Nanofluid with nonlinear Rosseland thermal radiation and mixed convection

Nanouid with nonlinear Rosseland thermal radiation and mixed convection

Nazish Iftikhar^a, Muhammad Bilal Riaz^b, Azhar Ali Zafar^c, Syed Muhammad Husnine^a

aDepartment of Sciences & Humanities, National University of Computer & Emerging Sciences, Lahore Campus, Lahore, Pakistan.

bDepartment of Mathematics, University of Management and Technology, Lahore, Pakistan.

Institute of Grounderwater Studies, University of the Free State, South Africa.

cDepartment of Mathematics, Government College University, Lahore, Pakistan.

Abstract

Two dimensional ow of mixed convection nanouid on horizontal plate with the eect of nonlinear Rosseland thermal radiation has been investigated. Mathematical model of the problem is based on partial dierential equations and optimal homotopy analysis method (OHAM) is applied to sort out solutions. Moreover, comprehensive study of inuence of emerging parameters is carried out via graphical interpretation and tables.

Keywords: Nanouid Mixed Convection Nonlinear Rosseland Thermal Radiation

1. Introduction

Nanouids have a lot of applications in medical industry, engine cooling, detergency, pharmaceutical pro- cesses, heat exchanger and space technology. Nanouids contain nanometer sized particles called nanopar- ticles. Nanouids consist of base uid which is usually water or oil with nanoparticles like metals, oxides, carbides and carbon. Some commonly used nanouids are T iO2 (Titanium dioxide) in water, CuO (Copper oxide) in water, Al2O₃ (Aluminium oxide) in water, ZnO (Zinc oxide) in ethylene glycol. Choi [1] established the concept of nanouids. Nanotechnology gain attention in the heat transfer process due to its character- istics of thermal conductivity.

Email addresses: [email protected] (Nazish Iftikhar), [email protected] (Muhammad Bilal Riaz), [email protected] (Azhar Ali Zafar), [email protected] (Syed Muhammad Husnine)

Received June 14, 2020; Accepted: May 29, 2021; Online: June 1, 2021.

Table 1: Nomenclature Symbol Quantity

Uw Plate velocity T Fluid temperature T_w Plate temperature T∞ Ambient temperature

C Nanoparticle volume fraction

C_w Nanoparticle volume fraction at plate

C∞ Nanoparticle volume fraction away from the plate g Gravitational acceleration

λ Local buoyancy parameter N t Thermophoresis parameter N b Parameter of Brownian motion N r Radiation paramter

N1 Concentration buoyancy parameter β_T Coecient of thermal expansion β_C Coecient concentration expansion DT Coecient of Thermophoretic diusion λ Retardation time

P r Prandtl number Sc Schmidt number

D_B Coecient of Brownian diusion θ_r Temperature parameter

aR Rosseland mean spectral absorption coecient σ_SB Stefan-Boltzmann constant

C_p Specic heat

Grx Local Grashof number

u, v Velocity along x-axis and y-axis

Mixed convection is a phenomenon occurred due to free convection and forced convection. Flow problems having mixed convection has great importance in applied perspective especially in industrial, technical pro- cesses. Pal and Mandal studied [2] three types of nanouids along with the thermal radiation and mixed convection. Hayat et al. [3] investigated coupled stress nanouid ow with nonlinear thermal radiation past a stretching surface. Thermal radiation eect in uid ow problems with mixed convection and convective condition are discussed in [4]-[6].

Nonlinear thermal radiation eect has great importance in engineering, nuclear reactors, missiles, and satel- lites. Hayat et al. [7] studied nonlinear thermal radiation eect in viscoelastic nanouid. Shehzad et al. [8]

studied thermophoresis eect and brownian motion in Jerey nanouid with thermal radiation. Pantokra- toras [9] investigated natural convection on isothermal plate with the impact of linear or nonlinear Rosseland radiation convection along with radiation parameter. Work has been done in this area by researchers [10]- [14]. Farooq et al. investigated heat transfer phenomena in viscoelastic nanouid with nonlinear radiative eects [15]. Hayat et al. [16] analyze heat transfer in nanouid with nonlinear thermal radiation and inclined magnetic eld. Many researchers pay attention towards nonlinear thermal radiation [17]-[24]. Pantokratoras and Fang [25] studies Blasius ow in the presence of nonlinear Rosseland thermal radiation. Some important phenomenons regarding nonlinear thermal radiations considered by researchers [26]-[29].

