- Email: [email protected]
Mathematical modeling and analysis of SWCNT-Water and MWCNT-Water flow over a stretchable sheet
Mathematical modeling and analysis of SWCNT-Water and MWCNT-Water flow over a stretchable sheet
Journal Pre-proof
Mathematical modeling and analysis of SWCNT-Water and MWCNT-Water flow over a stretchable sheet Muhammad Ibrahim, M. Ijaz Khan PII: DOI: Reference:
S0169-2607(19)31683-9 https://doi.org/10.1016/j.cmpb.2019.105222 COMM 105222
To appear in:
Computer Methods and Programs in Biomedicine
Received date: Revised date: Accepted date:
1 October 2019 13 November 2019 15 November 2019
Please cite this article as: Muhammad Ibrahim, M. Ijaz Khan, Mathematical modeling and analysis of SWCNT-Water and MWCNT-Water flow over a stretchable sheet, Computer Methods and Programs in Biomedicine (2019), doi: https://doi.org/10.1016/j.cmpb.2019.105222
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Highlights • Mixed convective of CNTs based flow of viscous material is addressed. • Both single and multi-walls carbon nanotubes is discussed. • Darcy’s law is used to characterize porous medium. • Viscous dissipation is used for heat transport in energy equation.
1
Mathematical modeling and analysis of SWCNT-Water and MWCNT-Water flow over a stretchable sheet Muhammad Ibrahim1,∗and M. Ijaz Khan2 1 School of Mathematics and Physics, University of Science and Technology Beijing, , Beijing 100083, China 2 Department of Mathematics, Quaid-I-Azam University 45320, Islamabad November 19, 2019
Abstract In this article we focused on the mixed convection flow of SWCNT-Water and MWCNT-Water over a stretchable permeable sheet. The nanofluid occupied porous medium. Darcy’s law is used to characterize porous medium. The impact of viscous dissipation is considered. Transformation procedure is adopted to transform the governing PDE’s system into dimensionless form. In order to solve the dimensionless PDE’s system we used numerical method known as Finite difference method. Effects of flow variables i.e porosity parameter, suction parameter, Grashof number and Reynolds number on velocity, skin friction, temperature and Nusselt number are described graphically. The obtained results shows that velocity is dominant in SWCNT-Water over MWCNT-Water. Temperature is dominant in MWCNT-Water over SWCNT-Water.
Keywords: Finite difference method; SWCNT-water and MWCNT-Water nanofluids; Stretching plate; Porous medium; Mixed convection; Viscous dissipation.
1
Introduction
In present time, nanofluids study gain the attention of researchers because they have higher thermal conductivity as compered to base fluids. The suspension of nanoparticles (1nm − 100nm) in the base fluid is called nanofluids. Base fluid includes polymeric solutions, oils, bio-fluids, water, refrigerants, lubricants, ethylene glycol etc. nanoparticles includes metallic particles (Cu, Al, T i, F e, Ag) and their oxides. Beside the above mentioned nanoparticles CN T (carbon nanotubes) is a nanoparticles. CN T is ∗
Corresponding author email: [email protected] (M. Ibrahim)
2
made by carbon. Various types of carbon nanotubes exists, but generally they categorized as SW CN T (single-walled carbon nanotube) or M W CN T (multi-walled carbon nanotube). SW CN T (single-walled carbon nanotube) has only one wall or layer and the other hand M W CN T (multi-walled carbon nanotube) are the collection of nested tubes of continuously enhances diameter. Due to the thermophysical properties (physical, optical, thermal and electrical) of CN T materials have various applications in industrial processes at macro as well as micro level. These applications includes electronics, biosensor, sensors, ultra-capacitors, conductive plastics, nuclear reactors, gas storage, flatpanel displays, technical textiles, medical equipments, solar collection, catalyst supports, nonporous filters and coatings of all sorts. First time Choi [1] introduce the concept of nanofluid. Two-dimensional flow of nanofluid in a enclosure is investigated by Khanfer et al. [2]. Effects of natural convection and magnetic field on Cu−Water nanofluid flow is studied by Sheikholeslami et al. [3]. Flow of Sutterby nanofluid due to a rotating disk with heat absorption/generation is expressed by Hayat et al. [4]. Entropy generation of Cu − H2 O nanoliquid is examined by Ellahi et al. [5]. Entropy generation of CN T s−H2 O nanofluid flow through porous medium is explored by Hayat et al. [6]. For further studies the readers are refereed to see [7 − 15]. Here we focused mixed convection flow of SWCNT-Water and MWCNT-water nanofluid through porous medium. The effect of viscous dissipation accounted. The governing PDE’s system is transform to dimensionless form by appropriate transformation. Numerical method known as Finite difference method [16 − 20] is used for numerical solution. The effect of flow variables porosity parameter, suction parameter, Grashof number, Reynolds number and nanoparticles volume fraction on velocity, surface drag force, temperature and temperature gradient are discussed graphically.
2
Mathematical modeling
Mixed convection flow of SWCNT-Water and MWCNT-Water nanofluids due to a permeable stretchable sheet is modeled in this section. The nanofluid fulfill the porous medium. For porous medium we used Darcy’s law. Energy equation contained the viscous dissipation effect. For flow diagram see F ig. 1. Mathematical expressions are given by: ∂e u ∂e v + = 0, ∂x ∂y 2 ∂e u ∂e u ∂e u ∂ u e ∂ 2u e νnf +u e + ve = νnf + − u e, ∂t ∂x ∂y ∂x2 ∂y 2 δ 2 νnf ∂e v ∂e v ∂e v ∂ ve ∂ 2 ve +u e + ve = νnf + − ve − gβnf (Te − T∞ ), ∂t ∂x ∂y ∂x2 ∂y 2 δ 2e ∂ Te ∂ Te ∂ Te knf ∂ T ∂ 2 Te µnf +u e + ve = + + 2 2 ∂t ∂x ∂y (ρcp )nf ∂x ∂y (ρcp )nf 2 2 2 ∂e u ∂e u ∂e v ∂e v 2 + + +2 , ∂x ∂y ∂x ∂y 3
(1) (2) (3)
(4)
Constraints are given by ve = 0, Te = Tw u e = 0, at t = 0, ve = −V0 , u e = Uw = ax, Te = Tw , at y = 0, ve = 0, u e = 0, Te = T∞ , as y → ∞.
