1 Introduction

In recent decade, because of rising demands of modern technology, including chemical production, power station, and microelectronics, there is a need to develop new types of fluids that will be more effective in terms of heat exchange performance. Nanofluids are produced by dispersing the nanometer-scale solid particles into base liquids with low thermal conductivity such as water, ethylene glycol (EG) and oils. [1]. The term “nanofluid” was first coined by Choi [2] to describe this new class of fluids. The materials with sizes of nanometers possess unique physical and chemical properties [3]. Using nanofluid instead of base fluid increases the effective thermal conductivity of the fluid and consequently enhances the heat transfer characteristics. Sheikholeslami et al. [4] studied the magnetic field effect on CuO–water nanofluid flow and heat transfer in an enclosure which is heated from below. They found that effect of Hartmann number and heat source length is more pronounced at high Rayleigh number. Rashidi et al. [5] considered the analysis of the second law of thermodynamics applied to an electrically conducting incompressible nanofluid fluid flowing over a porous rotating disk. They concluded that magnetic rotating disk drives have important applications in heat transfer enhancement in renewable energy systems. Sheikholeslami et al. [6] studied the problem of MHD free convection in an eccentric semi-annulus filled with nanofluid. They showed that Nusselt number decreases with increase of position of inner cylinder at high Rayleigh number. Hatami et al. [7] investigated the magnetohydrodynamic Jeffery–Hamel nanofluid flow in non-parallel walls. They found that skin friction coefficient is an increasing function of Reynolds number, opening angle and nanoparticle volume friction but decrease function of Hartmann number. Therefore, numerous methods have been taken to improve the thermal conductivity of these fluids by suspending nano/micro-sized particle materials in liquids. Several numerical studies have been published recently on the modeling of natural convection heat transfer in nanofluids such as [8–39].

Steady flow in a viscous and incompressible fluid outside of a stretching hollow cylinder in an ambient fluid at rest has been performed by Wang [40]. The problem is governed by a third-order nonlinear ordinary differential equation that leads to exact similarity solutions of the Navier–Stokes equations. The flow over a cylinder can be considered as two-dimensional flow if the body radius is large compared to the boundary layer thickness. On the other hand for a thin or slender cylinder, the radius of the cylinder may be of the same order as that of the boundary layer thickness. The study of convective heat transfer in fluid-saturated porous media has many important applications in technology of geothermal energy recovery such as oil recovery, food processing, fiber and granular insulation, porous burner and heater, combustion of low-calorific fuels to diesel engines and design of packed bed reactors. In general, suction tends to increase the skin friction and heat transfer coefficients, whereas injection acts in the opposite manner [41]. The effects of suction/injection on the flow and heat transfer over a slender cylinder have attracted many researchers to make further investigations. The effect of slot suction/injection over a thin cylinder was studied by Datta et al. [42] and Kumari and Nath [43]. These papers may be useful in the cooling of nuclear reactors during emergency shutdown, where a part of the surface can be cooled by injecting a coolant. Ishak et al. [44] studied uniform suction/blowing effect on flow and heat transfer due to a stretching cylinder which is useful as a simple model in understanding more complicated applications to practical problems, such as cooling of nuclear reactors.

The objective of the present paper is to study the nanofluid flow and heat transfer due to a stretching cylinder with uniform suction. The effective thermal conductivity and viscosity of the nanofluid are calculated by KKL (Koo–Kleinstreuer–Li) correlation. In this model, the effect of Brownian motion on the effective thermal conductivity is considered. The reduced ordinary differential equations are solved numerically using the fourth-order Runge–Kutta integration scheme featuring a shooting technique. The effects of the parameters governing the problem are studied and discussed.

2 Problem formulation and equations

Consider the steady laminar nanofluid flow caused by a stretching tube with radius a in the axial direction in a fluid (Fig. 1), where the z-axis is measured along the axis of the tube and the r-axis is measured in the radial direction. It is assumed that the surface of the tube has constant temperature T w and the ambient fluid temperature is T ∞ (T w > T ∞). The viscous dissipation is neglected as it is assumed to be small. It is assumed that the base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. Under these assumptions the governing equations are:

Fig. 1
figure 1

Geometry of the problem

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Vector form:

$$ \mathop \nabla \limits^{ \to } .\mathop V\limits^{ \to } = 0 $$
(1)
$$ \rho_{\text{nf}} \left( {\frac{{\partial \mathop V\limits^{ \to } }}{\partial t} + \left( {\mathop V\limits^{ \to } .\mathop \nabla \limits^{ \to } } \right)\mathop V\limits^{ \to } } \right) = - \mathop \nabla \limits^{ \to } P + \mu_{\text{nf}} \nabla^{2} \mathop V\limits^{ \to } $$
(2)
$$ \left( {\rho C_{\text{p}} } \right)_{\text{nf}} \left( {\frac{\partial T}{\partial t} + \left( {\mathop V\limits^{ \to } .\mathop \nabla \limits^{ \to } } \right)T} \right) = k_{\text{nf}} \nabla^{2} T $$
(3)

where \( \overrightarrow {V} = \left( {u,v,w} \right) \) is velocity vector, Tis temperature, Pis pressure, \( \rho_{\text{nf}} ,\mu_{\text{nf}} ,\left( {C_{\text{p}} } \right)_{\text{nf}} ,k_{\text{nf}} \) are density, viscosity, heat capacitance, thermal conductivity of nanofluid. Also the operation of \( \mathop \nabla \limits^{ \to } \) can be defined as\( \mathop \nabla \limits^{ \to } = \left( {\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}} \right) \).

Scalar form:

$$ \frac{\partial (rw)\,}{\partial z} + \frac{\partial (ru)}{\partial r} = 0, $$
(4)
$$ \rho_{{{\text{n}}\,{\text{f}}}} \left( {w\frac{\partial w}{\partial z} + u\frac{\partial w}{\partial r}} \right) = \mu_{{{\text{n}}\,{\text{f}}}} \left( {\frac{{\partial^{2} w}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial w}{\partial r}} \right), $$
(5)
$$ \rho_{{{\text{n}}\,{\text{f}}}} \left( {w\frac{\partial u}{\partial z} + u\frac{\partial u}{\partial r}} \right) = - \frac{\partial P}{\partial r} + \mu_{{{\text{n}}\,{\text{f}}}} \left( {\frac{{\partial^{2} u}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial u}{\partial r} - \frac{u}{{r^{2} }}} \right), $$
(6)
$$ \left( {w\frac{\partial T}{\partial z} + u\frac{\partial T}{\partial r}} \right) = \frac{{k_{{{\text{n}}\,{\text{f}}}} }}{{\left( {\rho C_{\text{p}} } \right)_{{{\text{n}}\,{\text{f}}}} }}\left( {\frac{{\partial^{2} T}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial T}{\partial r}} \right), $$
(7)

subject to the boundary conditions

$$ \begin{gathered} r = a\,\,\,\,\,\,:\,\,\,\,\,\,u = U_{\text{w}} ,\,\,\,\,\,\,\,w = w_{\text{w}} \,\,\,,\,\,\,\,\,\,\,T = T_{\text{w}} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\ r \to \infty :\,\,\,\,\,\,\,\,w \to 0,\,\,\,\,\,\,\,T \to T_{\infty } \hfill \\ \end{gathered} $$
(8)

where \( U_{\text{w}} = - {\text{ca}}\gamma ,w_{\text{w}} = 2 {\text{cz}}\,\, \) and \( c \) is a positive constant. \( \gamma \) is a constant in which \( \gamma > 0 \) and \( \gamma < 0 \) corresponding to mass suction and mass injection, respectively.

The effective density\( \left( {\rho_{{{\text{n}}\,{\text{f}}}} } \right) \) and the heat capacitance \( \left( {\rho C_{\text{p}} } \right)_{{{\text{n}}\,{\text{f}}}} \) of the nanofluid are given as:

$$ \rho_{{{\text{n}}\,{\text{f}}}} = \rho_{\text{f}} (1 - \phi ) + \rho_{\text{s}} \phi $$
(9)
$$ \left( {\rho C_{\text{p}} } \right)_{{{\text{n}}\,{\text{f}}}} = \left( {\rho C_{\text{p}} } \right)_{\text{f}} (1 - \phi ) + \left( {\rho C_{\text{p}} } \right)_{\text{s}} \phi $$
(10)

Here, \( \phi \) is the solid volume fraction.