In this article, nanouid with nonlinear Rosseland thermal radiation and mixed convection has been inves- tigated. Mathematical model involve partial dierential equations. OHAM is used to investigate solutions.

In addition, results are highlighted by tables and graphs.

direction of a horizontal plate with components of velocity u and v. Velocity of the plate is Uw (see gure 1). Mathematical model is given below:

Figure 1: Model of the problem

∂u

∂x +∂v

∂y = 0, (1)

uux+ vuy = νuyy+ g[βT(T − T∞) + βC(C − C∞)], (2)

u∂T

∂x + v∂T

∂y = α∂²T

∂y² + κDB

∂T

∂y

∂C

∂y + κDτ

T∞

(∂T

∂y)²+ 4σSB

3ρCpaR

(∂²T

∂y²), (3)

u∂C

∂x + v∂C

∂y = DB

∂²C

∂y² +DT

T∞

(∂²T

∂y²). (4)

The boundary conditions are considered as:

u = U_w, v = 0, T = T_w, C = C_w, at y = 0, (5)

u → 0, τ → T∞, C → C∞, as y → ∞. (6)

Introducing similarity transformations:

u = Uwg⁰(η), v = 1 2

rUwυ

x [ηg⁰(η) − g(η)], (7)

θ(η) = T − T∞

T_w− T_∞, η = rUw

xυy, φ(η) = C − C∞

C_w− C_∞. (8)

By putting values of u and v in equation (1), it satised. Moreover substituting equations (7) and (8) into equations (2), (3) and (4), we get

g⁰⁰⁰+1

2gg⁰⁰+ λ(θ + N1φ) = 0, (9)

m c^g₀ c^θ₀ c^φ₀ ε^m_t CPU Time 2.0 -1.27 -0.51 -0.49 6.3×10⁻³ 19.277 4.0 -1.28 -0.57 -0.59 3.2×10⁻³ 182.615 6.0 -1.03 -0.43 -1.05 3.5×10⁻⁴ 1348.11

Table 2: Values of errors according to Optimal convergence control parameters via BVPh2.0

N rθ⁰⁰+ N bP rN rθ⁰φ⁰+ N tP rN r(θ⁰)²+1

2N rP rgθ⁰ +

1

3[(θ(θr− 1) + 1)⁴]⁰⁰

θ_r− 1 = 0, (10)

φ⁰⁰+1

2Scgφ⁰+ N t

N bθ⁰⁰= 0, (11)

φ(0) = 1, φ(∞) = 0, θ(0) = 1, θ(∞) = 0,

g(0) = 0, g⁰(0) = 1, g⁰(∞) = 0. (12)

Dimensionless numbers with parameters are given below:

N t = DT(Tw− T_∞)κ νT∞

, N b = κDB(Cw− C_∞)

ν , α = k

ρC_p, Sc = ν

DB

, λ = Gr_x

Re²_x, θ_r= T_w T∞

, N r = ka_R

4σSBT_∞³ , Re = U_wx υ , Grx = (T_f − T_∞)x³gβ_T

ν² , N1 = (C_w− C∞)β_C (Tf − T_∞)βT

. (13)

Local Sherwood number is given below Sh/Re

1

x2 = −φ⁰(0), (14)

and local Reynold is given by Re_x= xU_w

ν . (15)

Assume initial approximations are φ₀(η) = e^−η,

θ₀(η) = e^−η,

g₀(η) = 1 − e^−η. (16)

Let auxiliary operators are Lφ= φ⁰⁰− φ,

L_θ = θ⁰⁰− θ,

L_g = g⁰⁰⁰− g⁰. (17)