(5)
In which t represents time, x, y space coordinates, u e, ve velocity components, knf nanofluid thermal conductivity, νnf nanofluid kinematic viscosity, δ porous medium permeability, ρnf nanofluid density, (cp )nf specific heat of nanofluid, Te temperature, µnf nanofluid dynamic viscosity, g gravity, Uw stretching velocity, V0 suction velocity, Tw sheet temperature, βnf nanofluid thermal expension coefficient, and T∞ ambient temperature. Nanofluid thermophysical properties are given by [21] : knf kCN T kCN T + kf kf kCN T + kf = 1 − φ + 2kf + 2φ In 1 − φ + 2φ In , kf kCN T − kf 2kf kCN T − kf 2kf (ρcp )nf ρnf = (1 − φ) + φ(ρcp )CN T (ρcp )f , = (1 − φ) + φρCN T ρf , (ρcp )f ρf µnf (ρβ)nf = (1 − φ) + φ(ρβ)CN T (ρβ)f , = 1 (1 − φ)2.5 , (6) (ρβ)f µf Note that νf stand for water kinematic viscosity, νCN T Carbon nanotube kinematic viscosity, βf water thermal expansion coefficient, βCN T Carbon nanotube thermal expansion coefficient, ρf density of water, ρCN T density of Carbon nanotube, kf thermal conductivity of water, kCN T thermal conductivity of Carbon nanotube, (cp )f water specific heat, (cp )CN T Carbon nanotube specific heat and φ nanoparetical volume fraction.
Table 1. Numerically estimated values of ρ, β, k and cp for H2 O and Carbon nanotube [22]. k(W/mk) ρ(kg/m3 ) 0.613 997.1 6600 2600 3000 1600
H2 O (Water) SWCNT MWCNT
β × 105 21 2.6 2.8
cp (J/kgk) 4179 0425 0769
Considering y L L , u(η, ξ, ζ) = u e, v(η, ξ, ζ) = ve, L νnf νnf νnf x Te − T∞ T (η, ξ, ζ) = , η = 2 t, ξ = , Tw − T∞ L L ζ=
4
(7)
Implementing Eg. (7), mathematical expressions of flow becomes ∂u ∂v + = 0, ∂ξ ∂ζ ∂u ∂u ∂u ∂ 2u ∂ 2u +u +v = 2 + 2 − Reλu, ∂η ∂ξ ∂ζ ∂ξ ∂ζ 2 ∂v ∂v ∂ 2v ∂ 2v (ρβ)nf ρnf µf ∂v +u +v = 2 + 2 − Reλv − Gr T, ∂η ∂ξ ∂ζ ∂ξ ∂ζ µnf (ρβ)f ρf 2 ∂T ∂T ∂T 1 knf (ρcp )f νf ∂ T ∂ 2T +u +v = + ∂η ∂ξ ∂ζ P r kf (ρcp )nf νnf ∂ξ 2 ∂ζ 2 2 2 2 (ρcp )f µnf νnf ∂u ∂u ∂v ∂v +Ec + 2 + +2 , (ρcp )nf µf νf ∂ξ ∂ζ ∂ξ ∂ζ
T = 1, u = 0, v = 0, at η = 0, νf νf u = Ra ξ, T = 1, v = −α , νnf νnf v = 0, u = 0, T = 0, at ζ → ∞.
(8) (9) (10)
(11)
at ζ = 0, (12)
νf LV0 Here λ = aδ represents porosity parameter, α = νf suction parameter, gβf (Tw −T∞ )L3 (µcp )f aL2 Gr = Grashof number, P r = kf Prandtl number, Re = νf νf2 νf2 Reynolds number and Ec = L2 (cp )f (Th −Tc ) Eckert number.