The Brownian motion has a significant impact on the effective thermal conductivity. Koo and Kleinstreuer [45] proposed that the effective thermal conductivity is composed of the particle’s conventional static part and a Brownian motion part. This 2-component thermal conductivity model takes into account the effects of particle size, particle volume fraction and temperature dependence as well as types of particle and base fluid combinations.

$$ k_{\text{eff}} = k_{\text{static}} + k_{\text{Brownian}} $$
(11)
$$ \frac{{k_{\text{static}} }}{{k_{\text{f}} }} = 1 + \frac{{3\left( {\frac{{k_{\text{p}} }}{{k_{\text{f}} }} - 1} \right)\phi }}{{\left( {\frac{{k_{\text{p}} }}{{k_{\text{f}} }} + 2} \right) - \left( {\frac{{k_{\text{p}} }}{{k_{\text{f}} }} - 1} \right)\phi }}{\kern 1pt} , $$
(12)

where k static is the static thermal conductivity based on Maxwell classical correlation. The enhanced thermal conductivity component generated by micro-scale convective heat transfer of a particle’s Brownian motion and affected by ambient fluid motion is obtained via simulating Stokes’ flow around a sphere (nanoparticle). By introducing two empirical functions (β and f) Koo [46] combined the interaction between nanoparticles in addition to the temperature effect in the model, leading to:

$$ k_{\text{Brownian}} = 5 \times 10^{4} \beta \phi \rho_{\text{f}} c_{\text{p,f}} \sqrt {\frac{{\kappa_{\text{b}} T}}{{\rho_{\text{p}} d_{\text{p}} }}} f(T,\phi ){\kern 1pt} . $$
(13)

In recent years, there has been an increasing trend to emphasize the importance of the interfacial thermal resistance between nanoparticles and base fluids (see for example, Prasher et al. [47] and Jang and Choi, [48] ). The thermal interfacial resistance (Kapitza resistance) is believed to exist in the adjacent layers of the two different materials; the thin barrier layer plays a key role in weakening the effective thermal conductivity of the nanoparticle.

Li [49] revisited the model of Koo and Kleinstreuer [50] and combined β and f functions to develop a new \( g^{\prime} \)-function which captures the influences of particle diameter, temperature and volume fraction. The empirical \( g^{\prime} \)-function depends on the type of nanofluid [50]. Also, by introducing a thermal interfacial resistance\( R_{\text{f}} = 4 \times 10^{ - 8} {\text{km}}^{ 2} / {\text{W}} \) the original k p in Eq. (9) was replaced by a new k p, eff in the form:

$$ R_{\text{f}} + \frac{{d_{\text{p}} }}{{k_{\text{p}} }} = \frac{{d_{\text{p}} }}{{k_{\text{p,eff}} }}\,. $$
(14)

For different base fluids and different nanoparticles, the function should be different. Only water-based nanofluids are considered in the current study. For Al2O3–water and CuO–water nanofluids, this function follows the format:

$$ \begin{gathered} g^{\prime}\left( {T,\phi ,d_{\text{p}} } \right) = \left( {a_{1} + a_{2} \ln \left( {d_{\text{p}} } \right) + a_{3} \ln \left( \phi \right) + a_{4} \ln \left( \phi \right)\ln \left( {d_{\text{p}} } \right) + a_{5} \ln \left( {d_{\text{p}} } \right)^{2} } \right)\ln \left( T \right) \hfill \\ \,\,\,\,\,\,\, + \left( {a_{6} + a_{7} \ln \left( {d_{\text{p}} } \right) + a_{8} \ln \left( \phi \right) + a_{9} \ln \left( \phi \right)\ln \left( {d_{\text{p}} } \right) + a_{10} \ln \left( {d_{\text{p}} } \right)^{2} } \right) \hfill \\ \end{gathered} $$
(15)

where the coefficients a i (i = 0, 1,…, 10) are based on the type of nanoparticles and also with these coefficients, Al2O3–water and CuO–water nanofluids have an R 2 of 96 and 98 %, respectively [50] (Tables 1 and 2). Finally, the KKL (Koo–Kleinstreuer–Li) correlation is written as:

$$ k_{\text{Brownian}} = 5 \times 10^{4} \phi \rho_{\text{f}} c_{\text{p,f}} \sqrt {\frac{{\kappa_{\text{b}} T}}{{\rho_{\text{p}} d_{\text{p}} }}} g^{\prime}(T,\phi ,d_{\text{p}} )\,. $$
(16)
Table 1 Thermo-physical properties of water and nanoparticles [45]
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Table 2 The coefficient values of Al2O3–water nanofluids and CuO–water nanofluids [45]
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Koo and Kleinstreuer [45] further investigated laminar nanofluid flow in micro heat-sinks using the effective nanofluid thermal conductivity model they had established (Koo and Kleistreuer [45] ). For the effective viscosity due to micro mixing in suspensions, they proposed:

$$ \mu_{\text{eff}} = \mu_{\text{static}} + \mu_{\text{Brownian}} = \mu_{\text{static}} + \frac{{k_{\text{Brownian}} }}{{k_{\text{f}} }} \times \frac{{\mu_{\text{f}} }}{{\Pr_{\text{f}} }} $$
(17)

where \( \mu_{\text{static}} = \frac{{\mu_{\text{f}} }}{{\left( {1 - \phi } \right)^{2.5} }} \) is viscosity of the nanofluid, as given originally by Brinkman.

Following Wang [40] we take the similarity transformation:

$$ \eta = (r/a)^{2} ,\,u = - {\text{ca}}[f(\eta )\,/\sqrt \eta ],w = 2cf^{\prime}(\eta )z,\,\theta (\eta ) = (T - T_{\infty } )/(T_{\text{w}} - T_{\infty } ), $$
(18)

where prime denotes differentiation with respect to η.

Substituting Eq. (18) into Eqs. (5) and (7), we get the following ordinary differential equations:

$$ {\text{Re}}.A_{1} .\left( {1 - \phi } \right)^{2.5} (f^{{{\prime }2}} - ff^{{{\prime \prime }}} ) = \eta f^{{{\prime \prime \prime }}} + f^{{{\prime \prime }}} , $$
(19)
$$ \eta \theta^{\prime\prime} + (1 + \text{Re} \,\Pr f.A_{2} /A_{3} )\theta^{\prime} = 0, $$
(20)

where \( {\text{Re}} = {\text{ca}}^{2} /2\upsilon_{\text{f}} \) is the Reynolds number, \( \upsilon_{\text{f}} = \mu_{\text{f}} /\rho_{\text{f}} \) the kinematic viscosity, \( { \Pr } = \mu_{\text{f}} \left( {\rho C_{\text{p}} } \right)_{\text{f}} /\left( {\rho_{\text{f}} \,k_{\text{f}} } \right) \) is the Prandtl number and \( A_{ 1} ,A_{ 2} ,A_{ 3} ,A_{ 4} \) are parameters having the following forms:

$$ A_{1} = (1 - \phi ) + \frac{{\rho_{\text{s}} }}{{\rho_{\text{f}} }}\phi $$
(21)
$$ A_{2} = (1 - \phi ) + \frac{{(\rho C_{{{\text{p}}\,}} )_{\text{s}} }}{{(\rho C_{{{\text{p}}\,}} )_{\text{f}} }}\phi $$
(22)
$$ A_{3} = \frac{{k_{\text{nf}} }}{{k_{\text{f}} }} = \frac{{k_{\text{s}} + 2k_{\text{f}} - 2\phi (k_{\text{f}} - k_{\text{s}} )}}{{k_{\text{s}} + 2k_{\text{f}} + \phi (k_{\text{f}} - k_{\text{s}} )}} $$
(23)

The boundary conditions (8) become

$$ \begin{gathered} \,f(1) = \gamma ,\,\,\,\,\,\,\,\,\,f^{\prime}(1) = 1,\,\,\,\,\,\,\,\,\theta (1) = 1, \hfill \\ f^{\prime}\left( \infty \right) \to 0,\,\,\,\,\,\theta \left( \infty \right) \to 0. \hfill \\ \end{gathered} $$
(24)

The pressure (P) can now be determined from Eq. (6) in the form

$$ \frac{P}{{\rho_{{{\text{n}}\,{\text{f}}}} }} = \frac{{P_{\infty } }}{{\rho_{{{\text{n}}\,{\text{f}}}} }} - \frac{{c^{2} a^{2} }}{2\eta }f^{2} (\eta ) - 2c\,\upsilon_{{{\text{n}}\,{\text{f}}}} f^{'} (\eta ), $$
(25)

i.e.,

$$ \frac{{P - P_{\infty } }}{{\rho_{{{\text{n}}\,{\text{f}}}} c\,\,\upsilon_{{{\text{n}}\,{\text{f}}}} }} = - \frac{\text{Re}}{\eta }.A_{1} .\left( {1 - \phi } \right)^{2.5} f^{2} (\eta ) - 2f^{\prime}(\eta ). $$
(26)

