Introduction

In recent years, there has been being a need to improve new forms of liquids that will be more influential in heat transfer efficiency, as increasing requests of such modernistic technology as microelectronics, chemical products and power station. The general definition of nanofluid is a fluid including nano-size solid particles existing in such basis fluid with slight thermal conductivity such as oils, ethylene glycol and water [1,2,3,4,5]. Choi and Eastman [6] proposed the phrase of ‘nanofluid’ when they were representing a new method to augment thermal feature of HTF (heat transfer fluid). In the past years, the endeavors for improving the conduction of fluid were conducted by mixing solid particles in micrometer or millimeter sizes into typical carrier fluid; nonetheless, such method because of suspending millimeter- or micrometer-sized solid particles in fluid brings about some challenges—including channel blocking, sedimentation and abrasion—which are then resolved by reducing the size of such particles to nanometer scales, which is around 1 to 100 nm. The types of utilized nanofluids are typically metals such as Cu and Au, oxide metals such as CuO, TiO2, Fe3O4 and Al2O3 and nonmetallic element such as carbon [7,8,9,10,11,12]. The inclusion of such nanoparticles leads the thermal conductivity of nanofluid to grow, as those nanoparticles have a greater level of thermal conductivity compared to the basis fluids [13,14,15,16,17,18,19,20,21].

Al2O3 nanofluid with 1.0% of mass concentration including a particle size of 50 nm has been utilized by Senthil et al. [22] who found that the maximum efficiency was acquired at 75% charging rate and 30° slop angle. Scientists paid attention to the most increase in thermal resistance, applying a various desirable mixture of disparate terms. Wei et al. [23] scrutinized the efficiency of nanomaterial within a pipe with mount of device with new shape for producing swirl flow. Ferrofluid migration with considering radiation was examined by Sheikholeslami and Shamlooei [24], and they concluded that Lorentz force makes velocity to decline. Huminic et al. [25] have conducted research about the usages of Fe2O3/H2O nanomaterial in a thermosyphon heat pipe. Nanomaterials of various fractions—such as 0%, 2% and 5.3%—have been charged in the pipe. Based on their results, the most rate of performance was related to the concentration of 5.3%. Sheikholeslami et al. [26] employed nanopowder for boiling phenomena and suggested empirical correlations. The efficiency of collector employing ZrO2/H2O and Ag/water nanomaterial was performed by Hussain et al. [27] who illustrated that those nanofluids grew the efficiency of unit than water, no more so than at a great inlet temperature; nevertheless, the performance of that collector silver nanopowder has been greater compared to ZrO2/water nanofluid. Other techniques were incorporated by researchers to augment performance [28,29,30,31,32,33,34,35,36,37,38,39,40,41]. The thermal efficiency as well as hydraulic efficiency of graphene nanoplatelets at different fluxes of heat flux has been computed by Arshad and Ali [42]. Based on their results, a growth in heat flux damages the nanofluid stability which is a reason for a reduction in thermal efficiency of nanofluid in a greater level of heat flux. The decrease in wall base temperature was 3.45%, and that reduction was 17.48% for thermal resistance compared to pure water. Melting of nanomaterial within a duct as a part of ventilation unit was simulated by Sheikholeslami et al. [43], and they provide second law analysis for all cases. Pressure drop and thermal feature of a coiled tube by applying alumina nanomaterial were studied by Kumar et al. [44] who found the Δp of 0.8% nanofluids to be 9%, greater compared to H2O. The empirical Nu for 0.1%, 0.4% and 0.8% nanofluids has been found to be 28%, 36% and 56%, respectively, greater than basis fluid. These improvements happened because of the greater rate of knf, efficient fluid combination and storage secondary stream style in duct. It is found that the amount of thermal efficiency coefficient was higher than unity. As a result, alumina nanomaterial can be utilized as a carrier fluid in helical pipe to improve the rate of heat transfer with slight pressure fall. Improving numerical approaches helps the designers to find the optimized design for heating systems [45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66]. To augment the efficiency of cooling unit, Sheikholeslami et al. [67] suggested inserting complex turbulator and they considered exergy loss in their modeling. Mixture of R-11 with titanium was applied by Naphon et al. [68], and they conduct some experiments for various tilt angles from 0° to 90°. Based on their results, the most thermal efficiency of heat pipe has been acquired at 50% charging mass and 60° tilt angle. To gain a more fundamental insight into the mechanisms of heat transfer in nanofluids, molecular dynamics simulations are proved to be a proper technique [69, 70]. Solar unit enhanced with twisted tape has been scrutinized by Farshad and Sheikholeslami [71], and they found the best model for turbulent modeling. The entropy production through nanofluid filled direct adsorption solar collector with Cu/water as operant fluid has been surveyed by Parvin et al. [72] in different solid mass fractions. The most efficient heat transfer factor improvement has been related to the greatest Re and mass fraction of 3%. This should be said that the efficiency of collector doubled.

