Continuum Mech. Thermodyn. DOI 10.1007/s00161-017-0563-0

O R I G I NA L A RT I C L E

R. V. M. S. S. Kiran Kumar · S. Vijaya Kumar Varma · C. S. K. Raju · S. M. Ibrahim · G. Lorenzini · E. Lorenzini

Magnetohydrodynamic 3D slip flow in a suspension of carbon nanotubes over a slendering sheet with heat source/sink Received: 11 January 2017 / Accepted: 14 March 2017 © Springer-Verlag Berlin Heidelberg 2017

Abstract Carbon nanotubes are allotropes of carbon with a cylindrical nanostructure. These cylindrical carbon molecules have unusual properties, which are valuable for nanotechnology, electronics, optics and other fields of materials science and technology. With this intention, we investigate the three-dimensional magnetohydrodynamic convective heat and mass transfer of nanofluid over a slendering stretching sheet filled with porous medium and heat source/sink. For balancing the flow, temperature and concentration slip mechanisms are also taken into account. In this investigation simulation performed by mixing the two types of carbon nanotubes, namely single- and multi-walled carbon nanotubes, into water as base fluid. The governing system of partial differential equations is transformed into nonlinear ordinary differential equations which answered by using R–K–Fehlberg-integration scheme. The impact of various pertinent parameters on velocity, temperature and concentration as well as the friction factor coefficient, local Nusselt and local Sherwood number is derived and discussed through graphs and tables for both single- and multi-walled carbon nanotubes cases. It is found that the momentum boundary layer thickness of SWCNTs is thicker than MWCNTs. These results can help us to conclude that SWCNTs are helpful for minimizing the friction between the particles, whereas MWCNTs are helpful for boosting the heat and mass transfer rate. Communicated by Andreas Öchsner. R. V. M. S. S. K. Kumar · S. V. K. Varma Department of Mathematics, Sri Venkateswara University, Tirupati, Andhra Pradesh, India E-mail: [email protected] S. V. K. Varma E-mail: [email protected] C. S. K. Raju Department Of Mathematics, VIT University, Vellore, Tamilnadu, India E-mail: [email protected] S. M. Ibrahim Department of Mathematics, GITAM University, Visakhapatnam 530045, India E-mail: [email protected] G. Lorenzini (B) Department of Engineering and Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy E-mail: [email protected] E. Lorenzini Department of Industrial Engineering, Alma Mater Studiorum-University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy E-mail: [email protected]

R. V. M. S. S. K. Kumar et al.

Keywords Carbon nanotubes · Magnetohydrodynamic · Porous medium · Multiple slips · Variable thickness sheet · Heat source Nomenclature u, v, w Cp f, g A m B(x) T k Dm Cs C Tm T∞ C∞ j1∗ j2∗ j3∗ f1 a b d m Pr QH B(x) M K Nt Le Nb j1 j2 j3 Cf N ux Sh x Rex

Velocity components in x, y and z directions Specific heat capacity at constant pressure Dimensionless velocities Coefficient related to stretching sheet Velocity power index parameter Magnetic field parameter Temperature of the fluid Thermal conductivity Molecular diffusivity of the species concentration Concentration susceptibility Concentration of the fluid Mean fluid temperature Temperature of the fluid in the free stream Concentration of the fluid in the free stream Dimensional velocity slip parameter Dimensional temperature jump parameter Dimensional concentration jump parameter Maxwell’s reflection coefficient Thermal accommodation coefficient Physical parameter related to stretching sheet Concentration accommodation coefficient Velocity power index parameter Prandtl number Heat source/sink parameter Dimensional magnetic field parameter Magnetic interaction parameter Porosity parameter Thermophoresis parameter Lewis number Brownian motion parameter Dimensionless velocity slip parameter Dimensionless temperature jump parameter Dimensionless concentration jump parameter Wall skin friction coefficient Local Nusselt number Local Sherwood number Local Reynolds number

Greek symbols φ η σ γ θ ρnf knf μnf υf δ

Dimensionless concentration Similarity variable Electrical conductivity of the fluid Ratio of specific heats Dimensionless temperature Density of the nanofluid Thermal conductivity of the nanofluid Dynamic viscosity of nanofluid Kinematic viscosity Wall thickness parameter