Physical quantities of interest are the skin friction coefficient (C f) and the Nusselt number(Nu), which are defined as

$$ C_{\text{f}} = \frac{{\tau_{\text{w}} }}{{\rho w_{\text{w}}^{ 2} / 2}},\,\,\,\,Nu = \frac{{a\,q_{\text{w}} }}{{k_{\text{f}} \left( {T_{\text{w}} - T_{\infty } } \right)}}\,\,, $$
(27)

with k f being the thermal conductivity of the base fluid. Further, τ w and q w are the surface shear stress and the surface heat flux, respectively, and they are given by

$$ \tau_{\text{w}} = \mu_{{{\text{n}}\,{\text{f}}}} \left( {\frac{\partial w}{\partial r}} \right)_{r = a} \,\,,\,\,\,\,q_{\text{w}} = - k_{{{\text{n}}\,{\text{f}}}} \left( {\frac{\partial T}{\partial r}} \right)_{r = a} \,, $$
(28)

i.e.,

$$ \tau_{\text{w}} = \frac{{4\mu_{{{\text{n}}\,{\text{f}}}} \,c\,z}}{a}f^{\prime\prime}(1)\,,\,\,\,\,q_{\text{w}} = - \frac{{2k_{{{\text{n}}\,{\text{f}}}} (T_{\text{w}} - T_{\infty } )}}{a}\theta^{\prime}(1). $$
(29)

Using variables (27), we have:

$$ C_{\text{f}} \equiv \left| {\frac{1}{{A_{1} \left( {1 - \phi } \right)^{2.5} }}f^{\prime\prime}(1)} \right|\,,\,\,\,\,Nu \equiv - 2\frac{{k_{{{\text{n}}\,{\text{f}}}} }}{{k_{\text{f}} }}\theta^{\prime}(1). $$
(30)

3 Numerical method

Before employing the Runge–Kutta integration scheme, first we reduce the governing differential equations into a set of first-order ODEs.

Let \( x_{1} = \eta ,x_{2} = f,x_{3} = f^{\prime},x_{4} = f^{\prime\prime},x_{5} = \theta ,x_{6} = \theta^{\prime} \). We obtain the following system:

$$ \left( \begin{gathered} x_{1}^{\prime } \hfill \\ x_{2}^{\prime } \hfill \\ x_{3}^{\prime } \hfill \\ x_{4}^{\prime } \hfill \\ x_{5}^{\prime } \hfill \\ x_{6}^{\prime } \hfill \\ \end{gathered} \right) = \left( \begin{gathered} 1 \\ x_{3} \\ x_{4} \\ \left[ {\text{Re} .A_{1} .\left( {1 - \phi } \right)^{2.5} (x_{3}^{2} - x_{4} x_{2} ) - x_{4} } \right]/\eta \\ x_{6} \\ x_{7} \\ - (1 + \text{Re} \,\Pr x_{2} A_{2} /A_{3} )x_{7} /\eta \\ \end{gathered} \right)\,\,\,\,\,\,\,\,\,\, $$
(31)

and the corresponding initial conditions are

$$ \left( \begin{gathered} x_{1} \hfill \\ x_{2} \hfill \\ x_{3} \hfill \\ x_{4} \hfill \\ x_{5} \hfill \\ x_{6} \hfill \\ \end{gathered} \right) = \left( \begin{gathered} 1 \\ \gamma \\ 1 \\ u_{1} \\ 1 \\ u_{2} \\ \end{gathered} \right) $$
(32)

The above nonlinear coupled ODEs along with initial conditions are solved using fourth-order Runge–Kutta integration technique. Suitable values of the unknown initial conditions

u 1 and u 2 are approximated through Newton’s method until the boundary conditions at \( f^{\prime}\left( \infty \right) \to 0,\,\,\,\,\,\theta \left( \infty \right) \to 0 \) are satisfied. The computations have been performed using MAPLE. The maximum value of\( \eta = \infty \), to each group of parameters, is determined when the values of unknown boundary conditions at x = 1 do not change to a successful loop with error less than 10-6.

4 Results and discussion

Nanofluid flow and heat transfer over a stretching porous cylinder are investigated. The governing equations and their boundary conditions are transformed to ordinary differential equations which are solved numerically using the fourth-order Runge–Kutta integration scheme featuring a shooting technique.

