Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 7-Day Trial for You or Your Team.

Learn More →

Homogeneous–heterogeneous reaction mechanism on MHD carbon nanotube flow over a stretching cylinder with prescribed heat flux using differential transform method

Homogeneous–heterogeneous reaction mechanism on MHD carbon nanotube flow over a stretching... Hydromagnetic nanofluid flow through an incompressible stretching cylinder accompanying with homogeneous–heterogeneous chemical reaction has been executed in current literature. SWCNTs (single-walled carbon nanotubes) and MWCNTs (multiwalled carbon nanotubes) as nanoparticles in appearance of prescribed heat flux are accounted here. Leading equations of the assumed model have been normalized through similarity practice and succeeding equations resolved numerically by spending RK-4 shooting practice and analytically by engaging differential transform method. The impulse of promising flow constraints on the flow characteristic is finalized precisely through graphs and charts. We perceived that velocity outlines and temperature transmission are advanced in MWCNT than SWCNT in every case. Keywords: carbon nanotubes; nanofluid; stretching cylinder; prescribed heat flux; homogeneous–heterogeneous chemical reaction; DTM 1. Introduction To utilize solar energy, researchers, scientists, and engineers are devoted to develop energy resources and the energy technologies due to significant dependence on it of human society. It is a well-known reality that improvement in thermal characteristic can be made by adding little amount of nanoparticles having high thermal characteristic. However, in recent times nanofluid (Bhatti, Abbas, & Rashidi, 2017; Daniel, Aziz, Ismail, & Salah, 2018; Dhlamini, Kameswaran, Sibanda, Motsa, & Mondal, 2019; Mondal, Almakki, & Sibanda, 2019), which is a new kind of fluid categorized due to solid–liquid arrangement in metal or nonmetal nanoparticle suspen- sion; originated by Choi (1995), to highten thermal conductivity of the fluid. Carbon nanotubes basically are the cylinder of single or multiple sheets of graphene. Centered on sheets of the graphene, carbon nanotubes are distinguished into two types viz. single and multiple-walled carbon nanotubes (SWCNTs and MWCNTs). CNT is generally used in electrodes, anodes, catalyst, and various Received: 24 June 2019; Revised: 9 September 2019; Accepted: 1 December 2019 The Author(s) 2020. Published by Oxford University Press on behalf of the Society for Computational Design and Engineering. This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact [email protected] 337 Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 338 MHD Carbon nanotube flow over a stretching cylinder using DTM medical and environmental applications due to its remarkable mechanical and electrical properties as well as thermal conductiv- ity. CNTs have wide applications in aerospace materials because it can also progress the properties of the progressive aerospace materials. A material containing CNT is highly electrically conductive, which proposes that they might defend aircraft from the light- ning strikes. Shah, Bonyah, Islam, and Gul (2019) explored impression of nonlinear radiation on MHD rotating flow of CNTs through stretching sheet and confirmed that heat-flux values enrich with physical quantities. CNT effects in a nanofluid flow with variable fluid properties were investigated by Shahzadi, Nadeem, and Rabiei ( 2017). Shah, Dawar, Islam, Khan, and Ching (2018) scrutinized Darcy–Forchheimer CNT flow in a rotating frame and convey that due to amplification in inertia coefficient transverse velocity dis- tribution has dual characteristic. Inspiration of velocity slip on CNT flow through a stretched rotating disk was inspected by Nasir, Shah, Islam, Khan, and Khan (2019a) and reported that velocity field reduced for larger values of velocity slip factor. Hussain, Haq, Khan, and Nadeem (2016) explored nanofluid flow of CNTs in a rotating channel. Darcy–Forchheimer unsteady thin film flow of CNTs through a stretching sheet was observed by Nasir, Shah, Islam, Bonyah, and Gul (2019b) and recounted that for larger values of poros- ity factor velocity field declines. Ahmed, Qahtani, Nadeem, and Saleem ( 2019) explored MHD CNT flow through a curved surface with nonuniform thermo-physical properties and confirmed that heat transmission rate heightened more in SWCNTs than MWCNTs. CNT flow on 3D rotating system was considered by Hayat, Nadeem, and Khan ( 2019) and they confirmed that augmenting temperature exponent declines temperature field. Ahmed and Nadeem ( 2019) studied CNT flow through squeezed channel and conclude that squeezed factor seemed to reduce temperature field. Nadeem, Khan, and Khan ( 2019a) analyzed CNT stagnation-point flow through an oscillatory sheet and acquired dual solution. Nadeem, Hayat, and Khan (2019b) explored 3D rotating CNT flow over heated surface and convey that heat transmission enriches with convective factor. Researchers recently have been attentive on the surface-driven flow to analyze flow features due to its prominence in manufac- turing progression such as plastic sheets, paper production, polymer processing, glass blowing, etc. Steady-flow exterior of cylinder by considering an ambient fluid at rest was coined by Wang ( 1998) and received exact solution also determined heat transmission. Ishak, Nazar, and Pop (2008) explored flow and heat transmission outside of stretching cylinder numerically by employing Keller-box method and recognized that injection is adequate to lessen skin friction. Hussain, Javed, and Nadeem (2019) analyzed numerical solution of Casson flow among concentric cylinders and verified that nanofluids are superior coolant than their base fluids. Tamoor, Waqas, Khan, Alsaedi, and Hayat (2017) inspected MHD Casson fluid flow through cylinder and fix the convergence area for the at- tained solutions. Sheikholeslami (2015) scrutinized the encouragement of unchanging suction happening on nanofluid flow through cylinder and compute effective thermal conductivity in addition viscosity by KKL scheme. Kardri, Bachok, Arifin, and Ali ( 2017) stud- ied flow and heat transfer features through cylinder and specify that dual solution be existent for stretching cylinder. Nadeem, Abbas, and Khan (2018) scrutinized the characteristics of 3D stagnation-point hybrid-nanofluid flow through circular cylinder and confirmed that heat transmission rate is higher in hybrid nanofluid than nanofluid. Heat flux is defined as the rate of heat energy transmission through a specified surface. Many researchers (Elbashbeshy, 1998; Bachok & Ishak, 2010; Munawar, Mehmood, & Ali, 2012) demonstrate flow and heat transmission over stretching cylinder using pre- scribed heat flux. Alavi, Hussanan, Kasim, Rosli, and Salleh ( 2017) considered prescribed heat flux to capture the characteristic of MHD flow over an exponentially stretching sheet. Heat transmission analysis using prescribed heat flux through stretching sheet was considered by Majeed, Zeeshan & Ellahi (2016). In numerous industrial manufacturing, biomedical invention, combustion, paint manufacturing, food processing, and metal ab- straction from naturally found ore, homogeneous–heterogeneous chemical reaction occurs. Bachok, Ishak, and Pop (2011) scrutinized stagnation flow in appearance of homogeneous–heterogeneous chemical reaction over an expanding surface by considering identi- cal diffusion rate of reactant and autocatalyst. Rana, Mehmood, and Akbar (2016) analyzed oblique flow of the Casson by accounting homogeneous–heterogeneous reactions. Muhammad, Nadeem, and Mustafa (2019) analyzed Ferro- hydrodynamic boundary-layer flow in the presence of homogeneous–heterogeneous reactions. Recently, nanofluid flow with homogeneous–heterogeneous reac- tions was studied by Kumar, Sood, Sheikholeslami, and Shehzad (2017). Nearly all mathematical modeling encloses linear or nonlinear differential equations and to resolve those equations perfectly, ap- proximately, or numerically several investigators are focused to hire numerous powerful analytic, semianalytic tools along with per- manent label of convergence. Specific techniques are differential transform method (DTM; Ayaz, 2013), modified DTM (Erfani, Rashidi, &Parsa, 2010; Rashidi, Laraqi, & Parsa, 2011a; Rashidi, Hayat, Keimanesh, & Yousefian, 2013), multistep DTM (Rashidi, Chamkha, & Keimanesh, 2011b), HAM (Sheikholeslami, Ellahi, Ashorynejad, & Hayat, 2014), ADM (Sheikholeslami, Ganji, & Ashorynejad, 2013), HPM (Giri, Das, & Kundu, 2018), VPM (Noor, Mohyud-Din, & Waheed 2008), etc. DTM is the iterative technique to acquire analytical series solution in the form of polynomial. In evaluation with other methods, solution achieved by DTM is of less size of polynomial and takes less CPU time for the same accuracy. As DTM is away from the any kind of linearization, discretization, and restricting as- sumption, one can directly apply this method without any type of affected error. Series solution achieved by DTM is divergent with the boundary conditions at infinity. DTM with Pad e´ approximation by Rashidi (2009) delivers an operative tool for solving boundary value differential equations on infinite/semiinfinite domains. Zhou ( 1986) first initiated DTM and in two dimensions it was initiated by Jang, Chen, and Liu (2001). Then several researchers (Rashidi & Erfani, 2011;Rashidi,Beg, ´ Asadi, & Rastegari, 2012a; Rashidi, Rahimzadeh, Ferdows, Uddin, & Beg, 2012b; Sheikholeslami & Ganji, 2015; Sheikholeslami, Rashidi, Alsaad, & Rokni, 2016) used DTM to resolve various problems. By receiving motivation from the above discussion in extant unit, we desire to explore hydromagnetic nanofluid flow through an incompressible stretching cylinder. SWCNTs and MWCNTs as nanoparticles in appearance of prescribed heat flux are accounted here. It has significant applications in electrodes, anodes, microelectronics, transistors, plastic sheets, paper production, polymer pro- cessing, glass blowing, combustion, paint industry, food processing, aerospace materials, various medical, industrial, and biomedical production, and environmental studies. We implement a similarity tactic to convert the primary Partial Differential Equations (PDEs) into Ordinary Differential Equations (ODEs) and resolved numerically by spending RK-4 shooting practice and analytically by engaging DTM to perform flow analysis. Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 339 Table 1: Physical properties of the base fluid and nanoparticles. Base fluid Nanoparticles Water SWCNT MWCNT ρ (kg/m ) 997.1 2600 1600 C (J/kg K) 4179 425 796 κ (W/mK) 0.613 6600 3000 Figure 1: Physical model and coordinate system. 2. Mathematical Formulation In this section, a steady-state hydromagnetic incompressible nanofluid flow through a stretching cylinder of radius R accompanying with homogeneous–heterogeneous chemical reaction is studied. SWCNT and MWCNT nanoparticles and water as a base fluid are considered here and their thermophysical properties are given in the Table 1. We accomplish the cylindrical co-ordinate system by fixing the x-axis along the axis of cylinder and r-axis normal to it as portrayed in Fig. 1. A uniform magnetic field of intensity B has been employed in a radial direction by assuming enough small magnetic Reynolds number to neglect the encouraged magnetic field. Stream carries two different chemically reacting components A and B and between them homogeneous and heterogeneous chemical reaction takes place. The isothermal cubic order homogeneous autocatalysis occurred as A + 2B → 3B,where Bplays the role of an autocatalyst and the rate of the phenomena is k a b . On the other side, at surface the occurrence of isothermal first- 1 1 order catalysis can be presented as A + B → 2B with rate of reaction k a . The concentrations of the chemical species A and B are s 1 a and b , respectively, and k and k are reaction rate factors accordingly. We support our study with the hypothesis that there is no 1 1 1 s added chemically responsive species, and all kinds of body forces along with viscous dissipation and joule heating are discounted. The cylindrical sheet is being stretched with a velocity U = U ( ) toward axial direction and the cylindrical surface is preserved at w 0 prescribed heat flux q = T ( )inwhich U and T are the constants and l signifies the characteristic length. w 0 0 0 With these considerations, governing equations are as follows (2017, 2016): ∂ ∂ (ru) + (rv) =0(1) ∂x ∂r ∂u ∂u μ 1 ∂ ∂u σ nf nf u + v = r − B u (2) ∂x ∂r ρ r ∂r ∂r ρ nf nf ∂T ∂T μ 1 ∂ ∂T nf u + v = r (3) ∂x ∂r (ρc ) r ∂r ∂r nf ∂a ∂a 1 ∂ ∂a 1 1 1 u + v = D r − k a b (4) A 1 1 ∂x ∂r r ∂r ∂r ∂b ∂b 1 ∂ ∂b 1 1 1 u + v = D r + k a b (5) B 1 1 1 ∂x ∂r r ∂r ∂r Boundary conditions are ∂T ∂a ∂b 1 1 ⎬ u = U ,v = 0,κ =−q , D = k a , D =−k a at r = R w f w A s 1 B s 1 (6) ∂r ∂r ∂r u = 0, T → T , a → a , b →0as r →∞ ∞ 1 0 1 Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 340 MHD Carbon nanotube flow over a stretching cylinder using DTM Here, (u,v) are the velocity components toward the (x, r)directionsand T is the temperature. D and D stand for the diffusion rate A B of chemical species A and B, respectively. Nanofluid density ( ρ ), effective heat capacity (ρc ) , dynamic viscosity (μ ), and thermal nf p nf nf conductivity (κ ) are given by (2018) nf ρ = (1 − φ) ρ + φρ , (ρc ) = (1 − φ) (ρc ) + φ(ρc ) nf f CNT p p p nf f CNT⎪ κ +κ κ CNT f CNT (1 − φ) + 2φ ln (7) μ κ f nf κ −κ 2κ CNT f f μ = , = ⎪ nf 2.5 κ κ +κ f CNT f ⎭ (1 − φ) f (1 − φ) + 2φ ln κ −κ 2κ CNT f f The subscripts CNT and f are used for separate values for nanoparticles and base fluid, respectively, and φ is the nanoparticle con- centration. We define the following dimensionless similarity transformations: 2 2 ⎪ r − R U ⎪ 2 ⎪ 2 ⎪ η = ,ψ = (v U x) Rf (η) f w ⎪ 2R v x (8) 1 ⎪ q v x a b f ⎪ w 2 1 1 T = T + θ (η) , g (η) = ,χ (η) = ⎭ κ U a a f w 0 0 1 ∂ψ 1 ∂ψ where f (η)and θ(η) are the dimensionless functions and steam function ψ satisfies equation ( 1), also defined as u = ,v =− . r ∂r r ∂x Using the substitution mentioned in equation (8) into the equations (2)–(5), we received subsequent similarity equations 2.5 CNT (1 + 2γη) f + 2γ f + (1 − φ) (1 − φ) + φ ff − f − Mf =0(9) nf 1 + 2γη θ + 2γθ + Pr f θ − f θ = 0 (10) [( ) ] ( ) (ρc ) CNT (1 − φ) + φ (ρc ) [(1 + 2γη) g + 2γ g ] + fg − λgχ = 0 (11) Sc [(1 + 2γη) χ + 2γχ ] + f χ + λgχ = 0 (12) Sc Accordingly, boundary conditions (6) become f (0) = 0, f (0) = 1,θ (0) =−1, g (0) = K g(0),δχ (0) =−K g(0) s s (13) f η → 0,θ η → 0, g η → 1,χ η →0as η →∞ ( ) ( ) ( ) ( ) 1 μ f (c p) v f l f σnf B0 l v f where γ = ( ) is the curvature parameter, Pr = is the Prandtl number, M = is the revised magnetic parameter, Sc = U l κ ρ U D 0 f f 0 A k a l v l 1 k f 0 s is the Schmidt number, λ = and K = are, respectively, the homogeneous and heterogeneous chemical reaction factors, U D U 0 A 0 and δ = is the ratio of mass diffusion coefficient. We assume that the diffusion rates of two chemical species A and B are of a comparable size. Further, we consider that D = D , A B i.e. δ = 1; so we consider from the assumption that g(η) + χ(η) = 1. Therefore, (11)and (12) converge to (1 + 2γη) g + 2γ g + Sc f g − λScg(1 − g) = 0 (14) with the reformed boundary conditions f (0) = 0, f (0) = 1,θ (0) =−1, g (0) = K g(0) (15) f (η) → 0,θ (η) → 0, g (η) →1as η →∞ For engineering and practical purposes, we are alert in the following factors: skin friction coefficient and Nusselt number. These are defined as 2μ ∂u nf C = f ⎪ 2 ⎬ ρ u ∂r nf w r =R (16) x ∂T Nu =− (T − T ) ∂r w ∞ r =R By using (8), we obtained reduced skin friction coefficient and reduced Nusselt number as C = Re .C = f (0)⎪ fr f ⎪ 2.5 (1 − φ) (17) 1 ⎪ κ 1 − nf ⎪ 2 ⎪ Nu = Re .Nu = ⎭ r x x κ θ (0) U x Re = is local Reynolds number. f Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 341 Table 2: Differential transform of some functions [34]. Original function Transformed function x(t) = αg(t) ± βh(t) X[k] = αG(k) ± β H(k) d f (t) (k+m)! x(t) = X[k] = F [k + m] dt k! x(t) = g(t)h(t) X[k] = G[m]H[k − m] m=0 1, if k = m x(t) = t X[k] = δ(k − m) = 0, if k = m Table 3: Variation of θ(0) for different values of Pr when M = 0,φ = 0,γ = 0 and in the absence of a chemical reaction. Pr Elbashbeshy (1998) Bachok & Ishak (2010) Present Result 0.72 1.2253 1.2367 1.236582 1.00 1.0000 1.0000 0.999999 6.70 — 0.3333 0.333303 10.0 0.2688 0.2688 0.268768 3. Numerical Experiment 3.1 DTM Let f (t) be analytic in a domain T and t ∈ T. Then, DTM of f (t)is 1 d f (t) F (k) = (18) k! dt t=t where F (k) is termed as T-function or a modestly transformed form of f (t). Now, inversion differential transform of F (k)isdefinedas f (t) = (t − t ) F (k) (19) k=0 From (19) with the help of (18), we have 1 d f (t) f (t) = (t − t ) (20) k! dt t=t k=0 We rewrite the equations (9)and (10)as (1 + 2γη) f + 2γ f + A ff − f − BM f = 0 (21) C [(1 + 2γη) θ + 2γθ ] + Pr ( f θ − f θ) = 0 (22) nf ρ f 2.5 CNT 2.5 where A = (1 − φ) ((1 − φ) + φ( )), B = (1 − φ) ,and C = . (ρc ) ρ p f CNT [(1−φ)+φ ] (ρc ) Applying DTM ( Basic operations of DTM are listed in Table 2)on(21), we get (k + 1)(k + 2)(k + 3) F [k + 3] + 2γ  (k − m − 1)(m + 1)(m + 2)(m + 3) F [m + 3]⎪ m=0 ⎪ +2γ (k + 1)(k + 2) F [k + 2] + A F [k − m](m + 1)(m + 2) F [m + 2] (23) m=0 ⎪ −A (k − m + 1) F [k − m + 1](m + 1) F [m + 1] − BM (k + 1) F [k + 1] = 0 m=0 1,m=1 where (m) ={ . 0,m=1 The corresponding boundary conditions are transformed to F [0] = 0, F [1] = 1, F [2] = a (24) Similarly, from (22)wehave C (k + 1)(k + 2)  [k + 2] + 2γ C  (k − m − 1)(m + 1)(m + 2)  [m + 2] m=0 (25) k k +2γ C (k + 1)  [k + 1] + Pr F [k − m](m + 1)  [m + 1] − Pr  [k − m](m + 1) F [m + 1] = 0 m=0 m=0 Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 342 MHD Carbon nanotube flow over a stretching cylinder using DTM Table 4: The values of f (1) for various values of M. Values of f (1) by DTM Values of f (1) by RK − 4 M SWCNT MWCNT SWCNT MWCNT 1.0 −0.309864 −0.310918 −0.309864107 −0.310917717 2.0 −0.289198 −0.290432 −0.289197717 −0.290431594 3.0 −0.265845 −0.267031 −0.265845308 −0.267030972 4.0 −0.243334 −0.244418 −0.243333817 −0.244418114 Table 5: The values of θ (1) for various values of M. Values of θ (1) by DTM Values of θ (1) by RK − 4 M SWCNT MWCNT SWCNT MWCNT 1.0 −0.118424 −0.118324 −0.118423879 −0.118324356 2.0 −0.127243 −0.127255 −0.127243216 −0.127254875 3.0 −0.134169 −0.134243 −0.134168985 −0.134242744 4.0 −0.139950 −0.140063 −0.139950380 −0.140063290 Table 6: The values of g (1) for various values of K . Values of g (1) by DTM Values of g (1) by RK − 4 K SWCNT MWCNT SWCNT MWCNT 1.0 0.091936 0.095787 0.091936031 0.095786678 2.0 0.956196 0.123987 0.119561956 0.123986728 3.0 0.133502 0.138204 0.133501858 0.138204086 4.0 0.141956 0.141669 0.141955901 0.141669055 Figure 2: Effect of curvature parameter (γ ) on velocity distribution. Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 343 Figure 3: Effect of curvature parameter (γ ) on temperature distribution. Figure 4: Effect of curvature parameter (γ ) on concentration profiles. The relevant boundary conditions become [0] = b, [1] =−1 (26) Again applying DTM on (14), we get (k + 1)(k + 2)G [k + 2] + 2γ  [k − m − 1](m + 1)(m + 2)G [m + 2]⎪ m=0 ⎪ +2γ (k + 1)G [k + 1] + Sc F [k − m](m + 1)G [m + 1] − λScG [k] (27) m=0 ⎪ k k k−t +2λSc G [k − m]G [m] − λSc G [t]G [m] G [k − m − t] = 0 m=0 t=0 m=0 Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 344 MHD Carbon nanotube flow over a stretching cylinder using DTM Table 7: Reduced skin friction coefficient and reduced Nusselt number for various parameters. C Nu f r γ M φ SWCNT MWCNT SWCNT MWCNT 0.0 −3.742635 −3.712047 3.686848 3.526191 0.1 −3.847419 −3.817071 3.746504 3.583227 0.2 −3.954626 −3.901835 3.835886 3.668690 0.3 −4.049939 −4.019968 3.865710 3.697204 0.0 −2.801107 −2.741811 4.203039 4.021273 1.0 −3.369516 −3.332956 4.025346 3.851202 2.0 −4.049939 −4.019968 3.865710 3.697204 4.0 −5.124731 −5.101383 3.611410 3.452540 0.00 −3.647186 −3.647186 2.754994 2.754994 0.05 −4.049939 −4.019968 3.865710 3.697204 0.10 −4.406153 −4.342640 4.852997 4.501512 0.15 −4.682546 −4.580576 5.718392 5.169770 Figure 5: Effect of magnetic-field parameter ( M) on velocity distribution. The relevant boundary conditions become G [0] = β, G [1] = K β (28) where F [k],[k], and G[k] are the differential transform of f (η),θ(η), and g(η)at η = 0, respectively, and a, b, and β are the constants that have to be determined by using boundary conditions. Now the iteration schemes are as follows: F[3] = (BM + A − 4γ a) 3! F[4] = 2Aa + 2BMa − 2BMγ − 2Aγ + 8aγ 4! 2 3 2 2 2 2 2 F[5] = 16aγ − 16aγ + 4Aa + 4Aγ − 4Aγ − 4aAγ − 8aBMγ − 4BMγ + B M + BMA + 4BMγ 5! ⎛ ⎞ 2 2 3 4 2 −24aAγ − 24aBMγ + 32BMγ + 32Aγ − 128aγ + 32aγ − 24a Aγ ⎝ ⎠ F[6] = 6! 