3
Quantities for engineering interest
Mathematically Cfx (skin friction) and N ux (Nusselt number) are given by Cf x =
xqw −2τw , , N ux = , 2 ρnf Uw knf (Tw − T∞ )
in which τw and qw are defined as ∂e v ∂e u τw = τyx = µnf + ∂x ∂y y=0
, y=0
e ∂T ∂ Te qw = −knf + . ∂x ∂y y=0
In dimensionless form skin friction and Nusselt number can be written as 2 −2 νnf ∂v ∂u ∂T ∂T Cf x = + , N ux = −ξ + . Re2 ξ 2 νf ∂ξ ∂η ζ=0 ∂ξ ∂η ζ=0 5
(13)
(14)
(15)
4
Finite difference scheme for solution
In order to solve the PDE’s system we chose Finite difference method. For given PDE’s system FD toolkit can be defined as: n un+1 uni+1,j − uni,j ∂u uni,j+1 − uni,j ∂u ∂u i,j − ui,j = , = , = , ∂ξ 4ξ ∂η 4η ∂ζ 4ζ n+1 n n n n n Ti,j − Ti,j − Ti,j − Ti,j Ti+1,j Ti,j+1 ∂T ∂T ∂T = , = , = , ∂ξ 4ξ ∂η 4η ∂ζ 4ζ n+1 n n n n n vi,j − vi,j vi+1,j − vi,j vi,j+1 − vi,j ∂v ∂v ∂v = , = , = , ∂ξ 4ξ ∂η 4η ∂ζ 4ζ n n n + vi,j − 2vi,j+1 vi,j+2 u2i+2,j − 2uni+1,j + uni,j ∂ 2v ∂ 2u = , = , ∂ζ 2 (4ζ)2 ∂ξ 2 (4ξ)2 n n n n n n + Ti,j + Ti,j − 2Ti,j+1 − 2Ti+1,j Ti,j+2 Ti+2,j ∂ 2T ∂ 2T = , = , ∂ζ 2 (4ζ)2 ∂ξ 2 (4ξ)2 n n n + vi,j − 2vi+1,j uni,j+2 − 2uni,j+1 + uni,j ∂ 2 v vi+2,j ∂ 2u = , = , ∂ζ 2 (4ζ)2 ∂ξ 2 (4ξ)2
(16)
Using Eq. (16), the PDE’s system reduced to the form n n − vi,j uni+1,j − uni,j vi,j+1 + = 0, (17) 4ξ 4ζ n n n un+1 uni+2,j − 2uni+1,j + uni,j ui+1,j − uni,j ui,j+1 − uni,j i,j − ui,j n n + ui,j + vi,j = 4η 4ξ 4ζ (4ξ)2 uni,j+2 − 2uni,j+1 + uni,j + − Reλuni,j , (18) (4ζ)2 n n n+1 n n n n n n vi,j − vi,j vi+1,j − vi,j vi,j+1 − vi,j vi+2,j − 2vi+1,j + vi,j n n + ui,j + vi,j = 4η 4ξ 4ζ (4ξ)2 2 n n n vi,j+2 − 2vi,j+1 + vi,j ρnf µf (ρβ)nf n n − Reλv − Gr + Ti,j , (19) i,j 2 (4ζ) ρf µnf (ρβ)f n n n+1 n n n Ti,j − Ti,j Ti+1,j − Ti,j Ti,j+1 − Ti,j 1 knf (ρcp )f νf n n + ui,j + vi,j = 4η 4ξ 4ζ P r kf (ρcp )nf νnf n n n n n n Ti+2,j − 2Ti+1,j + Ti,j Ti,j+2 − 2Ti,j+1 + Ti,j µnf (ρcp )f νnf + + Ec (4ξ)2 (4ζ)2 µf (ρcp )nf νf n 2 2 2 ui+1,j − uni,j uni,j+1 − uni,j uni,j+1 − uni,j uni,j+1 − uni,j 2 + + +2 , (20) 4ξ 4ζ 4ζ 4ζ
with
νf νf n = 0, = 0, = 1, = −α , ui,0 = Re (ξi+1 − ξi ), νnf νnf n n n Ti,0 = 1, uni,∞ = 0, vi,∞ = 0, Ti,∞ = 0. (21)
0 vi,j
u0i,j
0 Ti,j
n vi,0
6
5
Skin friction and Nusselt number reduced to th form n n uni,j+1 − uni,j vi+1,j − vi,j −2 νnf , C fx = + Re2 (ξi+1 − ξi )2 νf 4ξ 4ζ ζ=0 n n n n Ti+1,j − Ti,j Ti,j+1 − Ti,j . N ux = − ξi+1 − ξi + 4ξ 4ζ ζ=0
(22) (23)
Discussion
In this section we intend to analyze the impacts of flow variables on velocity, skin friction, temperature and Nusselt number. Flow variables include porosity parameter, suction parameter, Grashof number, Reynolds number and nanoparticle volume fraction. Effects of porosity parameter, Reynolds number, time, suction parameter and Grashof number on u(η, ξ, ζ) and v(η, ξ, ζ) are described in F igs. (2 − 9). Behavior of porosity parameter on u(η, ξ, ζ) is portrayed in F ig. 2. Here we noted that velocity decreases with porosity. For larger porosity parameter porous medium permeability is decreased. Therefore u(η, ξ, ζ) decreases. u(η, ξ, ζ) is dominant in SWCNT-Water over MWCNTWater. Variation of u(η, ξ, ζ) through Reynolds number is sketched in F ig. 3. u(η, ξ, ζ) increases for larger Reynolds number. Larger Reynolds number shows that inertial force is dominant over viscous force. Therefore u(η, ξ, ζ) increases. u(η, ξ, ζ) with time portrayed in F ig. 4. u(η, ξ, ζ) shows increasing behavior with time. Impact of suction parameter on v(η, ξ, ζ) is analyzed in F ig. 5. v(η, ξ, ζ) increases towards sheet for larger values of suction parameter. Porosity parameter behavior on v(η, ξ, ζ) is disclosed in F ig. 6. v(η, ξ, ζ) decreases with porosity parameter. Variation of v(η, ξ, ζ) through Reynolds number is described in F ig. 7. v(η, ξ, ζ) decreases for larger Reynolds number. Effect of Grashof number on v(η, ξ, ζ) is sketched in F ig. 8. v(η, ξ, ζ) increases towards sheet for larger Grashof number. variation of v(η, ξ, ζ) with time is shown in F ig. 9. v(η, ξ, ζ) increases with time. Variation of T (η, ξ, ζ) through nanoparticle volume fraction and time is portrayed in F igs. (10 − 11). Nanoparticle volume fraction impact on T (η, ξ, ζ) is analyzed in F ig. 10. T (η, ξ, ζ) increases for higher estimation of nanoparticle volume fraction. Also we noted that temperature is dominant in MWCNT-Water over SWCNT-Water. Variation of T (η, ξ, ζ) with time is shown in F ig. 11. Here we noted that T (η, ξ, ζ) decreases with time. Impacts of suction parameter, porosity parameter, Reynolds number and nanoparticles volume fraction on Cfx and N ux are portrayed in F igs. (12 − 15). Suction parameter behavior on Cfx is disclosed in F ig. 12. Cfx increases for larger suction parameter. Effect of porosity parameter on Cfx is sketched in F ig. 13. For higher estimation of porosity parameter Cfx increases. F ig. 14 depicts the behavior of Reynolds number on Cfx . For larger Reynolds number Cfx enhances. Variation of N ux through nanoparticles volume fraction is portrayed in F ig. 15. Larger values of nanoparticles volume fraction reduces the N ux .