Magnetohydrodynamic 3D slip flow in a suspension of carbon nanotubes

ξ1 , ξ 2 ξ3 , ξ4 Γ

Mean free path (constant) Mean free path (constant) Positive characteristic time

1 Literature survey The discovery of carbon nanotubes (CNTs) is initiated in 1991, and it is opened up a new era in materials science. These are the incredible structures that have an array of fascinating electronic, magnetic and mechanical properties. CNT is at least 100 times stronger than steel, but only one-sixth as heavy, so nanotube fibers could strengthen almost any material. Nanotubes heat conductivity is better than copper. CNTs are earlier used in polymers to control or enhance conductivity and are added to anti-static packaging. CNTs have various structures, differing in thickness, length and number of layers. The physical characteristics of nanotubes are different depending on how the graphene sheet is rolled up from the tube initiating it to act either metallic or a semiconductor. CNTs have numerous applications such as controlling the thermal conductivity, field emission, conductive properties, energy storage, conductive adhesive, molecular electronics based on CNTs, thermal materials and structural designing. The CNTs have inherently special physical and chemical properties like the thermal and magnetic performance and unique optical. Therefore, the CNTs can be observed as one of the most promising materials in all nanomaterials [1]. CNTs are a one-dimensional nanomaterial which has unique thermal, mechanical and optical properties [2]. Due to their high molecular weight they are considered as insoluble in almost known solvents. Nanotubes are categorized into two types, namely single-walled nanotubes (SWNTs) and multi-walled nanotubes (MWNTs). The most commonly used CNTs are single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). The SWNTs have only one single layer of grapheme cylinders, while the MWNTs have many layers [3]. By mixing them in the solid and fluid the authors [4–11] highlighted that CNTs have excellent thermal conductivity and mechanical properties. A new model of effective thermal conductivity of CNTs is presented by Xue [12]. He showed the theoretical results of the effective thermal conductivity of CNTs/oil and CNTs decent suspensions. These results show good agreement with the experimental data. The experimental and theoretical investigation of the CNTs thermal conductivity behavior is studied by Walvekar et al. [13]. Numerous studies have been previously performed on the heat transfer enhancement of nanofluids flowing through a tube under laminar regime. It is well admitted that experimental convective heat transfer of nanofluids varied with growing values of the velocity and volume fraction, whereas the carbon nanotubes are recently more advanced due to high stability in the flow. Khan et al. [14] found that the SWCNTs exhibit the thermal conductivity up to almost 15% and significantly higher than the nanoparticle-based nanofluids. SWCNTs have unique mechanical and electronic properties which can be used in numerous applications, such as field-emission displays, nanocomposite materials, nanosensors and logic elements. The peristaltic flow of carbon nanotubes in an asymmetric channel with thermal and velocity slip effects by taking two types of carbon nanotubes (SWCNTs and MWCNTs) is studied by Akbar [15]. The effect of SWCNTs in the existence of seawater and water with solar radiation energy under variable stream condition is investigated by Kandasamy et al. [16]. According to Murshed et al. [17], CNTs provide round about six times better thermal conductivity as compared to other materials at the room temperature. Peristaltic flow is used to study the flow and heat transfer of carbon nanotubes in an asymmetric channel with thermal and velocity slip effects studied by Akbar et al. [18]. Later, Khan et al. [19] studied the flow and heat transfer of CNTs over a flat plate with uniform heat flux and Navier slip conditions. The flow of water-based carbon nanotubes with the existence of particle volume fraction and temperature ranging from 0 to 40 ◦ C is investigated by Halelfadl et al. [20]. The magnetohydrodynamic flow along a vertical cone with variable wall temperature and saltwater solutions in CNTs is investigated by Ellahi et al. [21]. The hydromagnetic heat transfer analysis of water-based CNTs over a static/moving wedge is studied by Khan et al. [22]. The effects of magnetohydrodynamic and volume fraction of CNTs on the flow and heat transfer in two lateral directions over a stretching sheet by considering three different types of base fluids such as water, ethylene glycol and engine oil with single- and multi-walled CNTs were investigated by Haq et al. [23]. Recently, Raju et al. [24,25] studied magnetic nanoparticle with flow over a cone with constant viscosity and variable viscosity. With this they highlighted that the magnetic nanoparticles control the flow. CNT nanofluids were more stabilized using gum arabic (GA) when compared with ferrous nanoparticles. The concentration of CNTs was varied from 0.01 to 0.1 wt%, while the concentration of GA was varied from 1 to 2.5 wt%, respectively. With this motivation, in the present study, we examine the 3D magnetohydrodynamic convective heat and mass transfer of nanofluid over a slendering stretching sheet filled with porous medium in the presence of

R. V. M. S. S. K. Kumar et al.

heat source/sink. Resulting set of nonlinear ordinary differential equations (ODEs) are solved by using R–K– Fehlberg-integration scheme. The effect of various flow parameters is discussed graphically as well as tabular form in detail.