12.0 1.68×10 2.82×10 3.79×10 74.1165 18.0 5.75×10⁻⁸ 2.30×10⁻⁶ 7.26×10⁻⁵ 270.534 Table 3: Error analysis from Table 2 at 6^th iteration

3. Convergence control parameters

Convergence of solution can be control in homotopy analysis method by using dierent parameters denoted by c^g₀, c^θ₀ and c^φ₀. Values of these parameters can be obtained by minimizing error. BVPh2.0 is applied in order to get minimum error. Three arrays are selected. First array is selected at 2^nd iteration, second array is selected at 4^th iteration and third array is selected at 6^th iteration. Table 2 shows error analysis at 6^th iteration.

4. Results and discussion

Analysis of graphs for dierent parameters are examined in this section. Figure 2 depicts that there is an increase in velocity as λ increases. Thermal buoyancy force enhances when λ increases due to which velocity is enhanced. Eect of Nb on θ(η) and φ(η) presents in gures 3 and 4. By increasing Nb, temperature increases on the other hand concentration decreases. Figure 5 demonstrated that an increase in Nt, enhances temperature. Thermal conductivity enhances as Nt increases due to which temperature increases. Figure 6 displayed that φ(η) increases as Nt increases. There is reduction in θ(η) with the increasing value of Nr (see gure 7). Physically it is because of production of heat in moving uid which is generated to increase in radiation as a result temperature raises. Figure 8 interprets the inuence of φ(η) for Sc. As Sc increases there is a decrease in concentration. Physically by increasing Schmidt number there is mass diusivity become less and hence φ(η) decreases. Figure 9 shows that temperature decreases as N1 increases. As θr increases there is an increase in temperature (see gure 10). Further, Table 2 shows values of parameters which are responsible for convergence. Table 3 depicts error for 6th iteration. Table 4 presents values of Sherwood number corresponding to the parameters.

5. Conclusion

Nanouid is considered over horizontal moving plate under the inuence of nonlinear Rosseland thermal radiation with mixed convection. Fundamental observations are given below.

• N t and Nb have the same and opposite eect on φ(η) and θ(η) respectively.

•Increasing value of Nr accelerates the θr.

•Enhancement in Sc leads to increase in φ(η).

Figure 4: Inuence of Nb on φ(η) with Nt=P r=1.0, Nr=Sc=θr=1.5 and λ=N1=0.1

λ N₁ Sc θ_r N t N r Sh/Re

1

x2

0.2 0.98640

0.4 0.98833

0.6 0.99025

0.2 0.98553

0.4 0.98570

0.6 0.98588

0.1 0.85950

0.3 0.87708

0.5 0.89480

0.1 0.79807

0.3 0.78990

0.5 0.79765

0.0 0.62183

0.2 0.67277

0.4 0.73461

0.0 0.43759 0.2 0.51064 0.4 0.58368 Table 4: Local Sherwood numbers for existing parameters

Figure 2: Inuence of λ on g⁰(η)with Nt=P r=1.0, Nr=Nb=Sc=θr=1.5 and N1=0.1

Figure 3: Inuence of Nb on θ(η) with Nt=P r=1.0, Nr=Sc=θr=1.5 and λ=N1=0.1

Figure 5: Inuence of Nt on θ(η) with P r=1.0, Nr=Nb=Sc=θr=1.5 and λ=N1=0.1

Figure 6: Inuence of Nt on φ(η) with P r=1.0, Nr=Nb=Sc=θr=1.5 and λ=N1=0.1

Figure 7: Inuence of Nr on θ(η) with Nt=P r=1.0, Nb=Sc=θr=1.5 and λ=N1=0.1

Figure 8: Inuence of Sc on φ(η) with Nt=P r=1.0, Nr=Nb=θr=1.5 and λ=N1=0.1

Figure 9: Inuence of N1 on θ(η) with Nt=P r=1.0, Nr=Nb=Sc=θr=1.5 and λ=0.1

Figure 10: Inuence of θr on θ(η) with Nt=P r=1.0, Nr=Nb=Sc=1.5 and λ=N1=0.1

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