7
6
Closing remarks
In this paper we studied mixed convection flow of SWCNT-Water and MWCNT-Water through porus medium over a stretchable permeable sheet. Major results are given below: ? (u(η, ξ, zeta)) reduces with porosity parameter and it increases with Reynolds number and time. ? (v(η, ξ, zeta)) towards sheet increases through suction parameter, Grashof number and time while it is reduces with porosity parameter and Reynolds number. ? T (η, ξ, ζ) enhances with nanoparticle volume fraction while it reduces with time. ? Cfx increases for larger values of suction parameter, porosity parameter and Reynolds number. ? N ux decreases for larger values of nanoparticle volume fraction. Acknowledges: This work was supported by the University of Science and Technology Beijing. Muhammad Ibrahim acknowledges the Office of China Postdoctoral Council (OCPC) for the postdoctoral international exchange program.
8
References [1] S. U. S. Choi, Enhancing thermal conductivity of fluids with nanoparticles developments and applications of non-Newtonian fluid flow, ASME FED 66 (1995) 99-105. [2] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transfer 46 (2003) 36393653. [3] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Soheil Soleimani, Effect of a magnetic field on natural convection in an inclined halfannulus enclosure filled with Cuwater nanofluid using CVFEM, Adv. Powder Technol. 24 (2013) 980991. [4] T. Hayat, S. Ahmad, M. I. Khan and A. Alsaedi, Modeling chemically reactive flow of Sutterby nanofluid by a rotating disk in presence of heat generation/absorption, Commun. Theor. Phys. 69 (2018) 569576. [5] R. Ellahi, M. Hassan and A. Zeeshan, Shape effects of nanosize particles in Cu − H2 O nanofluid on entropy generation, Int. J. Heat Mass Transf. 81 (2015) 449-456. [6] T. Hayat, M. I. Khana, T. A. Khand, M. I. Khan, S. Ahmad and A. Alsaedi, Entropy generation in Darcy-Forchheimer bidirectional flow of water-based carbon nanotubes with convective boundary conditions, J. Mol. Liq. 265 (2018) 629638. [7] Z. Li, I. Khan, A. Shafee, I. Tlili and T. Asifa, Energy transfer of JefferyHamel nanofluid flow between non-parallel walls using MaxwellGarnetts (MG) and Brinkman models, Energy Report. (2018) 393-399. [8] Hashim, A. Hamid and M. Khan, Multiple solutions for MHD transient flow of Williamson nanofluids with convective heat transport, J. Taiwan Inst. Chem. Eng. 103 (2019) 126-137. [9] D. V. Guzei, A. V. Minakov and V. Y. Rudyak, On efficiency of convective heat transfer of nanofluids in laminar flow regime, Int. J. Heat Mass Transf. 139 (2019) 180-192. [10] A. H. Rajabi, D. Toghraie and B. Mehmandoust, Numerical simulation of turbulent nanofluid flow in the narrow channel with a heated wall and a spherical dimple placed on it by using of single- phase and mixture- phase models, Int. Comm. Heat Mass Transf. 108 (2019) 104316, [11] T. Hayat, S. Ahmad, M. I. Khan and A. Alsaedi, Modeling and analyzing flow of third grade nanofluid due to rotating stretchable disk with chemical reaction and heat source, Phys. B: Cond. Matt. 537 (2018) 116-126. [12] H. E. Ahmed, B. H. Salman and A. S. Kerbeet, Heat transfer enhancement of turbulent forced nanofluid flow in a duct using triangular rib, Int. J. Heat Mass Transf. 134 (2019) 30-40.
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[13] T. Hayat, S. Ahmad, M. I. Khan and A. Alsaedi, Simulation of ferromagnetic nanomaterial flow of Maxwell fluid, Results Phys. 8 (2018) 34-40. [14] J. Prakash, E. P. Siva, D. Tripathi and M. Kothandapani, Nanofluids flow driven by peristaltic pumping in occurrence of magnetohydrodynamics and thermal radiation, Mat. Scie. Semicond. Process. 100 (2019) 290-300. [15] S. Ahmad, M. I. Khan, T. Hayat, M. I. Khan and A. Alsaedi, Entropy generation optimization and unsteady squeezing flow of viscous fluid with five different shapes of nanoparticles, Coll. Surf. A: Phys. Eng. Asp. 554 (2018) 197-210. [16] Z. H. Khan, O. D. Makinde, R. Ahmad and W. A. Khan, Numerical Study of Unsteady MHD Flow and Entropy Generation in a Rotating Permeable Channel with Slip and Hall Effects, Commun. Theor. Phys. 70 (2018) 641650. [17] M. T. Hoang and A. M. Nagy, Uniform asymptotic stability of a Logistic model with feedback control of fractional order and nonstandard finite difference schemes, Chaos Solit. Fract. 123 (2019) 24-34. [18] S. Ahmad, T. Hayat, A. Alsaedi, Z. H. Khan and M. W. A. Khan, Finite difference analysis of time-dependent viscous nanofluid flow between parallel plates, Commun. Theor. Phys. (In press). [19] O. D. Makinde, Z. H. Khan, R. Ahmad and W. A. Khan, Numerical study of unsteady hydromagnetic radiating fluid flow past a slippery stretching sheet embedded in a porous medium, Phys. Fluids, 30, (2018); doi: 10.1063/1.5046331. [20] O. Adekanye and T. Washington, Nonstandard finite difference scheme for a Tacoma Narrows Bridge model, Appl. Math. Mod. 62 (2018) 223-236. [21] H. F.Oztop and E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow, 29 (2008) 1326-1336. [22] N. S. Akbar, M. Raza and R. Ellahib, Influence of induced magneticfield and heatflux with the suspensionof carbon nanotubes for the peristalticflow in a permeable channel, J. Mag. Mag. Mat. 381 (2015) 405-415.