2 Formulation of the problem In investigation, we consider three-dimensional, electrically conducting hydromagnetic slip flow of nanofluid over a slendering stretching sheet by assuming Z = J (x + y + c)(1−n)0.5 n  = 1. The sheet is followed with a porous medium, and heat source/sink is taken into consideration. The variable heat source is defined as Q(x, y) = Q 0 (x + y + c)1/2 . The slip conditions on velocity, temperature and concentration are also considered. We consider the water as a base fluid and carbon nanotubes thermophysical properties. For this study, we define two types of carbon nanoparticles, namely SWNTs and MWNTs. The magnetic Reynolds number is assumed to be very low, thermophoresis and Brownian effects are also considered the CNTs, and base fluid is assumed to be in thermal equilibrium. Under the above assumptions and with the usual boundary layer approximations, the governing equations are (Khadar and Megahed [26] and Anjalidevi and Prakash [27]) as follows ∂u ∂v ∂w + + = 0, ∂x ∂y ∂z   ∂ 2u ∂v ∂w μnf ∂u ρnf u +v +w = μnf 2 − σ B 2 (x)u − u, ∂x ∂y ∂z ∂z k0   ∂ 2v ∂v ∂v μnf ∂v +v +w = μnf 2 − σ B 2 (x)v − v, ρnf u ∂x ∂y ∂z ∂z k0         ∂T ∂T DB ∂ T 2 ∂T ∂2T ∂C ∂ T , +v +w + = K nf 2 − Q(x, y)(T − T∞ ) + τ DB ρcp nf u ∂x ∂y ∂z ∂z ∂z ∂z T∞ ∂z u

∂C ∂ 2C DT ∂ 2 T ∂C ∂C , +v +w = DB 2 + ∂x ∂y ∂z ∂z T∞ ∂z 2

(1) (2) (3) (4) (5)

Fig. 1 The physical model of flow configuration Table 1 Thermophysical properties of the water and the carbon nanotube nanoparticles Physical properties

Pure water

SWCNT

MWCNT

ρ (kg/m3 ) Cp (J/kg K) k (W/m K)

997 4179 0.613

2600 425 6600

1600 796 3000

Magnetohydrodynamic 3D slip flow in a suspension of carbon nanotubes

Table 2 Comparison of the values of wall friction when M = K = j2 = j3 = Nt = Nb = Le = 0, m = 0.5 δ

j1

Khader and Megahed [26]

Present study

0.2 0.25 0.5

0 0.2 0.2

−0.924828 −0.733395 −0.759570

−0.924829 −0.733396 −0.759570

Table 3 The variations of wall skin friction, the rate of heat and mass transfer values for various physical governing parameters for single-walled carbon nanotube (SWCNT) case K

M

φ

Nb

δ

Nt

QH

SWCNT f  (0)

0.1 0.4 0.7 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

2 2 2 1 3 5 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.3 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.2 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.3 0.1 0.1 0.1 0.1 0.1 0.1

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.3 0.1 0.1 0.1

−0.895543 −0.963299 −1.022663 −0.905352 −0.932653 −0.958498 −1.022663 −0.905352 −0.902653 −0.958498 −0.918863 −0.857150 −0.734076 −0.921047 −0.921047 −0.921047 −0.921047 −0.934611 −0.921047 −0.921047 −0.921047

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 1 2 3

g  (0)

−θ  (0)

−φ  (0)

0.731337 0.711010 0.693201 0.728394 0.720204 0.712451 0.724341 0.742855 0.779777 0.723686 0.723686 0.723686 0.723686 0.723686 0.723686 0.727761 0.723686 0.719617 0.723686 0.723686 0.723686