3 2 3 2 2 2 2 −8Aγ + 8aBMγ − 8BMγ + 2aA + 2A γ − 2aB M + 2ABMγ and so on. [2] = (2γ C + Pr .b) 2!C Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 345 Figure 6: Effect of magnetic-field parameter ( M) on temperature distribution. Figure 7: Effect of curvature parameter (γ ) and magnetic-field parameter ( M) on Nusselt number. [3] = 2ab Pr −2bγ Pr −4γ C 3!C 2 2 3 2 2 [4] = bCBM Pr +AbC Pr −8γ C + 8γ C − 4bC γ Pr + 4bC γ Pr − 2aC Pr − 2C γ Pr − b(Pr) − 12abC γ Pr 4!C and so on. G [2] = λSc(1 − β) − 2γ K G [3] = ScK β(3λβ − 4λβ + λ − 1) − 2γ λSc(1 − β) − 2γ K s s 6 Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 346 MHD Carbon nanotube flow over a stretching cylinder using DTM Figure 8: Effect of nanoparticle concentration (φ) on velocity distribution. Figure 9: Effect of nanoparticle concentration (φ) on temperature distribution. ⎡    ⎤ λ 3 2 2 Sc + λβ Sc − 2γ − Sc − 2λβSc β λSc(1 − β) − 2γ K − γ ScK β(3λβ s s ⎢ ⎥ 2 2 ⎢ ⎥ G [4] = ⎣ ⎦ 12 2 2 2 −4λβ + λ − 1) − 2γ λSc(1 − β) − 2γ K − aScβK + (2 − 3β)λScβ K s s and so on. From (19), we get f (η), θ(η), and g(η) as follows: 1 1 2 3 2 4 f (η) = η + aη + (BM + A − 4γ a) η + 2Aa + 2BMa − 2BMγ − 2Aγ + 8aγ η 3! 4! 2 3 2 2 2 2 2 5 + 16aγ − 16aγ + 4Aa + 4Aγ − 4Aγ − 4aAγ − 8aBMγ − 4BMγ + B M + BMA + 4BMγ η 5! Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 347 Figure 10: Effect of homogeneous chemical reaction factor (λ) on concentration profiles. Figure 11: Effect of heterogeneous chemical reaction factor (K ) on concentration profiles. 2 2 3 4 2 3 2 + − 24aAγ − 24aBMγ + 32BMγ + 32Aγ − 128aγ + 32aγ − 24a Aγ − 8Aγ + 8aBMγ 6! 3 2 2 2 2 6 − 8BMγ + 2aA + 2A γ − 2aB M + 2ABMγ η + .... (29) 1 1 2 2 3 θ(η) = b − η + (2γ C + Pr .b) η + 2ab Pr −2bγ Pr −4γ C η 2!C 3!C # $ 2 2 3 2 2 bCBM Pr +AbC Pr −8γ C + 8γ C − 4bC γ Pr +4bC γ Pr −2aC Pr + η + ... (30) 2 2 4!C −2C γ Pr −b(Pr) − 12abC γ Pr Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 348 MHD Carbon nanotube flow over a stretching cylinder using DTM Figure 12: Effect of Schmidt number (Sc) on concentration profiles. ⎡ ⎤ ScK β(3λβ − 4λβ + λ − 1) β 1 2 3 ⎣ ⎦ g(η) = β + K βη + λSc(1 − β) − 2γ K η +   η s s 2 6 −2γ λSc(1 − β) − 2γ K ⎡ ⎤ λ 3 2 2 Sc + λβ Sc − 2γ − Sc − 2λβSc β λSc(1 − β) − 2γ K ⎢ ⎥ 2 2 ⎢ ⎥ ⎢ ⎥ + ⎢ 2 ⎥ η + ..... (31) −γ ScK β(3λβ − 4λβ + λ − 1) − 2γ λSc(1 − β) − 2γ K s s 12 ⎢ ⎥ ⎣ ⎦ 2 2 −aScβK + (2 − 3β)λScβ K Considering γ = 0.3, M = 5.0, Pr = 1.0, and φ = 0.0 for the equations (29)and (30) and linking with the boundary conditions (15)at η →∞ (we accept η = 10 in large sense of η), we easily get a = 0.329877091, b = 0.596315085, and β = 0.778332251. 3.2 Solution by RK − 4 method Currently, we move onward to resolve the equations (9), (10), and (14) numerically using (15)byalteringthemtoaninitial-value problem. We transform the differential equations (9), (10)and (14), to a system of first-order differential equation as follows: y = y 2 ⎪ y = y %    & (32) 1 ρ ⎪ 2.5 CNT 2 ⎪ y = (1 − φ) My − 2γ y − (1 − φ) + φ y y − y 2 3 1 3 3 2 1 + 2γη ρ y = y ⎪ 4 ⎬ '   ( 1 κ (ρc ) (33) f p CNT y = − . Pr . (1 − φ) + φ (y y − y y ) − 2γ y .⎪ 1 5 2 4 5 1 + 2γη κ (ρc ) nf p y = y ⎬ 1 (34) y = λScy (1 − y ) − 2γ y − Scy y . 6 6 7 1 7 1 + 2γη where y , y , y , y , y , y , and y are used for f (η), f (η), f (η), θ(η), θ (η), g(η), and g (η), respectively, and the prime signifies differenti- 1 2 3 4 5 6 7 ation with respect to η. Thus, boundary condition (15) reduces to y = 0, y = 1, y =−1, y = y K at η = 0 1 2 5 7 6 s (35) y → 0, y → 0, y →1as η →∞ 2 4 6 To get the complete solution of the IVP (32), (33), and (34) we have required the values of y (0), i.e. f (0), y (0), i.e. θ(0) and y (0), 3 4 6 i.e. g(0), which are absent in boundary conditions (35). Hence, to begin integration outline we make a guess of y (0), y (0), and y (0). 3 4 6 Hence, reduced ODE (32), (33), and (34) with boundary condition (35) is resolved numerically aiming the optimizing of the solution Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 349 with y (0), y (0), and y (0). We adjust the values y (0), y (0), and y (0) for the improved approximation of solution. We adopt RK − 4 3 4 6 3 4 6 method for different step sizes as η = 0.01, 0.001, 0.0001, etc. The overhead technique is repeated to acquire the results with a desired −6 accuracy of 10 . 3.3 Testing of code To check the precision of numerical results plus code involved in extant unit, we compared the dissimilar values of θ(0) for several values of Pr with similar demonstration in diverse research works like Elbashbeshy (1998) and Bachok and Ishak (2010)when M = 0,φ = 0, and γ = 0 and in the absence of a chemical reaction. In Table 3 we have organized those values of θ(0) that are outstandingly approved with preceding research work stated above. Assessment among the numerical results achieved by RK − 4 method and analytical solution completed by DTM is arranged in Tables 4–6. 4. Results and Discussions Current article includes parametric workout of the flow region. Comprehensive discussion we have exemplified via tables and graph, to scrutinize the outcome of diverse emerging factors like curvature factor γ , modified magnetic-field factor M, and nanoparticles concentration φ. Default values of emerging factors throughout model are reflected as Pr = 6.2,M = 2.0,γ = 0.3,φ = 0.05, and λ = Sc = K = 1.0exceptifnot specified. 4.1 Effect of curvature parameter γ Figure 2 sketched on the circumstance that how curvature parameter acts on fluid velocity. We perceived the augmentation in fluid velocity when γ amplifies. Curvature parameter and radius of curvature are inversely proportional to each other. Therefore γ increases implies radius of curvature, i.e. the radius of the cylinder decreases consequently less resistance offers toward the fluid flow due to lessens of contact surface area with the fluid and hence fluid velocity progresses. Temperature distribution due to curvature parameter γ has been portrayed in Fig. 3. Temperature distribution amplifies like velocity profiles for cumulative γ . Since Kelvin temperature of substances is defined as average kinetic energy of the particles of substances, so as velocity profiles enhanced with γ accordingly kinetic energy upsurges and consequently temperature profiles amplify. Velocity profiles and temperature distribution are higher in MWCNT than SWCNT. Figure 4 demonstrates that concentration outlines uphold in an identical fashion like temperature outlines when γ escalates and subsequently concentration boundary layers get thicker. Concentration field intensifies due to species diffusion in flow regime. From Table 7, we detected that Nu upsurges when γ escalates but C diminishes and Nu is advanced in SWCNT r f r than MWCNT but the converse was perceived for C 4.2 Effect of magnetic-field parameter M Figure 5 depicts the response of the velocity fields due to magnetic-field parameter M. The graphical interpretation endorses the decline of fluid velocity when the strength of magnetic field gets enlarged. Physical explanation of such a result is that, when the strength of engaged magnetic field upsurges in nanofluid, it starts to spread resistive force specifically Lorentz force, which obstructs fluid motion. Figure 6 exposed how magnetic-field factor M acts on temperature field. We appreciate that temperature distribution boosted with M. Since supplementary work done needed to stretch out nanofluid in opposition to the object of employed magnetic field, which heats up carrying nanofluid and enhances temperature profiles. Velocity profiles and temperature outlines are higher in MWCNT than SWCNT. For growing M from Table 7, we appreciated that C and Nu are reduced and Nu is advanced in SWCNT than fr r r MWCNT equivalent can be established from Fig. 7 but the converse was perceived for C . 4.3 Effect of nanoparticle concentration φ Response of the velocity field due to nanoparticle concentration has been manifested in Fig. 8. We perceived the enhancement in fluid velocity for cumulative φ and hence momentum boundary layer gets thicker. Temperature distribution due to nanoparticles concentration φ has been exposed in Fig. 9 and it disclosed that temperature outlines upsurge with φ. This agreed with physical characteristic that φ increases indicates thermal conductivity accelerates and correspondingly thermal boundary-layer thickness gets increased. Velocity profiles and temperature distribution are higher in MWCNT than SWCNT. Table 7 acknowledged that Nu upsurges with φ and is reverse in C . fr 4.4 Effect of chemical reaction parameters Homogeneous chemical reaction factor (λ) amplifies the spices B, which suspended nanoparticles in it. So nanoparticles’ production enhances. This justifies that g is augmented in the system as elucidated in Fig. 10.InFig. 11, we uncover the clout of heterogeneous chemical reaction factor (K ) on the flow. First-order heterogeneous reaction takes place in the existence of solid absorbing heated flow surface, which absorbs A and B and then a reaction takes place and resultant spices B came out of the wall thermally charged nanoparticle in it. So g upturns. Concentration profiles are higher in SWCNT than MWCNT. Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 350 MHD Carbon nanotube flow over a stretching cylinder using DTM 4.5 Effect of Schmidt number (Sc) Schmidt number is identified as a ratio of the rate of momentum diffusivity to the rate of mass diffusivity; therefore, higher values of Sc imply small mass diffusion and as an upshot concentration profiles decline, which are portrayed in Fig. 12. Concentration profiles are higher in SWCNT than MWCNT. 