — — 0.5 Pr 0.7 — — 0.1 0.2 0.3 — 0.7 — — — 0.7 A1 1.3 — — — 1.3 — 0.1 0.2 0.3 — — 1.3 A2 1.6 — — — 1.6 — — 1.6 — 0.1 0.2 0.3 Nux 0.322700 0.322700 0.322700 0.324481 0.323778 0.323074 0.326815 0.325704 0.324593 0.324593 0.324222 0.323852 TABLE 2: Values of Sherwood number at the surface Nt 0.1 0.2 0.3 — 0.5 — — 0.5 — — — 0.5 Nb 0.7 — — 0.1 0.2 0.3 — 0.7 — — — 0.7 Sc 1.6 — — — 1.6 — 0.1 0.2 0.3 — — 1.6 Da 2.4 — — — 2.4 — — 2.4 — 0.1 0.2 0.3 Shx 0.0242947 0.0485449 0.0727949 0.2182950 0.1091700 0.0727949 0.0511027 0.0565257 0.0619488 0.0727949 0.0727949 0.0727949 mixed convected constraint. Physically, the thermal buoyancy forces increase for larger mixed convected constraint. It improves the ultimate thermal field. Figure 5(d) displays the influence of the Prandtl number on temperature profile. The augmented Prandtl number values cause an enhancing trend of temperature. Physically, the large Prandtl fluids have smaller thermal diffusivity. Therefore, the higher values of Prandtl number cause a reduction in temperature. The effects of wall concentration, Brownian motion, thermophoresis parameter, and Schmidt number on concentration distribution of a nanofluid flow in a cone are shown in Figs. 6(a)–6(d). The effect of wall concentration on nanofluid flow through the cone is shown in Fig. 6(a). By increasing wall concentration, concentration distribution increases. Figure 6(b) depicts the effect of Brownian motion on the concentration profile. The Brownian parameter in the Buongiorno model has an opposite relation with the nanofluid flow concentration distribution. As a result, the higher values of Brownian motion parameter lower the nanofluid’s flow concentration. The thermophoresis parameter’s impact on the concentration distribution for the cone is seen in Fig. 6(c). The concentration profile increases Volume 26, Issue 5, 2023 12 Khan, Nazeer, & Shehzad by increasing the values of thermophoresis parameter due to the effect that thermal conductivity enhances for higher values of thermophoresis parameter. The effect of Schmidt number on the concentration profile is shown in Fig. 6(d). A reducing influence in concentration profile is observed here. When the Schmidt number is increased, the Brownian diffusion coefficient and nanofluid flow diffusion through the boundary layer are reduced, and as a result the concentration profile declines. Figures 7 and 8 address the comparison of results with shooting method and HAM for various values of Brownian motion and thermophoresis parameter. A good agreement is achieved between HAM and shooting solutions. 4. CONCLUSIONS The non-linear mixed convected boundary-driven TH nanofluid flow through a cone is reported and discussed. DarcyForchheimer effects have also been taken into account in mathematical modeling and analysis. The governing equations have been constructed through the laws of conservation. The modeled flow problem is converted into a set of ordinary differential equations with the help of proposed similarity transformations. To interpret the modified system of equations, the homotopy analysis method (HAM) is applied. The key findings of our work are as follows: • When we increase the values of Gr, Gc , We, and Da, the velocity profile enhances. • The velocity profile declines when the values of S, α, δ, Fs , and M enhance. • The temperature profile enhances when we enhance the values of A1 , A2 , r1 , and Pr. • The increase in the values r2 and Nt give rise to the concentration profile. • The concentration is decreased by augmenting the Nb and Sc values. • The results computed by HAM and shooting technique have good comparison with each other. • The increasing values of Nt and Sc enhance the values of Sherowed number, while the opposite is observed by increasing the value of Nb . • Generally, the mathematical model is investigated, which is very useful in controlling and adjusting the heat transportation rate and velocity of fluid during the various industrial and technological processes. The utilization of Prandtl number is very prominent in the heat transportation rate for cooling processes. 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