1.148087 1.123460 1.100828 1.144598 1.134753 1.125246 1.139792 0.906558 0.692967 1.138731 0.941365 0.762565 1.038174 0.872693 0.738059 0.982688 1.041507 1.099786 1.116788 1.182537 1.240836

1.546825 1.523756 1.502375 1.543569 1.534360 1.525436 1.539083 1.633162 1.729343 1.538063 1.724017 1.758771 1.678201 1.583628 1.565169 1.318539 1.376228 1.431322 1.356919 1.339784 1.324373

Bold values indicate the variation of the specific parameter, while remaining parameters are fixed Table 4 The variations of wall skin friction, the rate of heat and mass transfer values for various physical governing parameters for single-walled carbon nanotube (MWCNT) case K

M

φ

Nb

Nt

δ

QH

MWCNT f  (0)

0.1 0.4 0.7 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

2 2 2 1 3 5 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.3 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.2 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.3 0.1 0.1 0.1 0.1 0.1 0.1

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.3 0.1 0.1 0.1

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 1 2 3

−0.946761 −1.008454 −1.062987 −0.921006 −1.011174 −1.086814 −0.968094 −0.977534 −0.897523 −0.969266 −0.969266 −0.969266 −0.969266 −0.969266 −0.969266 −0.957461 −0.969266 −0.981046 −0.969266 −0.969266 −0.969266

g  (0)

−θ  (0)

−φ  (0)

0.715972 0.697464 0.681104 0.723698 0.696648 0.673956 0.709572 0.716740 0.730743 0.709220 0.709220 0.709220 0.709220 0.709220 0.709220 0.712762 0.709220 0.705686 0.709220 0.709220 0.709220

1.132041 1.108792 1.087317 1.141435 1.107742 1.077648 1.124145 0.868115 0.645537 1.123530 0.926088 0.747914 1.022845 0.858707 0.725785 0.967438 1.026587 1.085256 1.104642 1.172410 1.232236

1.528807 1.506879 1.486461 1.537615 1.505884 1.477217 1.521383 1.591649 1.678531 1.520784 1.708494 1.743527 1.662264 1.564380 1.541359 1.299864 1.358263 1.414064 1.338255 1.320586 1.304752

Bold values indicate the variation of the specific parameter, while remaining parameters are fixed

R. V. M. S. S. K. Kumar et al.

Fig. 2 Effect of K on f  (ζ )

Fig. 3 Effect of K on g  (ζ )

The appropriate boundary conditions are    ∂u ∗ ∂v u (x, y) = u w (x) + , v (x, y) = vw (x) + j1 ∂z ∂z     ∂ T ∂C , C (x, y) = Cw (x) + j3∗ T (x, y) = Tw (x) + j2∗ ∂z ∂z j1∗



(6)

and u = 0, v = 0, T = T∞ , C = C∞ at z = ∞, where   ξ1 2 − f1 2γ ξ1 (x + y + c)(1−n)0.5 , ξ2 = f1 γ + 1 Pr    2−b 2γ ξ2 j2∗ = ξ2 (x + y + c)(1−n)0.5 , ξ3 = b γ + 1 Pr  n−1 2−d ξ3 (x + y + c)(1−n)0.5 , B(x) = B0 (x + y + c) 2 j3∗ = d

j1∗ =



(7) (8) (9)

Magnetohydrodynamic 3D slip flow in a suspension of carbon nanotubes

Fig. 4 Effect of K on θ(ζ )

Fig. 5 Effect of K on φ(ζ ) (ρC )

u w (x) = a(x + y + c)(n−1)0.5 , vw (x) = a(x + y + c)n , τ = (ρCpp )fs Tw (x) = T∞ + T0 (x + y + c)(1−n)0.5 , Cw (x) = C∞ + C0 (x + y + c)(1−n)0.5

for n  = 1 (10)

In this work we used the following definitions (ρCp )nf 1 ρnf μnf = = 1 − φ + φr, = (1 − φ) + φe, , μf (1 − 2.5φ) ρf (ρCp ) f (ρCp )s ks σs ρs , k= , σ = where r = , e = ρf (ρCp )f kf σf Now, we introduce the following similar transformations.   (n + 1) a 0.5 ξ =z (x + y + c)(n−1)0.5 2υ T = T∞ + (Tw (x) − T∞ ) θ, C = C∞ + (Cw (x) − C∞ ) Φ ∂f ∂g u = a (x + y + c)n , v = a (x + y + c)n ∂ξ ∂ξ

knf 3(k − 1)φ =1+ , kf k+2 (11)