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Fig. 1: Flow diagram
Fig. 2: u(η, ξ, ζ) w.r.t λ.
Fig. 3: u(η, ξ, ζ) w.r.t Re.
11
Fig. 4: u(η, ξ, ζ) w.r.t η.
Fig. 5: v(η, ξ, ζ) w.r.t α.
Fig. 6: v(η, ξ, ζ) w.r.t λ.
12
Fig. 7: v(η, ξ, ζ) w.r.t Re.
Fig. 8: v(η, ξ, ζ) w.r.t Gr.
Fig. 9: v(η, ξ, ζ) w.r.t η.
13
Fig. 10: T (η, ξ, ζ) w.r.t φ.
Fig. 11: T (η, ξ, ζ) w.r.t η.
Fig. 12: Cfx w.r.t α.
14
Fig. 13: Cfx w.r.t λ.
Fig. 14: Cfx w.r.t Re.
Fig. 15: N ux w.r.t φ.
15
Conflict of interest The authors declared that they have no conflict of interest and the paper presents their own work which does not been infringe any third-party rights, especially authorship of any part of the article is an original contribution, not published before and not being under consideration for publication elsewhere. Regards Dr. M. I. Khan
16
Journal Pre-proof
Mathematical modeling and analysis of SWCNT-Water and MWCNT-Water flow over a stretchable sheet Muhammad Ibrahim, M. Ijaz Khan PII: DOI: Reference:
S0169-2607(19)31683-9 https://doi.org/10.1016/j.cmpb.2019.105222 COMM 105222
To appear in:
Computer Methods and Programs in Biomedicine
Received date: Revised date: Accepted date:
1 October 2019 13 November 2019 15 November 2019
Please cite this article as: Muhammad Ibrahim, M. Ijaz Khan, Mathematical modeling and analysis of SWCNT-Water and MWCNT-Water flow over a stretchable sheet, Computer Methods and Programs in Biomedicine (2019), doi: https://doi.org/10.1016/j.cmpb.2019.105222
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Highlights • Mixed convective of CNTs based flow of viscous material is addressed. • Both single and multi-walls carbon nanotubes is discussed. • Darcy’s law is used to characterize porous medium. • Viscous dissipation is used for heat transport in energy equation.
1
Mathematical modeling and analysis of SWCNT-Water and MWCNT-Water flow over a stretchable sheet Muhammad Ibrahim1,∗and M. Ijaz Khan2 1 School of Mathematics and Physics, University of Science and Technology Beijing, , Beijing 100083, China 2 Department of Mathematics, Quaid-I-Azam University 45320, Islamabad November 19, 2019
Abstract In this article we focused on the mixed convection flow of SWCNT-Water and MWCNT-Water over a stretchable permeable sheet. The nanofluid occupied porous medium. Darcy’s law is used to characterize porous medium. The impact of viscous dissipation is considered. Transformation procedure is adopted to transform the governing PDE’s system into dimensionless form. In order to solve the dimensionless PDE’s system we used numerical method known as Finite difference method. Effects of flow variables i.e porosity parameter, suction parameter, Grashof number and Reynolds number on velocity, skin friction, temperature and Nusselt number are described graphically. The obtained results shows that velocity is dominant in SWCNT-Water over MWCNT-Water. Temperature is dominant in MWCNT-Water over SWCNT-Water.
Keywords: Finite difference method; SWCNT-water and MWCNT-Water nanofluids; Stretching plate; Porous medium; Mixed convection; Viscous dissipation.
1
Introduction
In present time, nanofluids study gain the attention of researchers because they have higher thermal conductivity as compered to base fluids. The suspension of nanoparticles (1nm − 100nm) in the base fluid is called nanofluids. Base fluid includes polymeric solutions, oils, bio-fluids, water, refrigerants, lubricants, ethylene glycol etc. nanoparticles includes metallic particles (Cu, Al, T i, F e, Ag) and their oxides. Beside the above mentioned nanoparticles CN T (carbon nanotubes) is a nanoparticles. CN T is ∗
Corresponding author email: [email protected] (M. Ibrahim)
2
made by carbon. Various types of carbon nanotubes exists, but generally they categorized as SW CN T (single-walled carbon nanotube) or M W CN T (multi-walled carbon nanotube). SW CN T (single-walled carbon nanotube) has only one wall or layer and the other hand M W CN T (multi-walled carbon nanotube) are the collection of nested tubes of continuously enhances diameter. Due to the thermophysical properties (physical, optical, thermal and electrical) of CN T materials have various applications in industrial processes at macro as well as micro level. These applications includes electronics, biosensor, sensors, ultra-capacitors, conductive plastics, nuclear reactors, gas storage, flatpanel displays, technical textiles, medical equipments, solar collection, catalyst supports, nonporous filters and coatings of all sorts. First time Choi [1] introduce the concept of nanofluid. Two-dimensional flow of nanofluid in a enclosure is investigated by Khanfer et al. [2]. Effects of natural convection and magnetic field on Cu−Water nanofluid flow is studied by Sheikholeslami et al. [3]. Flow of Sutterby nanofluid due to a rotating disk with heat absorption/generation is expressed by Hayat et al. [4]. Entropy generation of Cu − H2 O nanoliquid is examined by Ellahi et al. [5]. Entropy generation of CN T s−H2 O nanofluid flow through porous medium is explored by Hayat et al. [6]. For further studies the readers are refereed to see [7 − 15]. Here we focused mixed convection flow of SWCNT-Water and MWCNT-water nanofluid through porous medium. The effect of viscous dissipation accounted. The governing PDE’s system is transform to dimensionless form by appropriate transformation. Numerical method known as Finite difference method [16 − 20] is used for numerical solution. The effect of flow variables porosity parameter, suction parameter, Grashof number, Reynolds number and nanoparticles volume fraction on velocity, surface drag force, temperature and temperature gradient are discussed graphically.
