J. Cent. South Univ. (2019) 26: 865-872

DOI: https://doi.org/10.1007/s11771-019-4055-1

Numerical treatment for Darcy-Forchheimer flow of carbon nanotubes due to an exponentially stretching curved surface

Tasawar HAYAT^{1, 2}, Farwa HAIDER¹, Taseer MUHAMMAD³, Ahmed ALSAEDI²

1. Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan;

2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia;

3. Department of Mathematics, Government College Women University, Sialkot 51310, Pakistan

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: This article gives a numerical report to two dimensional (2D) Darcy-Forchheimer flow of carbon-water nanofluid. Flow is instigated by exponential extending curved surface. Viscous liquid in permeable space is described by Darcy-Forchheimer. The subsequent arrangement of partial differential equations is changed into ordinary differential framework through proper transformations. Numerical arrangements of governing frameworks are set up by NDSolve procedure. Outcomes of different sundry parameters on temperature and velocity are examined. Skin friction and heat transfer rate are also shown and inspected.

Key words: carbon nanotube; Darcy-Forchheimer flow; exponentially stretching curved surface; numerical solution

Cite this article as: Tasawar HAYAT, Farwa HAIDER, Taseer MUHAMMAD, Ahmed ALSAEDI. Numerical treatment for Darcy-Forchheimer flow of carbon nanotubes due to an exponentially stretching curved surface [J]. Journal of Central South University, 2019, 26(4): 865–872. DOI: https://doi.org/10.1007/s11771-019-4055-1.

1 Introduction

Carbon nanotubes (CNTs) are the minimum complex engineered creation and atomic holding outline of graphene sheet structures climbed in a condition of barrel. CNTs have extraordinary physical, electrical, warm, engineered and mechanical properties, on account of the blend of their little size, tube formed structure and tremendous surface zone. Dependent upon the amount of graphene layers, carbon nanotubes have been subdivided into single-walled (SWCNTs) and multi-walled (MWCNTs) carbon nanotubes. A typical thermal conductivity change in oil based nanofluids containing carbon nanotube is portrayed by CHOI et al [1]. XUE [2] proposed a model in light of Maxwell speculation for transport properties of CNTs-based composites. Heat transfer of liquid suspensions of multi-walled carbon nanotubes is analyzed by DING et al [3]. Further pertinent commitments toward this path can be investigated through the attempts [4–20] and different examinations therein.

Convective fluid flow and heat transfer subject to permeable space is encountered in chemical reactors, geology, drying and liquid composite molding, biological and combustion applications. Darcy built up a pioneering semi-experimental equation substantial for flow under low porosity and small velocity conditions. For high Reynolds number, nonlinearity shows up in semi- observational equation which is because of expanding part of inertial force. FORCHHEIMER [21] anticipated a changed expression in particular Darcy-Forchheimer expression by presenting nonlinear term of velocity in energy equation. MUSKAT [22] named it as Forchheimer term. Darcy-Forchheimer mixed convection in a fluid saturated porous space with thermophoresis and viscous dissipation is analyzed by SEDDEEK [23]. Having such in see, promoting important attempts on Darcy-Forchheimer model can be cited through examinations [24–30] and numerous investigations therein.

Researchers at present are engaged in examining flow instigated by an extending surface. It is because of its applications in manufacturing and industry mechanisms, for example, condensation process, annealing and tinning of copper wire, cooling of an endless metallic plate in a cooling shower, continuous stretching of plastic films, extrusion of plastic sheets and several others. Initially, CRANE [31] considered the fluid flow caused by a stretching surface. Much efforts in past have been devoted to planer stretching surface. Few studies have been only undertaken for curved stretching sheet. For example, flow of viscous fluid induced by a curved extending sheet is investigated by SAJID et al [32]. ROSCA et al [33] studied flow because of porous bended extending/contracting surface. Effects of magnetic field are present. Radiative flow of micropolar liquid by a bended extending sheet is examined by NAVEED et al [34]. ABBAS et al [35] analyzed hydromagnetic slip flow of nanofluid initiated by a bended extending sheet with heat generation and radiation. IMTIAZ et al [36] investigated impacts of homogeneous- heterogeneous responses in MHD flow caused by a bended extending sheet. Flow of micropolar liquid actuated by a bended extending surface with homogeneous-heterogeneous responses is analyzed by HAYAT et al [37]. SANNI et al [38] provided a numerical solution for flow of viscous liquid actuated by a nonlinearly extending bended surface. Flow of viscous liquid initiated by an exponentially extending bended sheet is introduced by OKECHI et al [39]. HAYAT et al [40] numerically contemplated Darcy-Forchheimer flow actuated by a bended extending sheet with Cattaneo-Christov theory and homogeneous-heterogeneous responses. Further recent examinations on bended extending surface can be cited through the investigations [41–46].