(12) (13) (14)

R. V. M. S. S. K. Kumar et al.

Fig. 6 Effect of M on f  (η)

Fig. 7 Effect of M on g  (η)



2av w=− n+1

0.5

(n−1)0.5

(x + y + c)



n+1 ( f + g) + ξ 2



n−1 2



∂f ∂g + ∂ξ ∂ξ

 (15)

3 Method of solution In view of above similarity transformations the governing partial differential equations are transformed to the following nonlinear ordinary differential equations       ∂ f ∂g ∂f 2 n+1 ∂3 f n + 1 ∂3 f 1 − [(1 − φ) + φr ] n +n − ( f + g) 3 ∂ξ ∂ξ ∂ξ 2 ∂ξ (1 − φ)2.5 2 ∂ξ 3 −(M + K )

∂f =0 ∂ξ

(16)

Magnetohydrodynamic 3D slip flow in a suspension of carbon nanotubes

Fig. 8 Effect of M on θ(ζ )

Fig. 9 Effect of M on φ(ζ )

      ∂g 2 n+1 n + 1 ∂3g ∂ f ∂g ∂3g − [(1 − φ) + φr ] n +n − ( f + g) 3 ∂ξ ∂ξ ∂ξ 2 ∂ξ (1 − φ)2.5 2 ∂ξ 3 1

∂g =0 ∂ξ       n+1 knf ∂ 2 θ 2 ∂θ ∂g 1 − n ∂f − Pr − φ) + φe] + f + g) θ − [(1 ( ks ∂ξ 2 n−1 2 ∂ξ ∂ξ 2 ∂ξ  2 ∂θ ∂θ ∂ −Q H θ + Nb + Nt =0 ∂ξ ∂ξ ∂ξ    ∂f Nt ∂ 2 θ ∂g 1 − n ∂φ ∂ 2φ + − Le + φ − f + g) = 0, ( ∂ξ 2 Nb ∂ξ 2 ∂ξ ∂ξ 1 + n ∂ξ −(M + K )

(17)

(18) (19)

R. V. M. S. S. K. Kumar et al.

Fig. 10 Effect of φ on f  (ζ )

Fig. 11 Effect of volume fraction of nanoparticle on g  (ζ )

and corresponding boundary conditions are 



⎫   2 ⎪ ∂2 f 1 (0) = 1 + j ∂ f f (0) = A 1−n , f ,⎪ 1 + j ⎪ 1 ∂ξ 2 1 ∂ξ 2 ⎪ n+1 ⎪ ξ =0 ξ =0 ⎪ ⎪   ⎪

 ⎪ ⎪ 2 2

∂ g ∂ g 1−n 1 ⎬ g(0) = A n+1 1 + j1 ∂ξ 2 , g (0) = 1 + j1 ∂ξ 2 , ⎪ ξ =0 ξ =0   ⎪



⎪ ⎪

∂φ ⎪ ⎪ θ (0) = 1 + j2 ∂θ , φ(0) = 1 + j ,

3 ∂ξ ⎪ ∂ξ ξ =0 ⎪ ξ =0 ⎪ ⎪ ⎪ ⎪ ∂f ∂g ⎭ = 0, = 0, θ = 0, φ = 0 as ξ → ∞ ∂ξ ∂ξ

(20)

Magnetohydrodynamic 3D slip flow in a suspension of carbon nanotubes

Fig. 12 Effect of volume fraction of nanoparticle on θ(ζ )

Fig. 13 Effect of volume fraction of nanoparticle on φ(ζ )

where σf B02 μf (Cp )f τ DB C 0 , Nb = , , Pr = ρf a kf kf υf μf Q0 Le = , K = . , QH = DB ρf k0 a akf M=

Nt =

τ DT T0 , T∞ kf (21)

The engineering design quantities of physical interest include the friction factor coefficient, the rate of heat and mass transfer coefficients and are given by

    1 n + 1 0.5 ∂ 2 f

n + 1 0.5 ∂θ

− 21 2 , N u x (Re) = − , Cf (Re) = 2 2 ∂ζ 2 ζ =0 2 ∂ζ ζ =0

  1 ∂φ n + 1 0.5

Sh x = − (22) (Re) 2 2 ∂ζ ζ =0

R. V. M. S. S. K. Kumar et al.

Fig. 14 Effect of Nb on θ(ζ )