5. Final Remarks In extant article, we cultivate hydromagnetic incompressible nanofluid flow through a stretching cylinder with homogeneous– heterogeneous chemical reaction. SWCNTs as well as MWCNTs nanoparticles are accounted here. We rehabilitated PDEs to ODEs by paying similarity approach and then resolved analytically by engaging DTM as well as numerically by employing Runge–Kutta– Fehlberg quadrature method through shooting practice. Through our discussion, we found many outcomes, among which specific conclusions are noted below: Impacts of γ and φ on velocity profiles are identical. Temperature profiles are amplified with γ, M, and φ. Velocity profiles and temperature distribution are higher in MWCNT than SWCNT and reverse for concentration profiles in each case. Concentration profiles amplify for chemical reaction factor. C in MWCNT is advanced than SWCNT in each case but we acknowledged reverse in Nu . fr r C is reduced when γ, M, and φ amplify. Acknowledgement The authors wish to express their cordial thanks to reviewers for giving valuable suggestions and comments to improve the presenta- tion of this article. Finally, the authors wish to express their sincere thanks to the editor in chief and the production office for primary consideration and correcting the mistakes for publication in Journal of Computational Design and Engineering. Conflict of interest statement Declarations of interest: none. References Ahmed, Z., & Nadeem, S. (2019). Flow of a micropolar CNT-based nanofluid across a squeezing channel. Physica Scripta, 94, 10. Ahmed, Z., Qahtani, A. A., Nadeem, S., & Saleem, S. (2019). Computational study of MHD nanofluid flow possessing micro-rotational inertia over a curved surface with variable thermophysical properties. Processes, 387(7), 1–15. Alavi, S. Q., Hussanan, A., Kasim, A. R. M., Rosli, N., & Salleh, M. Z. (2017). MHD stagnation point flow towards an exponentially stretching sheet with prescribed wall temperature and heat flux. International Journal of Applied and Computational Mathematics, 3(4), 3511–3523. Ayaz, F. (2013). On the two-dimensional differential transformation method. Applied Mathematics and Computation, 143(2–3), 361–374. Bachok, N., & Ishak, A. (2010). Flow and heat transfer over a stretching cylinder with prescribed surface heat flux. Malaysian Journal of Mathematical Sciences, 4, 159–169. Bachok, N., Ishak, A., & Pop, I. (2011). On the stagnation-point flow towards a stretching sheet with homogeneous–heterogeneous reactions effects. Communication in Nonlinear Science and Numerical Simulation, 16, 4296–4302. Bhatti, M. M., Abbas, T., & Rashidi, M. M., (2017). Entropy generation as a practical tool of optimization for non-Newtonian nanofluid flow through a permeable stretching surface using SLM. Journal of Computational Design and Engineering, 4, 21–28. Choi, S. U. S. (1995). Developments and application of non-Newtonian flows .New York,USA:ASMEPress. Daniel, Y. S., Aziz, Z. A., Ismail, Z., & Salah, F., (2018). Thermal stratification effects on MHD radiative flow of nanofluid over nonlinear stretching sheet with variable thickness. Journal of Computational Design and Engineering, 5, 232–242. Dhlamini, M., Kameswaran, P. K., Sibanda, P., Motsa, S., & Mondal, H., (2019). Activation energy and binary chemical reaction effects in mixed convective nanofluid flow with convective boundary conditions. Journal of Computational Design and Engineering, 6, 149–158. Elbashbeshy, E. M. A. (1998). Heat transfer over a stretching surface with variable surface heat flux. Journal of Physics D: Applied Physics, 31, 1951–1954. Erfani, E., Rashidi, M. M., & Parsa, A. B. (2010). The modified Differential Transform Method for solving off-centered stagnation flow towards a rotating disc. International Journal of Computational Methods, 7(4), 655–670. Giri, S. S., Das, K., & Kundu, P. K. (2018). Framing the features of a Darcy–Forchheimer nanofluid flow past a Riga plate with chemical reaction by HPM. The European Physical journal Plus, 133(365), 1–17. Hayat, T., Nadeem, S., & Khan, A. U. (2019). Aspects of 3D rotating hybrid CNT flow for a convective exponentially stretched surface. Applied Nanoscience, https://doi.org/10.1007/s13204-019-01036-y. Hussain, A., Javed, F., & Nadeem, S. (2019). Numerical Solution of a Casson nanofluid flow and heat transfer analysis between con- centric cylinders. Journal of Power Technologies, 99(1), 25–30. Hussain, S. T., Haq, R. U., Khan, Z. H., & Nadeem, S. (2016). Water-driven flow of carbon nanotubes in a rotating channel. Journal of Molecular Liquids, 214, 136–144. Ishak, A, Nazar, R., & Pop, I. (2008). Uniform suction/blowing effect on flow and heat transfer due to a stretching cylinder. Applied Mathematical Modelling, 32, 2059–2066. Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 351 Jang, M. J., Chen, C. L., & Liu, Y. C. (2001). Two-dimensional differential transform for partial differential equations. Applied Mathematical Computing, 121, 261–270. Kardri, M. A., Bachok, N., Arifin, N. M., & Ali, F. M. (2017). Boundary layer flow and heat transfer over a stretching cylinder in a copper- water nanofluid. American Institute of Physics, 1795, 020012. Kumar, R., Sood, S., Sheikholeslami, M., & Shehzad, S. A. (2017). Nonlinear thermal radiation and cubic autocatalysis chemical reaction effects on the flow of stretched nanofluid under rotational oscillations. Journal of Colloid and Interface Science, 505, 253–265. Majeed, A., Zeeshan, A., & Ellahi, R. (2016). Unsteady ferromagnetic liquid flow and heat transfer analysis over a stretching sheet with the effect of dipole and prescribed heat flux. Journal of Molecular liquids, 223, 528–533. Mondal, H., Almakki, M., & Sibanda, P., (2019). Dual solutions for three-dimensional magnetohydrodynamic nanofluid flow with en- tropy generation. Journal of Computational Design and Engineering, 6, 657–665. Muhammad, N., Nadeem, S., & Mustafa, M. T. (2019). Hybrid isothermal model for the ferrohydrodynamic chemically reactive species. Communication in Theoretical Physics, 71, 384–392. Munawar, S., Mehmood, A., & Ali, A. (2012). Time-dependent flow and heat transfer over stretching cylinder. Chinese Journal of Physics, 50(5), 828–848. Nadeem, S., Abbas, N., & Khan, A. U. (2018). Characteristics of three dimensional stagnation point flow of hybrid nanofluid past a circular cylinder. Results in Physics, 8, 829–835. Nadeem, S., Hayat, T., & Khan, A. U. (2019b). Numerical study on 3D rotating hybrid SWCNT-MWCNT flow over a convectively heated stretching surface with heat generation/absorption. Physica Scripta, 94,7. Nadeem, S., Khan, M. R., & Khan, A. U. (2019a). MHD oblique stagnation point flow of nanofluid over an oscillatory stretching/shrinking sheet: Existence of dual solutions. Physica Scripta, 94,7. Nasir, S., Shah, Z., Islam, S., Bonyah, E., & Gul, T. (2019b). Darcy Forchheimer nanofluid thin film flow of SWCNTs and heat transfer analysis over an unsteady stretching sheet. AIP Advances, 9, 015223. Nasir, S., Shah, Z., Islam, S., Khan, W., & Khan, S. N. (2019a). Radiative flow of magneto hydrodynamics single-walled carbon nanotube over a convectively heated stretchable rotating disk with velocity slip effect. Advances in Mechanical Engineering, 11(3), 1–11. Noor, M. A., Mohyud-Din, S. T., & Waheed, A. (2008). Variation of parameter method for solving fifth-order boundary value problems. Applied Mathematics and Information Sciences, 2, 135–141. Rana, S., Mehmood, R., & Akbar, N. S. (2016). Mixed convective oblique flow of a Casson fluid with partial slip, internal heating and homogeneous–heterogeneous reactions. Journal of Molecular Liquids, 222, 1010–1019. Rashidi, M. M. (2009). The modified differential transform method for solving MHD boundary-layer Equations. Computer Physics Com- munication, 180 (11), 2210–2217. Rashidi, M. M., Beg, ´ O. A., Asadi, M., & Rastegari, M. T. (2012a). DTM-Pade´ modeling of natural convective boundary layer flow of a nanofluid past a vertical surface. International Journal of Thermal & Environmental Engineering, 4(1), 13–24. Rashidi, M. M., Chamkha, A. J., & Keimanesh, M. (2011b). Application of multi-step differential transform method on flow of a second- grade fluid over a stretching or shrinking sheet. American Journal of Computational Mathematics, 1(2), 119–128. Rashidi, M. M., & Erfani, E. (2011). The modified differential transform method for investigating nano boundary-layers over stretching surfaces. International Journal of Numerical Methods for Heat & Fluid Flow, 21(7), 864–883. Rashidi, M. M., Hayat, T., Keimanesh, M., & Yousefian, H. (2013). A Study on heat transfer in a second-grade fluid through a porous medium with the modified differential transform method. Heat Transfer—Asian Research, 42(1), 31–45. Rashidi, M. M., Laraqi, N., & Parsa, A. B. (2011a). Analytical Modeling of heat convection in magnetized micropolar fluid by using modified differential transform method. Heat Transfer—Asian Research, 40(3), 187–204. Rashidi, M. M., Rahimzadeh, N., Ferdows, M., Uddin, M. J., & Beg, O. A. (2012b). Group theory and differential transform analysis of mixed convective heat and mass transfer from a horizontal surface with chemical reaction effects. Chemical Engineering Communi- cations, 199(8), 1012–1043. Shah, Z., Bonyah, E., Islam, S., & Gul, T. (2019). Impact of thermal radiation on electrical MHD rotating flow of Carbon nanotubes over a stretching sheet. AIP Advances, 9, 015115. Shah, Z., Dawar, A., Islam, S., Khan, I., & Ching, D. L. C. (2018). Darcy–Forchheimer flow of radiative carbon nanotubes with microstruc- ture and inertial characteristics in the rotating frame. Case Studies in Thermal Engineering, 12, 823–832. Shahzadi, I., Nadeem, S., & Rabiei, F. (2017). Simultaneous effects of single wall carbon nanotube and effective variable viscosity for peristaltic flow through annulus having permeable walls. Results in Physics, 7, 667–676. Sheikholeslami, M. (2015). Effect of uniform suction on nanofluid flow and heat transfer over a cylinder. The Brazilian Society of Me- chanical Sciences and Engineering, 37, 1623–1633. Sheikholeslami, M., Ellahi, R., Ashorynejad, H. R., & Hayat, T. (2014). Effects of heat transfer in flow of nanofluids over a permeable stretching wall in a porous medium. Journal of Computational and Theoretical Nanoscience, 11, 486–496. Sheikholeslami, M., & Ganji, D. D. (2015). Nanofluid flow and heat transfer between parallel plates considering Brownian motion using DTM. Computer Methods in Applied Mechanics and Engineering, 283(1), 651–663. Sheikholeslami, M., Ganji, D. D., & Ashorynejad, H. R. (2013). Investigation of squeezing unsteady nanofluid flow using ADM. Powder Technology, 239, 259–265. Sheikholeslami, M., Rashidi, M. M., Alsaad, D. M., & Rokni, H. B. (2016). Steady nanofluid flow between parallel plates considering thermophoresis and Brownian effects. Journal of King Saud University – Science, 28, 380–389. Tamoor, M., Waqas, M., Khan, M. I., Alsaedi, A., & Hayat, T. (2017). Magnetohydrodynamic flow of Casson fluid over a stretching cylinder. Results in Physics, 7, 498–502. Wang, C. Y. (1998). Fluid flow due to a stretching cylinder. Physics of Fluids, 31, 466–468. Zhou, K. (1986). Differential transformation and its applications for electrical circuits. Wuhan, China: Huazhong University Press. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Computational Design and Engineering Oxford University Press