2
Mathematical modeling
Mixed convection flow of SWCNT-Water and MWCNT-Water nanofluids due to a permeable stretchable sheet is modeled in this section. The nanofluid fulfill the porous medium. For porous medium we used Darcy’s law. Energy equation contained the viscous dissipation effect. For flow diagram see F ig. 1. Mathematical expressions are given by: ∂e u ∂e v + = 0, ∂x ∂y 2 ∂e u ∂e u ∂e u ∂ u e ∂ 2u e νnf +u e + ve = νnf + − u e, ∂t ∂x ∂y ∂x2 ∂y 2 δ 2 νnf ∂e v ∂e v ∂e v ∂ ve ∂ 2 ve +u e + ve = νnf + − ve − gβnf (Te − T∞ ), ∂t ∂x ∂y ∂x2 ∂y 2 δ 2e ∂ Te ∂ Te ∂ Te knf ∂ T ∂ 2 Te µnf +u e + ve = + + 2 2 ∂t ∂x ∂y (ρcp )nf ∂x ∂y (ρcp )nf 2 2 2 ∂e u ∂e u ∂e v ∂e v 2 + + +2 , ∂x ∂y ∂x ∂y 3
(1) (2) (3)
(4)
Constraints are given by ve = 0, Te = Tw u e = 0, at t = 0, ve = −V0 , u e = Uw = ax, Te = Tw , at y = 0, ve = 0, u e = 0, Te = T∞ , as y → ∞.
(5)
In which t represents time, x, y space coordinates, u e, ve velocity components, knf nanofluid thermal conductivity, νnf nanofluid kinematic viscosity, δ porous medium permeability, ρnf nanofluid density, (cp )nf specific heat of nanofluid, Te temperature, µnf nanofluid dynamic viscosity, g gravity, Uw stretching velocity, V0 suction velocity, Tw sheet temperature, βnf nanofluid thermal expension coefficient, and T∞ ambient temperature. Nanofluid thermophysical properties are given by [21] : knf kCN T kCN T + kf kf kCN T + kf = 1 − φ + 2kf + 2φ In 1 − φ + 2φ In , kf kCN T − kf 2kf kCN T − kf 2kf (ρcp )nf ρnf = (1 − φ) + φ(ρcp )CN T (ρcp )f , = (1 − φ) + φρCN T ρf , (ρcp )f ρf µnf (ρβ)nf = (1 − φ) + φ(ρβ)CN T (ρβ)f , = 1 (1 − φ)2.5 , (6) (ρβ)f µf Note that νf stand for water kinematic viscosity, νCN T Carbon nanotube kinematic viscosity, βf water thermal expansion coefficient, βCN T Carbon nanotube thermal expansion coefficient, ρf density of water, ρCN T density of Carbon nanotube, kf thermal conductivity of water, kCN T thermal conductivity of Carbon nanotube, (cp )f water specific heat, (cp )CN T Carbon nanotube specific heat and φ nanoparetical volume fraction.
Table 1. Numerically estimated values of ρ, β, k and cp for H2 O and Carbon nanotube [22]. k(W/mk) ρ(kg/m3 ) 0.613 997.1 6600 2600 3000 1600
H2 O (Water) SWCNT MWCNT
β × 105 21 2.6 2.8
cp (J/kgk) 4179 0425 0769
Considering y L L , u(η, ξ, ζ) = u e, v(η, ξ, ζ) = ve, L νnf νnf νnf x Te − T∞ T (η, ξ, ζ) = , η = 2 t, ξ = , Tw − T∞ L L ζ=
4
(7)
Implementing Eg. (7), mathematical expressions of flow becomes ∂u ∂v + = 0, ∂ξ ∂ζ ∂u ∂u ∂u ∂ 2u ∂ 2u +u +v = 2 + 2 − Reλu, ∂η ∂ξ ∂ζ ∂ξ ∂ζ 2 ∂v ∂v ∂ 2v ∂ 2v (ρβ)nf ρnf µf ∂v +u +v = 2 + 2 − Reλv − Gr T, ∂η ∂ξ ∂ζ ∂ξ ∂ζ µnf (ρβ)f ρf 2 ∂T ∂T ∂T 1 knf (ρcp )f νf ∂ T ∂ 2T +u +v = + ∂η ∂ξ ∂ζ P r kf (ρcp )nf νnf ∂ξ 2 ∂ζ 2 2 2 2 (ρcp )f µnf νnf ∂u ∂u ∂v ∂v +Ec + 2 + +2 , (ρcp )nf µf νf ∂ξ ∂ζ ∂ξ ∂ζ
T = 1, u = 0, v = 0, at η = 0, νf νf u = Ra ξ, T = 1, v = −α , νnf νnf v = 0, u = 0, T = 0, at ζ → ∞.
(8) (9) (10)
(11)
at ζ = 0, (12)
νf LV0 Here λ = aδ represents porosity parameter, α = νf suction parameter, gβf (Tw −T∞ )L3 (µcp )f aL2 Gr = Grashof number, P r = kf Prandtl number, Re = νf νf2 νf2 Reynolds number and Ec = L2 (cp )f (Th −Tc ) Eckert number.