Our inspiration in current paper is covered by three novel features. Firstly to model and analyze two dimensional (2D) Darcy-Forchheimer flow of water-based carbon nanotubes initiated by an exponentially extending curved surface. Secondly to use XUE model [2] for nanofluid transport process. Thirdly to construct numerical solutions for velocity and temperature fields through NDSolve technique. The commitments of different physical parameters are examined and discussed. Furthermore, skin friction and heat transfer rate are computed and interpreted.

2 Statement

We assume steady two dimensional (2D) flow of carbon-water nanofluid. Flow is induced by an exponential extending bended sheet coiled in circle of radius R (see Figure 1). Permeable space by Darcy-Forchheimer model is considered. Here depicts the exponential velocity with a>0. Resulting boundary-layer relations for mass, momentum and energy are [44, 45]:

Figure 1 Geometry of problem

                     (1)

                           (2)

  (3)

       (4)

where one has the following prescribed conditions [39, 45]

                              (5)

       (6)

where u and v denote velocity components in s– and r– directions, K^* is the permeability of porous space, C_b is the drag coefficient, is the non-uniform inertia coefficient, T_w and T_∞ are the wall and ambient temperatures respectively. Theoretical model suggested by XUE [2] is defined by

,

,

           (7)

where f depicts the volume fraction of nanoparticle, μ_nf is the nanofluid effective nanofluids dynamic viscosity, ρ_nf is the nanofluids density, (ρc_p)_nf is the nanofluids heat capacity, k_nf is the nanofluids thermal conductivity, ρ_CNT is the carbon nanotubes density, ρ_f is the base fluid density, k_CNT is the CNTs thermal conductivity and k_f is the base fluid thermal conductivity. Table 1 represents thermo-physical features of carbon nanotubes and water.

Table 1 Thermo-physical features of carbon nanotubes and water [2]

Considering [39, 45]

          (8)

Now, Eq. (1) is automatically verified and Eqs. (2)–(7) yield

              (9)

         (10)

                  (11)

Eliminating pressure H from Eqs. (9) and (10), we get

              (12)

                 (13)

            (14)

In the above relations, Pr depicts the Prandtl number, F_r is the Forchheimer number, λ is the local porosity parameter and K is the curvature parameter. These variables are:

 (15)

Skin friction coefficient and local Nusselt number are

      (16)

               (17)

whereelucidates local Reynolds number.

3 Discussion

Local-similar arrangements of Eqs.(11) and (12) via (13) and (14) are figured numerically by utilizing NDSolve method. This portion outlines impacts of solid volume fraction of nanoparticles f local porosity parameter λ, curvature parameter K, Forchheimer number F_r and temperature exponent A on velocity f ′(ζ) and temperature θ (ζ) fields. The outcomes are accomplished for SWCNTs and MWCNTs. Features of f on velocity f ′(ζ) are plotted in Figure 2. Larger f causes an enhancement in velocity field f ′(ζ) for SWCNTs and MWCNTs. Figure 3 is interpreted to analyze impact of curvature parameter K on velocity field f ′(ζ). It is observed that an increment in K shows higher velocity field f ′(ζ) for both SWCNTs and MWCNTs. In Figure 4, it is clearly examined that lower velocity field f ′(ζ) is generated by using larger local porosity parameter λ for both SWCNTs and MWCNTs. Behavior of Forchheimer number Fr on velocity field f ′(ζ) is shown in Figure 5. Here velocity field f ′(ζ) reduces for higher Fr for both SWCNTs and MWCNTs. Figure 6 displays impact of f on temperature θ(ζ). It is revealed that increment in f enhances temperature θ(ζ) for SWCNTs and MWCNTs. The variation of curvature K on temperature field θ(ζ) is depicted in Figure 7. By increasing K, a reduction in temperature field θ(ζ) for SWCNTs and MWCNTs is observed. Figure 8 depicts that how local porosity λ affects temperature field θ(ζ). Here higher λ leads to enhancement in temperature θ(ζ) for SWCNTs and MWCNTs. Figure 9 displays that larger Forchheimer number Fr yields stronger temperature field θ(ζ) for SWCNTs and MWCNTs. Figure 10 is sketched to characterize the consequences of temperature exponent A on temperature field θ(ζ). It is noted that an increment in A causes weaker temperature field θ(ζ) for SWCNTs and MWCNTs.