Homogeneous–heterogeneous reaction mechanism on MHD carbon nanotube flow over a stretching cylinder with prescribed heat flux using differential transform method

Download PDF
15 pages

Loading next page...
 
/lp/oxford-university-press/homogeneous-heterogeneous-reaction-mechanism-on-mhd-carbon-nanotube-0NJnt0YQV2

References (52)

Publisher
Oxford University Press
Copyright
© The Author(s) 2020. Published by Oxford University Press on behalf of the Society for Computational Design and Engineering.
eISSN
2288-5048
DOI
10.1093/jcde/qwaa028
Publisher site
See Article on Publisher Site

Abstract

Hydromagnetic nanofluid flow through an incompressible stretching cylinder accompanying with homogeneous–heterogeneous chemical reaction has been executed in current literature. SWCNTs (single-walled carbon nanotubes) and MWCNTs (multiwalled carbon nanotubes) as nanoparticles in appearance of prescribed heat flux are accounted here. Leading equations of the assumed model have been normalized through similarity practice and succeeding equations resolved numerically by spending RK-4 shooting practice and analytically by engaging differential transform method. The impulse of promising flow constraints on the flow characteristic is finalized precisely through graphs and charts. We perceived that velocity outlines and temperature transmission are advanced in MWCNT than SWCNT in every case. Keywords: carbon nanotubes; nanofluid; stretching cylinder; prescribed heat flux; homogeneous–heterogeneous chemical reaction; DTM 1. Introduction To utilize solar energy, researchers, scientists, and engineers are devoted to develop energy resources and the energy technologies due to significant dependence on it of human society. It is a well-known reality that improvement in thermal characteristic can be made by adding little amount of nanoparticles having high thermal characteristic. However, in recent times nanofluid (Bhatti, Abbas, & Rashidi, 2017; Daniel, Aziz, Ismail, & Salah, 2018; Dhlamini, Kameswaran, Sibanda, Motsa, & Mondal, 2019; Mondal, Almakki, & Sibanda, 2019), which is a new kind of fluid categorized due to solid–liquid arrangement in metal or nonmetal nanoparticle suspen- sion; originated by Choi (1995), to highten thermal conductivity of the fluid. Carbon nanotubes basically are the cylinder of single or multiple sheets of graphene. Centered on sheets of the graphene, carbon nanotubes are distinguished into two types viz. single and multiple-walled carbon nanotubes (SWCNTs and MWCNTs). CNT is generally used in electrodes, anodes, catalyst, and various Received: 24 June 2019; Revised: 9 September 2019; Accepted: 1 December 2019 The Author(s) 2020. Published by Oxford University Press on behalf of the Society for Computational Design and Engineering. This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact [email protected] 337 Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 338 MHD Carbon nanotube flow over a stretching cylinder using DTM medical and environmental applications due to its remarkable mechanical and electrical properties as well as thermal conductiv- ity. CNTs have wide applications in aerospace materials because it can also progress the properties of the progressive aerospace materials. A material containing CNT is highly electrically conductive, which proposes that they might defend aircraft from the light- ning strikes. Shah, Bonyah, Islam, and Gul (2019) explored impression of nonlinear radiation on MHD rotating flow of CNTs through stretching sheet and confirmed that heat-flux values enrich with physical quantities. CNT effects in a nanofluid flow with variable fluid properties were investigated by Shahzadi, Nadeem, and Rabiei ( 2017). Shah, Dawar, Islam, Khan, and Ching (2018) scrutinized Darcy–Forchheimer CNT flow in a rotating frame and convey that due to amplification in inertia coefficient transverse velocity dis- tribution has dual characteristic. Inspiration of velocity slip on CNT flow through a stretched rotating disk was inspected by Nasir, Shah, Islam, Khan, and Khan (2019a) and reported that velocity field reduced for larger values of velocity slip factor. Hussain, Haq, Khan, and Nadeem (2016) explored nanofluid flow of CNTs in a rotating channel. Darcy–Forchheimer unsteady thin film flow of CNTs through a stretching sheet was observed by Nasir, Shah, Islam, Bonyah, and Gul (2019b) and recounted that for larger values of poros- ity factor velocity field declines. Ahmed, Qahtani, Nadeem, and Saleem ( 2019) explored MHD CNT flow through a curved surface with nonuniform thermo-physical properties and confirmed that heat transmission rate heightened more in SWCNTs than MWCNTs. CNT flow on 3D rotating system was considered by Hayat, Nadeem, and Khan ( 2019) and they confirmed that augmenting temperature exponent declines temperature field. Ahmed and Nadeem ( 2019) studied CNT flow through squeezed channel and conclude that squeezed factor seemed to reduce temperature field. Nadeem, Khan, and Khan ( 2019a) analyzed CNT stagnation-point flow through an oscillatory sheet and acquired dual solution. Nadeem, Hayat, and Khan (2019b) explored 3D rotating CNT flow over heated surface and convey that heat transmission enriches with convective factor. Researchers recently have been attentive on the surface-driven flow to analyze flow features due to its prominence in manufac- turing progression such as plastic sheets, paper production, polymer processing, glass blowing, etc. Steady-flow exterior of cylinder by considering an ambient fluid at rest was coined by Wang ( 1998) and received exact solution also determined heat transmission. Ishak, Nazar, and Pop (2008) explored flow and heat transmission outside of stretching cylinder numerically by employing Keller-box method and recognized that injection is adequate to lessen skin friction. Hussain, Javed, and Nadeem (2019) analyzed numerical solution of Casson flow among concentric cylinders and verified that nanofluids are superior coolant than their base fluids. Tamoor, Waqas, Khan, Alsaedi, and Hayat (2017) inspected MHD Casson fluid flow through cylinder and fix the convergence area for the at- tained solutions. Sheikholeslami (2015) scrutinized the encouragement of unchanging suction happening on nanofluid flow through cylinder and compute effective thermal conductivity in addition viscosity by KKL scheme. Kardri, Bachok, Arifin, and Ali ( 2017) stud- ied flow and heat transfer features through cylinder and specify that dual solution be existent for stretching cylinder. Nadeem, Abbas, and Khan (2018) scrutinized the characteristics of 3D stagnation-point hybrid-nanofluid flow through circular cylinder and confirmed that heat transmission rate is higher in hybrid nanofluid than nanofluid. Heat flux is defined as the rate of heat energy transmission through a specified surface. Many researchers (Elbashbeshy, 1998; Bachok & Ishak, 2010; Munawar, Mehmood, & Ali, 2012) demonstrate flow and heat transmission over stretching cylinder using pre- scribed heat flux. Alavi, Hussanan, Kasim, Rosli, and Salleh ( 2017) considered prescribed heat flux to capture the characteristic of MHD flow over an exponentially stretching sheet. Heat transmission analysis using prescribed heat flux through stretching sheet was considered by Majeed, Zeeshan & Ellahi (2016). In numerous industrial manufacturing, biomedical invention, combustion, paint manufacturing, food processing, and metal ab- straction from naturally found ore, homogeneous–heterogeneous chemical reaction occurs. Bachok, Ishak, and Pop (2011) scrutinized stagnation flow in appearance of homogeneous–heterogeneous chemical reaction over an expanding surface by considering identi- cal diffusion rate of reactant and autocatalyst. Rana, Mehmood, and Akbar (2016) analyzed oblique flow of the Casson by accounting homogeneous–heterogeneous reactions. Muhammad, Nadeem, and Mustafa (2019) analyzed Ferro- hydrodynamic boundary-layer flow in the presence of homogeneous–heterogeneous reactions. Recently, nanofluid flow with homogeneous–heterogeneous reac- tions was studied by Kumar, Sood, Sheikholeslami, and Shehzad (2017). Nearly all mathematical modeling encloses linear or nonlinear differential equations and to resolve those equations perfectly, ap- proximately, or numerically several investigators are focused to hire numerous powerful analytic, semianalytic tools along with per- manent label of convergence. Specific techniques are differential transform method (DTM; Ayaz, 2013), modified DTM (Erfani, Rashidi, &Parsa, 2010; Rashidi, Laraqi, & Parsa, 2011a; Rashidi, Hayat, Keimanesh, & Yousefian, 2013), multistep DTM (Rashidi, Chamkha, & Keimanesh, 2011b), HAM (Sheikholeslami, Ellahi, Ashorynejad, & Hayat, 2014), ADM (Sheikholeslami, Ganji, & Ashorynejad, 2013), HPM (Giri, Das, & Kundu, 2018), VPM (Noor, Mohyud-Din, & Waheed 2008), etc. DTM is the iterative technique to acquire analytical series solution in the form of polynomial. In evaluation with other methods, solution achieved by DTM is of less size of polynomial and takes less CPU time for the same accuracy. As DTM is away from the any kind of linearization, discretization, and restricting as- sumption, one can directly apply this method without any type of affected error. Series solution achieved by DTM is divergent with the boundary conditions at infinity. DTM with Pad e´ approximation by Rashidi (2009) delivers an operative tool for solving boundary value differential equations on infinite/semiinfinite domains. Zhou ( 1986) first initiated DTM and in two dimensions it was initiated by Jang, Chen, and Liu (2001). Then several researchers (Rashidi & Erfani, 2011;Rashidi,Beg, ´ Asadi, & Rastegari, 2012a; Rashidi, Rahimzadeh, Ferdows, Uddin, & Beg, 2012b; Sheikholeslami & Ganji, 2015; Sheikholeslami, Rashidi, Alsaad, & Rokni, 2016) used DTM to resolve various problems. By receiving motivation from the above discussion in extant unit, we desire to explore hydromagnetic nanofluid flow through an incompressible stretching cylinder. SWCNTs and MWCNTs as nanoparticles in appearance of prescribed heat flux are accounted here. It has significant applications in electrodes, anodes, microelectronics, transistors, plastic sheets, paper production, polymer pro- cessing, glass blowing, combustion, paint industry, food processing, aerospace materials, various medical, industrial, and biomedical production, and environmental studies. We implement a similarity tactic to convert the primary Partial Differential Equations (PDEs) into Ordinary Differential Equations (ODEs) and resolved numerically by spending RK-4 shooting practice and analytically by engaging DTM to perform flow analysis. Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 339 Table 1: Physical properties of the base fluid and nanoparticles. Base fluid Nanoparticles Water SWCNT MWCNT ρ (kg/m ) 997.1 2600 1600 C (J/kg K) 4179 425 796 κ (W/mK) 0.613 6600 3000 Figure 1: Physical model and coordinate system. 2. Mathematical Formulation In this section, a steady-state hydromagnetic incompressible nanofluid flow through a stretching cylinder of radius R accompanying with homogeneous–heterogeneous chemical reaction is studied. SWCNT and MWCNT nanoparticles and water as a base fluid are considered here and their thermophysical properties are given in the Table 1. We accomplish the cylindrical co-ordinate system by fixing the x-axis along the axis of cylinder and r-axis normal to it as portrayed in Fig. 1. A uniform magnetic field of intensity B has been employed in a radial direction by assuming enough small magnetic Reynolds number to neglect the encouraged magnetic field. Stream carries two different chemically reacting components A and B and between them homogeneous and heterogeneous chemical reaction takes place. The isothermal cubic order homogeneous autocatalysis occurred as A + 2B → 3B,where Bplays the role of an autocatalyst and the rate of the phenomena is k a b . On the other side, at surface the occurrence of isothermal first- 1 1 order catalysis can be presented as A + B → 2B with rate of reaction k a . The concentrations of the chemical species A and B are s 1 a and b , respectively, and k and k are reaction rate factors accordingly. We support our study with the hypothesis that there is no 1 1 1 s added chemically responsive species, and all kinds of body forces along with viscous dissipation and joule heating are discounted. The cylindrical sheet is being stretched with a velocity U = U ( ) toward axial direction and the cylindrical surface is preserved at w 0 prescribed heat flux q = T ( )inwhich U and T are the constants and l signifies the characteristic length. w 0 0 0 With these considerations, governing equations are as follows (2017, 2016): ∂ ∂ (ru) + (rv) =0(1) ∂x ∂r ∂u ∂u μ 1 ∂ ∂u σ nf nf u + v = r − B u (2) ∂x ∂r ρ r ∂r ∂r ρ nf nf ∂T ∂T μ 1 ∂ ∂T nf u + v = r (3) ∂x ∂r (ρc ) r ∂r ∂r nf ∂a ∂a 1 ∂ ∂a 1 1 1 u + v = D r − k a b (4) A 1 1 ∂x ∂r r ∂r ∂r ∂b ∂b 1 ∂ ∂b 1 1 1 u + v = D r + k a b (5) B 1 1 1 ∂x ∂r r ∂r ∂r Boundary conditions are ∂T ∂a ∂b 1 1 ⎬ u = U ,v = 0,κ =−q , D = k a , D =−k a at r = R w f w A s 1 B s 1 (6) ∂r ∂r ∂r u = 0, T → T , a → a , b →0as r →∞ ∞ 1 0 1 Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 340 MHD Carbon nanotube flow over a stretching cylinder using DTM Here, (u,v) are the velocity components toward the (x, r)directionsand T is the temperature. D and D stand for the diffusion rate A B of chemical species A and B, respectively. Nanofluid density ( ρ ), effective heat capacity (ρc ) , dynamic viscosity (μ ), and thermal nf p nf nf conductivity (κ ) are given by (2018) nf ρ = (1 − φ) ρ + φρ , (ρc ) = (1 − φ) (ρc ) + φ(ρc ) nf f CNT p p p nf f CNT⎪ κ +κ κ CNT f CNT (1 − φ) + 2φ ln (7) μ κ f nf κ −κ 2κ CNT f f μ = , = ⎪ nf 2.5 κ κ +κ f CNT f ⎭ (1 − φ) f (1 − φ) + 2φ ln κ −κ 2κ CNT f f The subscripts CNT and f are used for separate values for nanoparticles and base fluid, respectively, and φ is the nanoparticle con- centration. We define the following dimensionless similarity transformations: 2 2 ⎪ r − R U ⎪ 2 ⎪ 2 ⎪ η = ,ψ = (v U x) Rf (η) f w ⎪ 2R v x (8) 1 ⎪ q v x a b f ⎪ w 2 1 1 T = T + θ (η) , g (η) = ,χ (η) = ⎭ κ U a a f w 0 0 1 ∂ψ 1 ∂ψ where f (η)and θ(η) are the dimensionless functions and steam function ψ satisfies equation ( 1), also defined as u = ,v =− . r ∂r r ∂x Using the substitution mentioned in equation (8) into the equations (2)–(5), we received subsequent similarity equations 2.5 CNT (1 + 2γη) f + 2γ f + (1 − φ) (1 − φ) + φ ff − f − Mf =0(9) nf 1 + 2γη θ + 2γθ + Pr f θ − f θ = 0 (10) [( ) ] ( ) (ρc ) CNT (1 − φ) + φ (ρc ) [(1 + 2γη) g + 2γ g ] + fg − λgχ = 0 (11) Sc [(1 + 2γη) χ + 2γχ ] + f χ + λgχ = 0 (12) Sc Accordingly, boundary conditions (6) become f (0) = 0, f (0) = 1,θ (0) =−1, g (0) = K g(0),δχ (0) =−K g(0) s s (13) f η → 0,θ η → 0, g η → 1,χ η →0as η →∞ ( ) ( ) ( ) ( ) 1 μ f (c p) v f l f σnf B0 l v f where γ = ( ) is the curvature parameter, Pr = is the Prandtl number, M = is the revised magnetic parameter, Sc = U l κ ρ U D 0 f f 0 A k a l v l 1 k f 0 s is the Schmidt number, λ = and K = are, respectively, the homogeneous and heterogeneous chemical reaction factors, U D U 0 A 0 and δ = is the ratio of mass diffusion coefficient. We assume that the diffusion rates of two chemical species A and B are of a comparable size. Further, we consider that D = D , A B i.e. δ = 1; so we consider from the assumption that g(η) + χ(η) = 1. Therefore, (11)and (12) converge to (1 + 2γη) g + 2γ g + Sc f g − λScg(1 − g) = 0 (14) with the reformed boundary conditions f (0) = 0, f (0) = 1,θ (0) =−1, g (0) = K g(0) (15) f (η) → 0,θ (η) → 0, g (η) →1as η →∞ For engineering and practical purposes, we are alert in the following factors: skin friction coefficient and Nusselt number. These are defined as 2μ ∂u nf C = f ⎪ 2 ⎬ ρ u ∂r nf w r =R (16) x ∂T Nu =− (T − T ) ∂r w ∞ r =R By using (8), we obtained reduced skin friction coefficient and reduced Nusselt number as C = Re .C = f (0)⎪ fr f ⎪ 2.5 (1 − φ) (17) 1 ⎪ κ 1 − nf ⎪ 2 ⎪ Nu = Re .Nu = ⎭ r x x κ θ (0) U x Re = is local Reynolds number. f Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 341 Table 2: Differential transform of some functions [34]. Original function Transformed function x(t) = αg(t) ± βh(t) X[k] = αG(k) ± β H(k) d f (t) (k+m)! x(t) = X[k] = F [k + m] dt k! x(t) = g(t)h(t) X[k] = G[m]H[k − m] m=0 1, if k = m x(t) = t X[k] = δ(k − m) = 0, if k = m Table 3: Variation of θ(0) for different values of Pr when M = 0,φ = 0,γ = 0 and in the absence of a chemical reaction. Pr Elbashbeshy (1998) Bachok & Ishak (2010) Present Result 0.72 1.2253 1.2367 1.236582 1.00 1.0000 1.0000 0.999999 6.70 — 0.3333 0.333303 10.0 0.2688 0.2688 0.268768 3. Numerical Experiment 3.1 DTM Let f (t) be analytic in a domain T and t ∈ T. Then, DTM of f (t)is 1 d f (t) F (k) = (18) k! dt t=t where F (k) is termed as T-function or a modestly transformed form of f (t). Now, inversion differential transform of F (k)isdefinedas f (t) = (t − t ) F (k) (19) k=0 From (19) with the help of (18), we have 1 d f (t) f (t) = (t − t ) (20) k! dt t=t k=0 We rewrite the equations (9)and (10)as (1 + 2γη) f + 2γ f + A ff − f − BM f = 0 (21) C [(1 + 2γη) θ + 2γθ ] + Pr ( f θ − f θ) = 0 (22) nf ρ f 2.5 CNT 2.5 where A = (1 − φ) ((1 − φ) + φ( )), B = (1 − φ) ,and C = . (ρc ) ρ p f CNT [(1−φ)+φ ] (ρc ) Applying DTM ( Basic operations of DTM are listed in Table 2)on(21), we get (k + 1)(k + 2)(k + 3) F [k + 3] + 2γ  (k − m − 1)(m + 1)(m + 2)(m + 3) F [m + 3]⎪ m=0 ⎪ +2γ (k + 1)(k + 2) F [k + 2] + A F [k − m](m + 1)(m + 2) F [m + 2] (23) m=0 ⎪ −A (k − m + 1) F [k − m + 1](m + 1) F [m + 1] − BM (k + 1) F [k + 1] = 0 m=0 1,m=1 where (m) ={ . 0,m=1 The corresponding boundary conditions are transformed to F [0] = 0, F [1] = 1, F [2] = a (24) Similarly, from (22)wehave C (k + 1)(k + 2)  [k + 2] + 2γ C  (k − m − 1)(m + 1)(m + 2)  [m + 2] m=0 (25) k k +2γ C (k + 1)  [k + 1] + Pr F [k − m](m + 1)  [m + 1] − Pr  [k − m](m + 1) F [m + 1] = 0 m=0 m=0 Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 342 MHD Carbon nanotube flow over a stretching cylinder using DTM Table 4: The values of f (1) for various values of M. Values of f (1) by DTM Values of f (1) by RK − 4 M SWCNT MWCNT SWCNT MWCNT 1.0 −0.309864 −0.310918 −0.309864107 −0.310917717 2.0 −0.289198 −0.290432 −0.289197717 −0.290431594 3.0 −0.265845 −0.267031 −0.265845308 −0.267030972 4.0 −0.243334 −0.244418 −0.243333817 −0.244418114 Table 5: The values of θ (1) for various values of M. Values of θ (1) by DTM Values of θ (1) by RK − 4 M SWCNT MWCNT SWCNT MWCNT 1.0 −0.118424 −0.118324 −0.118423879 −0.118324356 2.0 −0.127243 −0.127255 −0.127243216 −0.127254875 3.0 −0.134169 −0.134243 −0.134168985 −0.134242744 4.0 −0.139950 −0.140063 −0.139950380 −0.140063290 Table 6: The values of g (1) for various values of K . Values of g (1) by DTM Values of g (1) by RK − 4 K SWCNT MWCNT SWCNT MWCNT 1.0 0.091936 0.095787 0.091936031 0.095786678 2.0 0.956196 0.123987 0.119561956 0.123986728 3.0 0.133502 0.138204 0.133501858 0.138204086 4.0 0.141956 0.141669 0.141955901 0.141669055 Figure 2: Effect of curvature parameter (γ ) on velocity distribution. Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 343 Figure 3: Effect of curvature parameter (γ ) on temperature distribution. Figure 4: Effect of curvature parameter (γ ) on concentration profiles. The relevant boundary conditions become [0] = b, [1] =−1 (26) Again applying DTM on (14), we get (k + 1)(k + 2)G [k + 2] + 2γ  [k − m − 1](m + 1)(m + 2)G [m + 2]⎪ m=0 ⎪ +2γ (k + 1)G [k + 1] + Sc F [k − m](m + 1)G [m + 1] − λScG [k] (27) m=0 ⎪ k k k−t +2λSc G [k − m]G [m] − λSc G [t]G [m] G [k − m − t] = 0 m=0 t=0 m=0 Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 344 MHD Carbon nanotube flow over a stretching cylinder using DTM Table 7: Reduced skin friction coefficient and reduced Nusselt number for various parameters. C Nu f r γ M φ SWCNT MWCNT SWCNT MWCNT 0.0 −3.742635 −3.712047 3.686848 3.526191 0.1 −3.847419 −3.817071 3.746504 3.583227 0.2 −3.954626 −3.901835 3.835886 3.668690 0.3 −4.049939 −4.019968 3.865710 3.697204 0.0 −2.801107 −2.741811 4.203039 4.021273 1.0 −3.369516 −3.332956 4.025346 3.851202 2.0 −4.049939 −4.019968 3.865710 3.697204 4.0 −5.124731 −5.101383 3.611410 3.452540 0.00 −3.647186 −3.647186 2.754994 2.754994 0.05 −4.049939 −4.019968 3.865710 3.697204 0.10 −4.406153 −4.342640 4.852997 4.501512 0.15 −4.682546 −4.580576 5.718392 5.169770 Figure 5: Effect of magnetic-field parameter ( M) on velocity distribution. The relevant boundary conditions become G [0] = β, G [1] = K β (28) where F [k],[k], and G[k] are the differential transform of f (η),θ(η), and g(η)at η = 0, respectively, and a, b, and β are the constants that have to be determined by using boundary conditions. Now the iteration schemes are as follows: F[3] = (BM + A − 4γ a) 3! F[4] = 2Aa + 2BMa − 2BMγ − 2Aγ + 8aγ 4! 2 3 2 2 2 2 2 F[5] = 16aγ − 16aγ + 4Aa + 4Aγ − 4Aγ − 4aAγ − 8aBMγ − 4BMγ + B M + BMA + 4BMγ 5! ⎛ ⎞ 2 2 3 4 2 −24aAγ − 24aBMγ + 32BMγ + 32Aγ − 128aγ + 32aγ − 24a Aγ ⎝ ⎠ F[6] = 6! 