Fig. 15 Effect of Nb on φ(ζ )

Here Re = u w (x)(x+y+c) , υf Nonlinear differential equations (16)–(19) with boundary conditions (20) are solved numerically using Runge–Kutta Fehlberg-integration methods. Initially, the set of nonlinear ordinary differential equations are converted to first-order differential equations, by using the substitution derivatives. Primarily, we guess the values which are not given at the initial conditions. Equations (16)–(19) are integrated with help of Runge– Kutta Feldberg-integration scheme with the consecutive iterative step length that is 0.01. In order to validate the precision of the present solutions compared with Khader and Megahed [26] solutions, we found worthy agreement with those solutions.

4 Results and discussion The dimensional less governing equations (16)–(19) subject to boundary conditions (20) are solved numerically by using R–K–Fehlberg-integration scheme. The influence of various flow parameters on velocity, temperature

Magnetohydrodynamic 3D slip flow in a suspension of carbon nanotubes

Fig. 16 Effect of Nt on θ(ζ )

and concentration as well as the friction factor coefficient, local Nusselt number and local Sherwood number for both single- and multi-walled carbon nanotubes is studied through graphs and tables. For numerical computations we fixed the nondimensional parameter values as Λ = 0.3, K = 0.2, j3 = 0.3, j1 = 0.3, j2 = 0.3, n = 0.65, M = 2, φ = 0.1, Le = 6, Nb = 0.2, Nt = 0.1, Pr = 6.2, δ = 0.1. The thermophysical properties of SWCNTs and MWCNTs are depicted in Table 1. The validations of present solutions with available 1 solution are presented in Table 2. The numerical values of the skin friction coefficients Cf (Re) 2 and the rate 1 1 of heat and mass transfer coefficients (N u x (Re)− 2 , Sh x (Re)− 2 ) for both the CNTs are presented in Tables 3 and 4. The effect of permeability parameter K on velocity profiles f  (ζ ) and g  (ζ ) is shown in Figs. 2 and 3. It is observed that there is a fall in the velocity distributions with increasing K . This is due to the fact that the presence of a porous medium increases the resistance to the flow, causing a decrease in the fluid velocity for both SWNTs MWNTs. Also, we found that the momentum boundary layer for SWNTs is thicker than that of MWNTs, while the opposite phenomenon is observed on temperature and concentration profiles (see Figs. 4, 5). This happens due to domination of higher particle-to-particle interaction. Figures 6 and 7 illustrate the velocity profiles f  (ζ ) and g  (ζ ) for various values of M. The rising values of M depreciate the velocity profiles for both the nanotubes (SWNT and MWNT). This is because the magnetic field parameter progresses the resistive-type drag force contradictory to the flow direction which weakens the velocity field and enhances the temperature and concentration fields (see Figs. 7, 8, 9). This is due to additional work expended in dragging the nanofluid in the boundary layer against the action of the Lorentz force. Figures 10, 11, 12 and 13 demonstrate the variations of velocity fields f  (ζ ) and g  (ζ ), temperature field θ (ζ ) and concentration field φ(ζ ) for different values of volume fraction parameter φ. It is clear that the rising values of φ improve the velocity, temperature and concentration profiles for both nanotubes (see Figs. 12, 13). This is due to the fact that as the nanoparticle volume fraction increases, the reaction becomes increasingly confined to a relatively narrow region far from the wall. In addition to this, when the volume fraction of the nanoparticle increases, the thermal conductivity increases. Brownian motion of the nanoparticles is a key parameter for studying the effect of nanoparticles on the temperature and concentration distributions. Hence, Figs. 14 and 15 represent the effect of Brownian motion parameter Nb on temperature and concentration of the nanofluids. It is observed that the temperature of the nanofluid increases with the increase of Nb . As the particle size scale approaches to the nanometer scale, the particle Brownian motion improves the more colloidal particle interaction. This leads to opposite trend observed on concentration field. Figures 16 and 17 display the effect of thermophoresis parameter Nt on temperature and concentration distributions. Increasing values of Nt improve the temperature and concentration profiles. We notice that positive Nt indicate a cold surface while negative to a hot surface. For hot surfaces, thermophoresis tends to blow the nanoparticle volume fraction boundary layer away from the surface since a hot surface repels the submicron-sized particles from it, thereby forming a relative particle-free layer near the surface. In