Effects of different types of nanofluid on velocity, temperature, skin friction coefficient and Nusselt number are shown in Fig. 2, Tables 3 and 4. It can be said that the shear stress and rate of heat transfer change with using different types of nanofluid. This means that type of nanofluid will be important in the cooling and heating processes. By selecting CuO as nanoparticle we can reach higher Nusselt number and smaller skin friction coefficient. So we use CuO–water to examine the effect of active parameters. Figure 3 shows the effect of nanoparticle volume fraction (ϕ) on velocity profile and temperature distribution. It has been found that when the volume fraction of the nanoparticle increases from 0 to 0.04, the thermal boundary layer thickness decreases while no sensible change occurs in velocity profile.

Fig. 2
figure 2

Velocity profile and temperature distribution for different types of nanofluids when \( \phi = 0.04,\,\,\,{\text{Re}} = 1\, \) and Pr = 6.8

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Table 3 Effects of different kind of nanoparticles on skin friction coefficient when ϕ = 0.04
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Table 4 Effects of different kind of nanoparticles on Nusselt number when ϕ = 0.04
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Fig. 3
figure 3

Effect of nanoparticle volume fraction (ϕ) on velocity profile and temperature distribution when Re = 1, γ = 1 and Pr = 6.8(CuO–water)

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Effects of Reynolds number on velocity profile, pressure distribution and temperature distribution are shown in Figs. 4 and 5. As Reynolds number increases velocity and temperature profiles decrease. Pressure distribution increases with increase of Reynolds number near the cylinder while opposite trend is observed for further distance. When γ = 1, effects of Reynolds number on velocity profile and temperature distribution become greater while the effect of Reynolds number on pressure distribution has been changed. It means that as Reynolds number increases pressure decreases for all values of distant from the surface.

Fig. 4
figure 4

Effect of Reynolds number on velocity profile, pressure distribution and temperature distribution when ϕ = 0.04, γ = 0 and Pr = 6.8(CuO–water)

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Fig. 5
figure 5

Effect of Reynolds number on velocity profile, pressure distribution and temperature distribution when ϕ = 0.04, γ = 1 and Pr = 6.8 (CuO–water)

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Figure 6 shows the effect of suction parameter on velocity profile, pressure distribution and temperature distribution. The velocity curves in this figure show that the velocity gradient at the surface increases as γ increases, which implies the increasing of the wall shear stress. The temperature is found to decrease as γ increases, also it decreases as the distance from the surface increases and finally vanishes in a large distance from the surface, which implies increasing in the wall temperature gradient and in turn increases the surface heat transfer rate. Hence, the Nusselt number increases as γ increases. Effect of suction parameter on pressure distribution is similar to that of Reynolds number in presence of suction.

Fig. 6
figure 6

Effect of suction parameter on velocity profile, pressure distribution and temperature distribution when ϕ = 0.04, Re = 1 and Pr = 6.8 (CuO–water)

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Figure 7 shows the effects of the Reynolds number, suction parameter and nanoparticle volume fraction on skin friction coefficient. It can be found that skin friction coefficient decreases with increase of nanoparticle volume fraction while it increases with increase of Reynolds number and suction parameter.

Fig. 7
figure 7

Effects of the Reynolds number, suction parameter and nanoparticle volume fraction on skin friction coefficient when Pr = 6.8 (CuO–water)

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Figure 8 depicts the effects of the Reynolds number, suction parameter and nanoparticle volume fraction on Nusselt number. Nusselt number is an increasing function of Reynolds number, suction parameter and nanoparticle volume fraction.

Fig. 8
figure 8

Effects of the Reynolds number, suction parameter and nanoparticle volume fraction on Nusselt number when Pr = 6.8 (CuO–water)

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5 Conclusions

In the present study, two-dimensional nanofluid flow due to a stretching permeable tube has been investigated. The equations are solved numerically using the fourth-order Runge–Kutta method. Effects of nanoparticle volume fraction, types of nanofluid, suction parameter and Reynolds number on the flow and heat transfer characteristics have been examined. Results show that skin friction coefficient has direct relationship with Reynolds number and suction parameter but it has reverse relationship with nanoparticle volume fraction. It is observed that thermal boundary layer thickness decreases with increase of nanoparticle volume fraction, Reynolds number and suction parameter. The type of nanofluid is a key factor for heat transfer enhancement. The higher values of Nusselt number are obtained by selecting CuO nanoparticles.