3 2 3 2 2 2 2 −8Aγ + 8aBMγ − 8BMγ + 2aA + 2A γ − 2aB M + 2ABMγ and so on. [2] = (2γ C + Pr .b) 2!C Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 345 Figure 6: Effect of magnetic-field parameter ( M) on temperature distribution. Figure 7: Effect of curvature parameter (γ ) and magnetic-field parameter ( M) on Nusselt number. [3] = 2ab Pr −2bγ Pr −4γ C 3!C 2 2 3 2 2 [4] = bCBM Pr +AbC Pr −8γ C + 8γ C − 4bC γ Pr + 4bC γ Pr − 2aC Pr − 2C γ Pr − b(Pr) − 12abC γ Pr 4!C and so on. G [2] = λSc(1 − β) − 2γ K G [3] = ScK β(3λβ − 4λβ + λ − 1) − 2γ λSc(1 − β) − 2γ K s s 6 Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 346 MHD Carbon nanotube flow over a stretching cylinder using DTM Figure 8: Effect of nanoparticle concentration (φ) on velocity distribution. Figure 9: Effect of nanoparticle concentration (φ) on temperature distribution. ⎡    ⎤ λ 3 2 2 Sc + λβ Sc − 2γ − Sc − 2λβSc β λSc(1 − β) − 2γ K − γ ScK β(3λβ s s ⎢ ⎥ 2 2 ⎢ ⎥ G [4] = ⎣ ⎦ 12 2 2 2 −4λβ + λ − 1) − 2γ λSc(1 − β) − 2γ K − aScβK + (2 − 3β)λScβ K s s and so on. From (19), we get f (η), θ(η), and g(η) as follows: 1 1 2 3 2 4 f (η) = η + aη + (BM + A − 4γ a) η + 2Aa + 2BMa − 2BMγ − 2Aγ + 8aγ η 3! 4! 2 3 2 2 2 2 2 5 + 16aγ − 16aγ + 4Aa + 4Aγ − 4Aγ − 4aAγ − 8aBMγ − 4BMγ + B M + BMA + 4BMγ η 5! Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 347 Figure 10: Effect of homogeneous chemical reaction factor (λ) on concentration profiles. Figure 11: Effect of heterogeneous chemical reaction factor (K ) on concentration profiles. 2 2 3 4 2 3 2 + − 24aAγ − 24aBMγ + 32BMγ + 32Aγ − 128aγ + 32aγ − 24a Aγ − 8Aγ + 8aBMγ 6! 3 2 2 2 2 6 − 8BMγ + 2aA + 2A γ − 2aB M + 2ABMγ η + .... (29) 1 1 2 2 3 θ(η) = b − η + (2γ C + Pr .b) η + 2ab Pr −2bγ Pr −4γ C η 2!C 3!C # $ 2 2 3 2 2 bCBM Pr +AbC Pr −8γ C + 8γ C − 4bC γ Pr +4bC γ Pr −2aC Pr + η + ... (30) 2 2 4!C −2C γ Pr −b(Pr) − 12abC γ Pr Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 348 MHD Carbon nanotube flow over a stretching cylinder using DTM Figure 12: Effect of Schmidt number (Sc) on concentration profiles. ⎡ ⎤ ScK β(3λβ − 4λβ + λ − 1) β 1 2 3 ⎣ ⎦ g(η) = β + K βη + λSc(1 − β) − 2γ K η +   η s s 2 6 −2γ λSc(1 − β) − 2γ K ⎡ ⎤ λ 3 2 2 Sc + λβ Sc − 2γ − Sc − 2λβSc β λSc(1 − β) − 2γ K ⎢ ⎥ 2 2 ⎢ ⎥ ⎢ ⎥ + ⎢ 2 ⎥ η + ..... (31) −γ ScK β(3λβ − 4λβ + λ − 1) − 2γ λSc(1 − β) − 2γ K s s 12 ⎢ ⎥ ⎣ ⎦ 2 2 −aScβK + (2 − 3β)λScβ K Considering γ = 0.3, M = 5.0, Pr = 1.0, and φ = 0.0 for the equations (29)and (30) and linking with the boundary conditions (15)at η →∞ (we accept η = 10 in large sense of η), we easily get a = 0.329877091, b = 0.596315085, and β = 0.778332251. 3.2 Solution by RK − 4 method Currently, we move onward to resolve the equations (9), (10), and (14) numerically using (15)byalteringthemtoaninitial-value problem. We transform the differential equations (9), (10)and (14), to a system of first-order differential equation as follows: y = y 2 ⎪ y = y %    & (32) 1 ρ ⎪ 2.5 CNT 2 ⎪ y = (1 − φ) My − 2γ y − (1 − φ) + φ y y − y 2 3 1 3 3 2 1 + 2γη ρ y = y ⎪ 4 ⎬ '   ( 1 κ (ρc ) (33) f p CNT y = − . Pr . (1 − φ) + φ (y y − y y ) − 2γ y .⎪ 1 5 2 4 5 1 + 2γη κ (ρc ) nf p y = y ⎬ 1 (34) y = λScy (1 − y ) − 2γ y − Scy y . 6 6 7 1 7 1 + 2γη where y , y , y , y , y , y , and y are used for f (η), f (η), f (η), θ(η), θ (η), g(η), and g (η), respectively, and the prime signifies differenti- 1 2 3 4 5 6 7 ation with respect to η. Thus, boundary condition (15) reduces to y = 0, y = 1, y =−1, y = y K at η = 0 1 2 5 7 6 s (35) y → 0, y → 0, y →1as η →∞ 2 4 6 To get the complete solution of the IVP (32), (33), and (34) we have required the values of y (0), i.e. f (0), y (0), i.e. θ(0) and y (0), 3 4 6 i.e. g(0), which are absent in boundary conditions (35). Hence, to begin integration outline we make a guess of y (0), y (0), and y (0). 3 4 6 Hence, reduced ODE (32), (33), and (34) with boundary condition (35) is resolved numerically aiming the optimizing of the solution Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 349 with y (0), y (0), and y (0). We adjust the values y (0), y (0), and y (0) for the improved approximation of solution. We adopt RK − 4 3 4 6 3 4 6 method for different step sizes as η = 0.01, 0.001, 0.0001, etc. The overhead technique is repeated to acquire the results with a desired −6 accuracy of 10 . 3.3 Testing of code To check the precision of numerical results plus code involved in extant unit, we compared the dissimilar values of θ(0) for several values of Pr with similar demonstration in diverse research works like Elbashbeshy (1998) and Bachok and Ishak (2010)when M = 0,φ = 0, and γ = 0 and in the absence of a chemical reaction. In Table 3 we have organized those values of θ(0) that are outstandingly approved with preceding research work stated above. Assessment among the numerical results achieved by RK − 4 method and analytical solution completed by DTM is arranged in Tables 4–6. 4. Results and Discussions Current article includes parametric workout of the flow region. Comprehensive discussion we have exemplified via tables and graph, to scrutinize the outcome of diverse emerging factors like curvature factor γ , modified magnetic-field factor M, and nanoparticles concentration φ. Default values of emerging factors throughout model are reflected as Pr = 6.2,M = 2.0,γ = 0.3,φ = 0.05, and λ = Sc = K = 1.0exceptifnot specified. 4.1 Effect of curvature parameter γ Figure 2 sketched on the circumstance that how curvature parameter acts on fluid velocity. We perceived the augmentation in fluid velocity when γ amplifies. Curvature parameter and radius of curvature are inversely proportional to each other. Therefore γ increases implies radius of curvature, i.e. the radius of the cylinder decreases consequently less resistance offers toward the fluid flow due to lessens of contact surface area with the fluid and hence fluid velocity progresses. Temperature distribution due to curvature parameter γ has been portrayed in Fig. 3. Temperature distribution amplifies like velocity profiles for cumulative γ . Since Kelvin temperature of substances is defined as average kinetic energy of the particles of substances, so as velocity profiles enhanced with γ accordingly kinetic energy upsurges and consequently temperature profiles amplify. Velocity profiles and temperature distribution are higher in MWCNT than SWCNT. Figure 4 demonstrates that concentration outlines uphold in an identical fashion like temperature outlines when γ escalates and subsequently concentration boundary layers get thicker. Concentration field intensifies due to species diffusion in flow regime. From Table 7, we detected that Nu upsurges when γ escalates but C diminishes and Nu is advanced in SWCNT r f r than MWCNT but the converse was perceived for C 4.2 Effect of magnetic-field parameter M Figure 5 depicts the response of the velocity fields due to magnetic-field parameter M. The graphical interpretation endorses the decline of fluid velocity when the strength of magnetic field gets enlarged. Physical explanation of such a result is that, when the strength of engaged magnetic field upsurges in nanofluid, it starts to spread resistive force specifically Lorentz force, which obstructs fluid motion. Figure 6 exposed how magnetic-field factor M acts on temperature field. We appreciate that temperature distribution boosted with M. Since supplementary work done needed to stretch out nanofluid in opposition to the object of employed magnetic field, which heats up carrying nanofluid and enhances temperature profiles. Velocity profiles and temperature outlines are higher in MWCNT than SWCNT. For growing M from Table 7, we appreciated that C and Nu are reduced and Nu is advanced in SWCNT than fr r r MWCNT equivalent can be established from Fig. 7 but the converse was perceived for C . 4.3 Effect of nanoparticle concentration φ Response of the velocity field due to nanoparticle concentration has been manifested in Fig. 8. We perceived the enhancement in fluid velocity for cumulative φ and hence momentum boundary layer gets thicker. Temperature distribution due to nanoparticles concentration φ has been exposed in Fig. 9 and it disclosed that temperature outlines upsurge with φ. This agreed with physical characteristic that φ increases indicates thermal conductivity accelerates and correspondingly thermal boundary-layer thickness gets increased. Velocity profiles and temperature distribution are higher in MWCNT than SWCNT. Table 7 acknowledged that Nu upsurges with φ and is reverse in C . fr 4.4 Effect of chemical reaction parameters Homogeneous chemical reaction factor (λ) amplifies the spices B, which suspended nanoparticles in it. So nanoparticles’ production enhances. This justifies that g is augmented in the system as elucidated in Fig. 10.InFig. 11, we uncover the clout of heterogeneous chemical reaction factor (K ) on the flow. First-order heterogeneous reaction takes place in the existence of solid absorbing heated flow surface, which absorbs A and B and then a reaction takes place and resultant spices B came out of the wall thermally charged nanoparticle in it. So g upturns. Concentration profiles are higher in SWCNT than MWCNT. Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 350 MHD Carbon nanotube flow over a stretching cylinder using DTM 4.5 Effect of Schmidt number (Sc) Schmidt number is identified as a ratio of the rate of momentum diffusivity to the rate of mass diffusivity; therefore, higher values of Sc imply small mass diffusion and as an upshot concentration profiles decline, which are portrayed in Fig. 12. Concentration profiles are higher in SWCNT than MWCNT. 