R. V. M. S. S. K. Kumar et al.

Fig. 17 Effect of Nt on φ(ζ )

Fig. 18 Effect of δ on θ(ζ )

particular, the effect of increasing the thermophoresis parameter Nt is limited to increasing slightly the wall slope of the nanoparticle volume fraction profiles, but decreasing the nanoparticle volume fraction. This is true only for small Lewis numbers for which the Brownian diffusion effect is large compared with the convection effect. However, for large Lewis numbers, the diffusion effect is minimal compared with the convection effect. Figures 18 and 19 illustrate the effect of wall thickness parameter δ on temperature and concentration fields of both single- and multi-walled CNTs. It is evident that increasing values of δ depreciate the temperature and concentration profiles. This validates the physical behavior of δ. Generally, the increasing values of wall thickness parameter generate higher pressure forces on the flow; this leads to improve the particle moment faster. Due to this the temperature and concentration fields are depreciated. Figure 20 explains the effect of heat source/sink Q H on temperature field for both the single- and multi-walled CNTs. It is evident that increasing values of Q H depreciate the temperature profiles. The heat source parameter absorbs the internal heat energy from the surface. Because of this we have seen decrement in the temperature profiles. Table 1 displays the thermophysical properties of water and carbon nanotube nanoparticles. The validations of current studies with already existing studies are presented in Table 2. The numerical values of the

Magnetohydrodynamic 3D slip flow in a suspension of carbon nanotubes

Fig. 19 Effect of δ on φ(ζ )

Fig. 20 Effect of Q H on θ(ζ )

friction factor coefficients ( f  (0) and g  (0)), the local Nusselt number (−θ  (0)) and the local Sherwood number (−φ  (0)) for both single- and multi-walled carbon nanotubes for various values of flow parameters like K , M, φ, Nb , Q H , Nt , and δ are presented in Tables 3 and 4. Due to the domination of opposing force, it is seen that an increase in K and M depreciates the friction factor coefficients as well as the rate of heat and mass transfer coefficients for both CNTs. From this, we conclude that the magnetic field and porosity parameters are controlling the fluid flow. The volume fraction parameter boosts up the friction factor coefficients and depreciates the rate of heat and mass transfer coefficients. The rising values of volume fraction increase the size of the nanoparticle; this leads to enhance the friction and reduce the local Nusselt and Sherwood numbers. Also, it is observed that the heat source parameter reduces the skin friction coefficients, while it improves the heat and mass transfer coefficients. The higher particle diffusion in the Brownian motion parameter increases 1 the skin friction coefficient f  (0) and the mass transfer rate (Sh x (Re)− 2 ), but it reduces the heat transfer rate 1 N u x (Re)− 2 . However, the thermophoresis parameter inhibited the growth of f  (0) as well as the rate of heat 1 1 and mass transfer coefficients (N u x (Re)− 2 , Sh x (Re)− 2 ). Furthermore, no effect of Nt and Nb is observed on g  (0).

R. V. M. S. S. K. Kumar et al.

5 Conclusions In this paper, we investigate the effect of heat source/sink on 3D convective heat and mass transfer of magnetohydrodynamic slip flow over a slendering stretching sheet filled with porous medium. The obtained similarity ordinary differential equations are solved by using R–K–Fehlberg-integration scheme. The impact of various flow parameters on velocity, temperature and concentrations as well as the friction factor coefficient and the rate of heat and mass transfer coefficients is derived and analyzed through graphs and tables. The conclusions are as follows: 1. The heat transfer rate is high for SWCNTs when compared with MWCNTs. This is because the thermal conductivity of SWCNT is higher than that of MWCNT. 2. The combined effect of thermophoresis and Brownian motion plays a dominant role on the flow field. The strength of thermophoresis declines for smaller nanoparticles such that the smaller nanoparticles are able to accumulate at the heated wall and enhance the heat transfer rate. 3. The heat source parameter absorbs the internal heat energy from the sheet. This can help to improve the heat transfer rate per unit volume. 4. The rising values of magnetic and porosity parameters are controlling the flow, heat and mass transfer coefficients. 5. The volume fraction of CNTs improves the velocity, temperature and concentration profiles.

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