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Quantities for engineering interest
Mathematically Cfx (skin friction) and N ux (Nusselt number) are given by Cf x =
xqw −2τw , , N ux = , 2 ρnf Uw knf (Tw − T∞ )
in which τw and qw are defined as ∂e v ∂e u τw = τyx = µnf + ∂x ∂y y=0
, y=0
e ∂T ∂ Te qw = −knf + . ∂x ∂y y=0
In dimensionless form skin friction and Nusselt number can be written as 2 −2 νnf ∂v ∂u ∂T ∂T Cf x = + , N ux = −ξ + . Re2 ξ 2 νf ∂ξ ∂η ζ=0 ∂ξ ∂η ζ=0 5
(13)
(14)
(15)
4
Finite difference scheme for solution
In order to solve the PDE’s system we chose Finite difference method. For given PDE’s system FD toolkit can be defined as: n un+1 uni+1,j − uni,j ∂u uni,j+1 − uni,j ∂u ∂u i,j − ui,j = , = , = , ∂ξ 4ξ ∂η 4η ∂ζ 4ζ n+1 n n n n n Ti,j − Ti,j − Ti,j − Ti,j Ti+1,j Ti,j+1 ∂T ∂T ∂T = , = , = , ∂ξ 4ξ ∂η 4η ∂ζ 4ζ n+1 n n n n n vi,j − vi,j vi+1,j − vi,j vi,j+1 − vi,j ∂v ∂v ∂v = , = , = , ∂ξ 4ξ ∂η 4η ∂ζ 4ζ n n n + vi,j − 2vi,j+1 vi,j+2 u2i+2,j − 2uni+1,j + uni,j ∂ 2v ∂ 2u = , = , ∂ζ 2 (4ζ)2 ∂ξ 2 (4ξ)2 n n n n n n + Ti,j + Ti,j − 2Ti,j+1 − 2Ti+1,j Ti,j+2 Ti+2,j ∂ 2T ∂ 2T = , = , ∂ζ 2 (4ζ)2 ∂ξ 2 (4ξ)2 n n n + vi,j − 2vi+1,j uni,j+2 − 2uni,j+1 + uni,j ∂ 2 v vi+2,j ∂ 2u = , = , ∂ζ 2 (4ζ)2 ∂ξ 2 (4ξ)2
(16)
Using Eq. (16), the PDE’s system reduced to the form n n − vi,j uni+1,j − uni,j vi,j+1 + = 0, (17) 4ξ 4ζ n n n un+1 uni+2,j − 2uni+1,j + uni,j ui+1,j − uni,j ui,j+1 − uni,j i,j − ui,j n n + ui,j + vi,j = 4η 4ξ 4ζ (4ξ)2 uni,j+2 − 2uni,j+1 + uni,j + − Reλuni,j , (18) (4ζ)2 n n n+1 n n n n n n vi,j − vi,j vi+1,j − vi,j vi,j+1 − vi,j vi+2,j − 2vi+1,j + vi,j n n + ui,j + vi,j = 4η 4ξ 4ζ (4ξ)2 2 n n n vi,j+2 − 2vi,j+1 + vi,j ρnf µf (ρβ)nf n n − Reλv − Gr + Ti,j , (19) i,j 2 (4ζ) ρf µnf (ρβ)f n n n+1 n n n Ti,j − Ti,j Ti+1,j − Ti,j Ti,j+1 − Ti,j 1 knf (ρcp )f νf n n + ui,j + vi,j = 4η 4ξ 4ζ P r kf (ρcp )nf νnf n n n n n n Ti+2,j − 2Ti+1,j + Ti,j Ti,j+2 − 2Ti,j+1 + Ti,j µnf (ρcp )f νnf + + Ec (4ξ)2 (4ζ)2 µf (ρcp )nf νf n 2 2 2 ui+1,j − uni,j uni,j+1 − uni,j uni,j+1 − uni,j uni,j+1 − uni,j 2 + + +2 , (20) 4ξ 4ζ 4ζ 4ζ
with
νf νf n = 0, = 0, = 1, = −α , ui,0 = Re (ξi+1 − ξi ), νnf νnf n n n Ti,0 = 1, uni,∞ = 0, vi,∞ = 0, Ti,∞ = 0. (21)
0 vi,j
u0i,j
0 Ti,j
n vi,0
6
5
Skin friction and Nusselt number reduced to th form n n uni,j+1 − uni,j vi+1,j − vi,j −2 νnf , C fx = + Re2 (ξi+1 − ξi )2 νf 4ξ 4ζ ζ=0 n n n n Ti+1,j − Ti,j Ti,j+1 − Ti,j . N ux = − ξi+1 − ξi + 4ξ 4ζ ζ=0
(22) (23)
Discussion
In this section we intend to analyze the impacts of flow variables on velocity, skin friction, temperature and Nusselt number. Flow variables include porosity parameter, suction parameter, Grashof number, Reynolds number and nanoparticle volume fraction. Effects of porosity parameter, Reynolds number, time, suction parameter and Grashof number on u(η, ξ, ζ) and v(η, ξ, ζ) are described in F igs. (2 − 9). Behavior of porosity parameter on u(η, ξ, ζ) is portrayed in F ig. 2. Here we noted that velocity decreases with porosity. For larger porosity parameter porous medium permeability is decreased. Therefore u(η, ξ, ζ) decreases. u(η, ξ, ζ) is dominant in SWCNT-Water over MWCNTWater. Variation of u(η, ξ, ζ) through Reynolds number is sketched in F ig. 3. u(η, ξ, ζ) increases for larger Reynolds number. Larger Reynolds number shows that inertial force is dominant over viscous force. Therefore u(η, ξ, ζ) increases. u(η, ξ, ζ) with time portrayed in F ig. 4. u(η, ξ, ζ) shows increasing behavior with time. Impact of suction parameter on v(η, ξ, ζ) is analyzed in F ig. 5. v(η, ξ, ζ) increases towards sheet for larger values of suction parameter. Porosity parameter behavior on v(η, ξ, ζ) is disclosed in F ig. 6. v(η, ξ, ζ) decreases with porosity parameter. Variation of v(η, ξ, ζ) through Reynolds number is described in F ig. 7. v(η, ξ, ζ) decreases for larger Reynolds number. Effect of Grashof number on v(η, ξ, ζ) is sketched in F ig. 8. v(η, ξ, ζ) increases towards sheet for larger Grashof number. variation of v(η, ξ, ζ) with time is shown in F ig. 9. v(η, ξ, ζ) increases with time. Variation of T (η, ξ, ζ) through nanoparticle volume fraction and time is portrayed in F igs. (10 − 11). Nanoparticle volume fraction impact on T (η, ξ, ζ) is analyzed in F ig. 10. T (η, ξ, ζ) increases for higher estimation of nanoparticle volume fraction. Also we noted that temperature is dominant in MWCNT-Water over SWCNT-Water. Variation of T (η, ξ, ζ) with time is shown in F ig. 11. Here we noted that T (η, ξ, ζ) decreases with time. Impacts of suction parameter, porosity parameter, Reynolds number and nanoparticles volume fraction on Cfx and N ux are portrayed in F igs. (12 − 15). Suction parameter behavior on Cfx is disclosed in F ig. 12. Cfx increases for larger suction parameter. Effect of porosity parameter on Cfx is sketched in F ig. 13. For higher estimation of porosity parameter Cfx increases. F ig. 14 depicts the behavior of Reynolds number on Cfx . For larger Reynolds number Cfx enhances. Variation of N ux through nanoparticles volume fraction is portrayed in F ig. 15. Larger values of nanoparticles volume fraction reduces the N ux .