The topic of this study is investigating the nanoparticle stream and its heat transfer because of a cylinder including an identical suction. Nanofluid viscosity and efficient thermal conductivity are computed by KKL equation. In this pattern, Brownian movement impact on the efficient thermal conductivity is taken into account. The declined ODEs are numerically resolved applying fourth-order Runge–Kutta method. The influences of factors governing on this problem are investigated and presented.

Mathematic description

Figure 1 illustrates a steady laminar nanofluid stream resulted from a stretching pipe (radius = a in axial axis). Z-direction as well as r-direction is evaluated along the direction of the pipe and in the radial one, respectively.

Fig. 1
figure 1

Geometry of cylinder in the current study

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The viscous waste is supposed to be negligible, and tube surface is considered to have a fixed temperature Tw which is larger than T∞. The basis fluid and the powders were in thermal balance, in which no any slip exists among them. The correlations are represented as follows under such presumptions [73]:

$$\frac{\partial P}{\partial r} + \rho_{\text{nf}} \left( {u\frac{\partial u}{\partial r} + w\frac{\partial u}{\partial z}} \right) = \left( { - \frac{u}{{r^{2} }} + \frac{{\partial^{2} u}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial u}{\partial r}} \right)\mu_{\text{nf}} ,$$
(1)
$$\rho_{\text{nf}} \left( {u\frac{\partial w}{\partial r} + w\frac{\partial w}{\partial z}} \right) = \left( {\frac{{\partial^{2} w}}{{\partial r^{2} }} + \frac{\partial w}{\partial r}\frac{1}{r}} \right)\mu_{\text{nf}} ,$$
(2)
$$\frac{\partial (ru)}{\partial r} + \frac{\partial (rw)}{\partial z} = 0,$$
(3)
$$\left( {u\frac{\partial T}{\partial r} + \frac{\partial T}{\partial z}w} \right) = \left( {\rho C_{\text{p}} } \right)^{ - 1}_{\text{nf}} \left( {\frac{1}{r}\frac{\partial T}{\partial r} + \frac{{\partial^{2} T}}{{\partial r^{2} }}} \right)k_{\text{nf}} ,$$
(4)
$$\begin{array}{*{20}l} {r \to \infty :} \hfill & {T \to T_{\infty } ,} \hfill & {w \to 0,} \hfill \\ {r = a:} \hfill & {T = T_{\rm w} ,\;w = w_{\rm w} ,} \hfill & {u = U_{\rm w} } \hfill \\ \end{array}$$
(5)

\(w_{\text{w}} = 2zc,U_{\text{w}} = - a\gamma c,\) and \(c\) is a positive fixed.

The carrier fluid is water, and the selected nanopowders are the same of [74], and also we utilized the KKL model for prediction of feature of nanomaterial. We supposed that mixture of copper oxide and base fluid is homogeneous. The resembling transformation is considered as follows based on Wang [75]:

$$\begin{array}{*{20}l} {u = - ac[\eta^{ - 0.5} f(\eta )],} \hfill & {\eta = (r/a)^{2} ,} \hfill \\ {\theta (\eta ) = (T_{\infty } - T)/(T_{\infty } - T_{\text{w}} ),} \hfill & {w = 2zcf^{\prime}(\eta ),} \hfill \\ \end{array}$$
(6)

ODEs are acquired as follows by replacing Eq. (8) into Eq. (5):

$${f^{\prime \prime }} + {f^{\prime \prime \prime }}\eta = Re \cdot {A_1} \cdot {\left( {1 - \phi } \right)^{2.5}}\left( { - f{f^{\prime \prime }} + {f^{\prime 2}}{\text{ }}} \right),$$
(7)
$$\eta \theta ^{\prime\prime} + (PrRe\,f\,A_{3}^{ - 1} A_{2} + 1)\theta ^{\prime} = 0,$$
(8)
$$\begin{array}{lll} f(1) = \gamma , \hfill \\ \theta \left( \infty \right) \to 0,\quad \theta (1) = 1, \hfill \\ f^{\prime}(1) = 1,\quad f^{\prime}\left( \infty \right) \to 0, \hfill \\ \end{array}$$
(9)

where parameters should be calculated as:

$$Pr = \mu_{\text{f}} \left( {k_{\text{f}} \rho_{\text{f}} } \right)^{ - 1} \left( {C_{\text{p}} \rho } \right)_{\text{f}} ,\;A_{3} = \frac{{k_{\text{nf}} }}{{k_{\text{f}} }}Re = ca^{2} /2\upsilon_{\text{f}} ,\;A_{2} = \frac{{(\rho C_{\text{p}} )_{\text{nf}} }}{{(\rho C_{\text{p}} )_{\text{f}} }},\;A_{1} = \frac{{\rho_{\text{nf}} }}{{\rho_{\text{f}} }}$$
(10)