Skin friction coefficient for various pertinent flow variables such as f, K and λ is plotted in Figures 11 and 12. Here skin friction is higher for increasing estimations of f for both SWCNTs and MWCNTs. Figures 13 and 14 elucidate local Nusselt number for SWCNTs and MWCNTs. We concluded that local Nusselt number is increased for higher f, K and λ. Table 2 is developed to validate present outcomes with previously published outcomes by OKECHI et al [39]. From this Table, we examined that present NDSolve solution is in good agreement with the previous solution by OKECHI et al [39].

Figure 2 Plots of f ′(ζ) for f

Figure 3 Plots of f ′(ζ) for K

Figure 4 Plots of f ′(ζ) for λ

Figure 5 Plots of f ′(ζ) for Fr

Figure 6 Plots of θ (ζ) for f

Figure 7 Plots of θ (ζ) for K

4 Conclusions

Darcy-Forchheimer flow of carbon nanotubes (CNTs) by an exponentially extending bended surface is examined. Fundamental perceptions of present examination are:

1) Opposite behavior is noted in velocity and temperature fields when local porosity parameter λ and Forchheimer number Fr are enhanced.

2) Both velocity and temperature fields show similar trend for larger solid volume fraction of nanoparticles f.

Figure 8 Plots of θ (ζ) for λ

Figure 9 Plots of θ (ζ) for F_r

Figure 10 Plots of θ (ζ) for A

Figure 11 Plots of for f and K

Figure 12 Plots of for f and λ

Figure 13 Plots of for f and K

3) An increment in curvature parameter K yields higher velocity field.

4) Larger temperature exponent A exhibits decreasing trend for temperature.

5) Both skin friction and heat transfer rate are increased for larger f.

Figure 14 Plots of for f and λ

Table 2 Comparative data of for varying K when f=λ=Fr=0

Nomenclature

u, v

Velocity components

r, s

Space coordinate

μ_f

Fluid dynaminc viscosity

v_f

Kinematic fluid viscosity

k_f

Basefluid thermal conductivity

α_f

Thermal diffusivity of base fluid

T_w

Wall temperature

p

Pressure

a

Positive constant

F

Non-uniform inertia coefficient

C_b

Drag coefficient

C_f

Skin friction coefficient

λ

Local porosity parameter

Fr

Forchheimer number

K

Curvature parameter

CNTs

Carbon nanotubes

Pr

Prandtl number

R

Radius

U_w

Surface stretching velocity

ρ_f

Fluid density

v_nf

Kinematic nanofluid viscosity

k_nf

Nanofluids thermal conductivity

α_nf

Thermal diffusivity of nanofluid

T_∞

Ambient temperature

k_CNT

CNTS thermal conductivity

K^*

Permeability of porous medium

f

Nanomaterial volume fraction

Re_s

Local Reynolds number

Nu_s

Local Nusselt number

ζ

Dimensionless variable

f′

Dimensionless velocities

θ

Dimensionless temperature

H

Dimensionless pressure

A

Temperature exponent

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(Edited by HE Yun-bin)

中文导读

碳纳米管指数拉伸曲面上Darcy-Forchheimer流动的数值处理

摘要：本文对碳水纳米流体的二维流动进行了数值模拟。流体是由指数扩展曲面引起的。本文用Darcy-Forchheimer方法描述了渗透空间中的黏性液体。通过适当的变换，将偏微分方程的后续排列转化为常微分框架。采用NDSolve程序建立了控制框架的数值排布。考察了不同参数对温度和速度的影响, 并对表面摩擦和传热速率进行监测。

关键词：碳纳米管；Darcy-Forchheimer流；指数拉伸曲面；数值解

Received date: 2018-05-06; Accepted date: 2018-10-23

Corresponding author: Taseer MUHAMMAD, PhD, Assistant Professor; Tel: +92-321-7132571; E-mail: taseer_qau@yahoo.com