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6
Closing remarks
In this paper we studied mixed convection flow of SWCNT-Water and MWCNT-Water through porus medium over a stretchable permeable sheet. Major results are given below: ? (u(η, ξ, zeta)) reduces with porosity parameter and it increases with Reynolds number and time. ? (v(η, ξ, zeta)) towards sheet increases through suction parameter, Grashof number and time while it is reduces with porosity parameter and Reynolds number. ? T (η, ξ, ζ) enhances with nanoparticle volume fraction while it reduces with time. ? Cfx increases for larger values of suction parameter, porosity parameter and Reynolds number. ? N ux decreases for larger values of nanoparticle volume fraction. Acknowledges: This work was supported by the University of Science and Technology Beijing. Muhammad Ibrahim acknowledges the Office of China Postdoctoral Council (OCPC) for the postdoctoral international exchange program.
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[13] T. Hayat, S. Ahmad, M. I. Khan and A. Alsaedi, Simulation of ferromagnetic nanomaterial flow of Maxwell fluid, Results Phys. 8 (2018) 34-40. [14] J. Prakash, E. P. Siva, D. Tripathi and M. Kothandapani, Nanofluids flow driven by peristaltic pumping in occurrence of magnetohydrodynamics and thermal radiation, Mat. Scie. Semicond. Process. 100 (2019) 290-300. [15] S. Ahmad, M. I. Khan, T. Hayat, M. I. Khan and A. Alsaedi, Entropy generation optimization and unsteady squeezing flow of viscous fluid with five different shapes of nanoparticles, Coll. Surf. A: Phys. Eng. Asp. 554 (2018) 197-210. [16] Z. H. Khan, O. D. Makinde, R. Ahmad and W. A. Khan, Numerical Study of Unsteady MHD Flow and Entropy Generation in a Rotating Permeable Channel with Slip and Hall Effects, Commun. Theor. Phys. 70 (2018) 641650. [17] M. T. Hoang and A. M. Nagy, Uniform asymptotic stability of a Logistic model with feedback control of fractional order and nonstandard finite difference schemes, Chaos Solit. Fract. 123 (2019) 24-34. [18] S. Ahmad, T. Hayat, A. Alsaedi, Z. H. Khan and M. W. A. Khan, Finite difference analysis of time-dependent viscous nanofluid flow between parallel plates, Commun. Theor. Phys. (In press). [19] O. D. Makinde, Z. H. Khan, R. Ahmad and W. A. Khan, Numerical study of unsteady hydromagnetic radiating fluid flow past a slippery stretching sheet embedded in a porous medium, Phys. Fluids, 30, (2018); doi: 10.1063/1.5046331. [20] O. Adekanye and T. Washington, Nonstandard finite difference scheme for a Tacoma Narrows Bridge model, Appl. Math. Mod. 62 (2018) 223-236. [21] H. F.Oztop and E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow, 29 (2008) 1326-1336. [22] N. S. Akbar, M. Raza and R. Ellahib, Influence of induced magneticfield and heatflux with the suspensionof carbon nanotubes for the peristalticflow in a permeable channel, J. Mag. Mag. Mat. 381 (2015) 405-415.
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Fig. 1: Flow diagram
Fig. 2: u(η, ξ, ζ) w.r.t λ.
Fig. 3: u(η, ξ, ζ) w.r.t Re.
11
Fig. 4: u(η, ξ, ζ) w.r.t η.
Fig. 5: v(η, ξ, ζ) w.r.t α.
Fig. 6: v(η, ξ, ζ) w.r.t λ.
12
Fig. 7: v(η, ξ, ζ) w.r.t Re.
Fig. 8: v(η, ξ, ζ) w.r.t Gr.
Fig. 9: v(η, ξ, ζ) w.r.t η.
13
Fig. 10: T (η, ξ, ζ) w.r.t φ.
Fig. 11: T (η, ξ, ζ) w.r.t η.
Fig. 12: Cfx w.r.t α.
14
Fig. 13: Cfx w.r.t λ.
Fig. 14: Cfx w.r.t Re.
Fig. 15: N ux w.r.t φ.
15
Conflict of interest The authors declared that they have no conflict of interest and the paper presents their own work which does not been infringe any third-party rights, especially authorship of any part of the article is an original contribution, not published before and not being under consideration for publication elsewhere. Regards Dr. M. I. Khan
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