5. Final Remarks In extant article, we cultivate hydromagnetic incompressible nanofluid flow through a stretching cylinder with homogeneous– heterogeneous chemical reaction. SWCNTs as well as MWCNTs nanoparticles are accounted here. We rehabilitated PDEs to ODEs by paying similarity approach and then resolved analytically by engaging DTM as well as numerically by employing Runge–Kutta– Fehlberg quadrature method through shooting practice. Through our discussion, we found many outcomes, among which specific conclusions are noted below: Impacts of γ and φ on velocity profiles are identical. Temperature profiles are amplified with γ, M, and φ. Velocity profiles and temperature distribution are higher in MWCNT than SWCNT and reverse for concentration profiles in each case. Concentration profiles amplify for chemical reaction factor. C in MWCNT is advanced than SWCNT in each case but we acknowledged reverse in Nu . fr r C is reduced when γ, M, and φ amplify. Acknowledgement The authors wish to express their cordial thanks to reviewers for giving valuable suggestions and comments to improve the presenta- tion of this article. Finally, the authors wish to express their sincere thanks to the editor in chief and the production office for primary consideration and correcting the mistakes for publication in Journal of Computational Design and Engineering. Conflict of interest statement Declarations of interest: none. References Ahmed, Z., & Nadeem, S. (2019). Flow of a micropolar CNT-based nanofluid across a squeezing channel. Physica Scripta, 94, 10. Ahmed, Z., Qahtani, A. A., Nadeem, S., & Saleem, S. (2019). Computational study of MHD nanofluid flow possessing micro-rotational inertia over a curved surface with variable thermophysical properties. Processes, 387(7), 1–15. Alavi, S. Q., Hussanan, A., Kasim, A. R. M., Rosli, N., & Salleh, M. Z. (2017). MHD stagnation point flow towards an exponentially stretching sheet with prescribed wall temperature and heat flux. International Journal of Applied and Computational Mathematics, 3(4), 3511–3523. Ayaz, F. (2013). On the two-dimensional differential transformation method. Applied Mathematics and Computation, 143(2–3), 361–374. Bachok, N., & Ishak, A. (2010). Flow and heat transfer over a stretching cylinder with prescribed surface heat flux. Malaysian Journal of Mathematical Sciences, 4, 159–169. Bachok, N., Ishak, A., & Pop, I. (2011). On the stagnation-point flow towards a stretching sheet with homogeneous–heterogeneous reactions effects. Communication in Nonlinear Science and Numerical Simulation, 16, 4296–4302. Bhatti, M. M., Abbas, T., & Rashidi, M. M., (2017). Entropy generation as a practical tool of optimization for non-Newtonian nanofluid flow through a permeable stretching surface using SLM. Journal of Computational Design and Engineering, 4, 21–28. Choi, S. U. S. (1995). Developments and application of non-Newtonian flows .New York,USA:ASMEPress. Daniel, Y. S., Aziz, Z. A., Ismail, Z., & Salah, F., (2018). Thermal stratification effects on MHD radiative flow of nanofluid over nonlinear stretching sheet with variable thickness. Journal of Computational Design and Engineering, 5, 232–242. Dhlamini, M., Kameswaran, P. K., Sibanda, P., Motsa, S., & Mondal, H., (2019). Activation energy and binary chemical reaction effects in mixed convective nanofluid flow with convective boundary conditions. Journal of Computational Design and Engineering, 6, 149–158. Elbashbeshy, E. M. A. (1998). Heat transfer over a stretching surface with variable surface heat flux. Journal of Physics D: Applied Physics, 31, 1951–1954. Erfani, E., Rashidi, M. M., & Parsa, A. B. (2010). The modified Differential Transform Method for solving off-centered stagnation flow towards a rotating disc. International Journal of Computational Methods, 7(4), 655–670. Giri, S. S., Das, K., & Kundu, P. K. (2018). Framing the features of a Darcy–Forchheimer nanofluid flow past a Riga plate with chemical reaction by HPM. The European Physical journal Plus, 133(365), 1–17. Hayat, T., Nadeem, S., & Khan, A. U. (2019). Aspects of 3D rotating hybrid CNT flow for a convective exponentially stretched surface. Applied Nanoscience, https://doi.org/10.1007/s13204-019-01036-y. Hussain, A., Javed, F., & Nadeem, S. (2019). Numerical Solution of a Casson nanofluid flow and heat transfer analysis between con- centric cylinders. Journal of Power Technologies, 99(1), 25–30. Hussain, S. T., Haq, R. U., Khan, Z. H., & Nadeem, S. (2016). Water-driven flow of carbon nanotubes in a rotating channel. Journal of Molecular Liquids, 214, 136–144. Ishak, A, Nazar, R., & Pop, I. (2008). Uniform suction/blowing effect on flow and heat transfer due to a stretching cylinder. Applied Mathematical Modelling, 32, 2059–2066. Downloaded from https://academic.oup.com/jcde/article/7/3/337/5818507 by DeepDyve user on 27 August 2020 Journal of Computational Design and Engineering, 2020, 7(3), 337–351 351 Jang, M. J., Chen, C. L., & Liu, Y. C. (2001). Two-dimensional differential transform for partial differential equations. Applied Mathematical Computing, 121, 261–270. Kardri, M. A., Bachok, N., Arifin, N. M., & Ali, F. M. (2017). Boundary layer flow and heat transfer over a stretching cylinder in a copper- water nanofluid. American Institute of Physics, 1795, 020012. Kumar, R., Sood, S., Sheikholeslami, M., & Shehzad, S. A. (2017). Nonlinear thermal radiation and cubic autocatalysis chemical reaction effects on the flow of stretched nanofluid under rotational oscillations. Journal of Colloid and Interface Science, 505, 253–265. Majeed, A., Zeeshan, A., & Ellahi, R. (2016). Unsteady ferromagnetic liquid flow and heat transfer analysis over a stretching sheet with the effect of dipole and prescribed heat flux. Journal of Molecular liquids, 223, 528–533. Mondal, H., Almakki, M., & Sibanda, P., (2019). Dual solutions for three-dimensional magnetohydrodynamic nanofluid flow with en- tropy generation. Journal of Computational Design and Engineering, 6, 657–665. Muhammad, N., Nadeem, S., & Mustafa, M. T. (2019). Hybrid isothermal model for the ferrohydrodynamic chemically reactive species. Communication in Theoretical Physics, 71, 384–392. Munawar, S., Mehmood, A., & Ali, A. (2012). Time-dependent flow and heat transfer over stretching cylinder. Chinese Journal of Physics, 50(5), 828–848. Nadeem, S., Abbas, N., & Khan, A. U. (2018). Characteristics of three dimensional stagnation point flow of hybrid nanofluid past a circular cylinder. Results in Physics, 8, 829–835. Nadeem, S., Hayat, T., & Khan, A. U. (2019b). Numerical study on 3D rotating hybrid SWCNT-MWCNT flow over a convectively heated stretching surface with heat generation/absorption. Physica Scripta, 94,7. Nadeem, S., Khan, M. R., & Khan, A. U. (2019a). MHD oblique stagnation point flow of nanofluid over an oscillatory stretching/shrinking sheet: Existence of dual solutions. Physica Scripta, 94,7. Nasir, S., Shah, Z., Islam, S., Bonyah, E., & Gul, T. (2019b). Darcy Forchheimer nanofluid thin film flow of SWCNTs and heat transfer analysis over an unsteady stretching sheet. AIP Advances, 9, 015223. Nasir, S., Shah, Z., Islam, S., Khan, W., & Khan, S. N. (2019a). Radiative flow of magneto hydrodynamics single-walled carbon nanotube over a convectively heated stretchable rotating disk with velocity slip effect. Advances in Mechanical Engineering, 11(3), 1–11. Noor, M. A., Mohyud-Din, S. T., & Waheed, A. (2008). Variation of parameter method for solving fifth-order boundary value problems. Applied Mathematics and Information Sciences, 2, 135–141. Rana, S., Mehmood, R., & Akbar, N. S. (2016). Mixed convective oblique flow of a Casson fluid with partial slip, internal heating and homogeneous–heterogeneous reactions. Journal of Molecular Liquids, 222, 1010–1019. Rashidi, M. M. (2009). The modified differential transform method for solving MHD boundary-layer Equations. Computer Physics Com- munication, 180 (11), 2210–2217. Rashidi, M. M., Beg, ´ O. A., Asadi, M., & Rastegari, M. T. (2012a). DTM-Pade´ modeling of natural convective boundary layer flow of a nanofluid past a vertical surface. International Journal of Thermal & Environmental Engineering, 4(1), 13–24. Rashidi, M. M., Chamkha, A. J., & Keimanesh, M. (2011b). Application of multi-step differential transform method on flow of a second- grade fluid over a stretching or shrinking sheet. American Journal of Computational Mathematics, 1(2), 119–128. Rashidi, M. M., & Erfani, E. (2011). The modified differential transform method for investigating nano boundary-layers over stretching surfaces. International Journal of Numerical Methods for Heat & Fluid Flow, 21(7), 864–883. Rashidi, M. M., Hayat, T., Keimanesh, M., & Yousefian, H. (2013). A Study on heat transfer in a second-grade fluid through a porous medium with the modified differential transform method. Heat Transfer—Asian Research, 42(1), 31–45. Rashidi, M. M., Laraqi, N., & Parsa, A. B. (2011a). Analytical Modeling of heat convection in magnetized micropolar fluid by using modified differential transform method. Heat Transfer—Asian Research, 40(3), 187–204. Rashidi, M. M., Rahimzadeh, N., Ferdows, M., Uddin, M. J., & Beg, O. A. (2012b). Group theory and differential transform analysis of mixed convective heat and mass transfer from a horizontal surface with chemical reaction effects. Chemical Engineering Communi- cations, 199(8), 1012–1043. Shah, Z., Bonyah, E., Islam, S., & Gul, T. (2019). Impact of thermal radiation on electrical MHD rotating flow of Carbon nanotubes over a stretching sheet. AIP Advances, 9, 015115. Shah, Z., Dawar, A., Islam, S., Khan, I., & Ching, D. L. C. (2018). Darcy–Forchheimer flow of radiative carbon nanotubes with microstruc- ture and inertial characteristics in the rotating frame. Case Studies in Thermal Engineering, 12, 823–832. Shahzadi, I., Nadeem, S., & Rabiei, F. (2017). Simultaneous effects of single wall carbon nanotube and effective variable viscosity for peristaltic flow through annulus having permeable walls. Results in Physics, 7, 667–676. Sheikholeslami, M. (2015). Effect of uniform suction on nanofluid flow and heat transfer over a cylinder. The Brazilian Society of Me- chanical Sciences and Engineering, 37, 1623–1633. Sheikholeslami, M., Ellahi, R., Ashorynejad, H. R., & Hayat, T. (2014). Effects of heat transfer in flow of nanofluids over a permeable stretching wall in a porous medium. Journal of Computational and Theoretical Nanoscience, 11, 486–496. Sheikholeslami, M., & Ganji, D. D. (2015). Nanofluid flow and heat transfer between parallel plates considering Brownian motion using DTM. Computer Methods in Applied Mechanics and Engineering, 283(1), 651–663. Sheikholeslami, M., Ganji, D. D., & Ashorynejad, H. R. (2013). Investigation of squeezing unsteady nanofluid flow using ADM. Powder Technology, 239, 259–265. Sheikholeslami, M., Rashidi, M. M., Alsaad, D. M., & Rokni, H. B. (2016). Steady nanofluid flow between parallel plates considering thermophoresis and Brownian effects. Journal of King Saud University – Science, 28, 380–389. Tamoor, M., Waqas, M., Khan, M. I., Alsaedi, A., & Hayat, T. (2017). Magnetohydrodynamic flow of Casson fluid over a stretching cylinder. Results in Physics, 7, 498–502. Wang, C. Y. (1998). Fluid flow due to a stretching cylinder. Physics of Fluids, 31, 466–468. Zhou, K. (1986). Differential transformation and its applications for electrical circuits. Wuhan, China: Huazhong University Press.

Journal

Journal of Computational Design and Engineering – Oxford University Press

Published: Jun 1, 2020

There are no references for this article.