Physical numbers of interest are the Nu and Cf, introduced as follows [73]:

$$\begin{aligned} Nu & \equiv \frac{{k_{\text{nf}} }}{{k_{\text{f}} }}\left( { - 2} \right)\theta ^{\prime}(1) \\ C_{\text{f}} & \equiv \left| {\left( {A_{1}^{ - 1} \left( {1 - \phi } \right)^{ - 2.5} } \right)f^{\prime\prime}(1)} \right|, \\ \end{aligned}$$
(11)

Results and discussion

On a stretching cylinder which is porous, nanofluid stream and its heat transfer are studied. The related correlations and the boundary conditions have been altered to ODEs, numerically solved by applying fourth-order Runge–Kutta method. Figures 2, 3 and 4 demonstrate the effects of \(\phi ,\lambda\) and Re on Cf. Also influences of scrutinized parameters on Nu are illustrated in Figs. 5, 6 and 7. Dispersion of nanoparticles becomes a significant approach in heating and cooling processes. When CuO is mixed with water, the greater Nusselt amount and lower surface friction factor can be reached. It is observed that the width of thermal boundary reduces and there is a ignorable change in the profile of velocity as the nanofluid mass fraction grows from 0 to 0.04. Temperature and velocity reduce when Re rises. This resulted that surface friction factor reduces with growing mass fraction of nanofluid; nonetheless, it rises when Re and suction factor grow. The impacts of Re, nanoparticle mass fraction and suction factor on Nu and Cf can be summarized in two formulas:

$$\begin{aligned} C_{\text{f}} & = 1.83 + 1.04Re + 0.65\lambda - 0.013\phi \\ & \quad - 1.39 \times 10^{ - 3} Re\;\phi + 1.98 \times 10^{ - 3} \phi \lambda - 0.45Re^{2} + 0.079\lambda^{2} \\ \end{aligned}$$
(12)
$$\begin{aligned} Nu & = 15.85 + 13.11Re + 12\lambda + 0.057\phi \\ & \quad - 0.032Re\phi - 0.089\phi \lambda - 0.14Re^{2} + 0.93\lambda^{2} \\ \end{aligned}$$
(13)
Fig. 2
figure 2

Influences of \(\lambda\) and Re on Cf when \(\phi = 0.02\)

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Fig. 3
figure 3

Influences of \(\phi\) and Re on Cf when \(\lambda = 1\)

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Fig. 4
figure 4

Influences of \(\lambda\) and \(\phi\) on Cf when \({\text{Re}} = 1.05\)

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Fig. 5
figure 5

Influences of \(\lambda\) and Re on Nu when \(f = 0.02\)

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Fig. 6
figure 6

Influences of \(\phi\) and Re on Nu when \(\lambda = 1\)

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Fig. 7
figure 7

Influences of \(\lambda\) and \(\phi\) on Nu when \(\text{Re} = 1.05\)

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As λ augments, velocity gradient enhances and it makes the Fanning factor to augment. Cf augments with the rise of Re which is attribute to higher velocity gradient. Influence of nanopowder concentration on Cf is negligible as depicted in outputs. As a consequence of augmenting \(\lambda\), thinner boundary layer will be obtained and it enhances the Nu. Nu tends to decline with the decrease of Re and similar trend was exhibited for fraction of nanofluid.

Conclusions

This paper surveyed two-dimensional nanofluid stream because of a stretching penetrable pipe. Numerically, the equations are resolved by utilizing the fourth-order Runge–Kutta procedure. The impacts of mass fraction of nanofluid, suction factor, Re on the stream and its heat transfer specifications were investigated. Based on results, there is a direct communication between surface friction factor and Re and suction factor; nevertheless, the inverse communication is observed with mass fraction of nanofluid. Also, the width of thermal boundary layer reduces when nanofluid mass fraction, suction factor and Re rise. Opting higher fraction of CuO leads to greater Nu.