The carbon-nanotube nanofluid sprayed on an unsteady stretching cylinder together with entropy generation

Abstract

The water-based single- and multiple-wall carbon nanotubes nanofluid over the surface of an unsteady stretched cylinder has been studied. The thin film of the carbon-nanotube nanofluid has been focused for the heat transfer enhancement applications. The well-known thermal conductivity model for the revolving tube materials like single- and multiple-walled carbon nanotubes defined by Xue were used. The modeled problem has been solved through the optimal homotopy analysis method using the BVPh 2.0 package. The distribution of the thin layer has been regulated through the pressure term using the variable thickness of the nanoliquid. The entropy generation has mainly focused during the motion of the thin layer for the both sorts of carbon nanotubes. The important features of the entropy generation and Bejan number under the influence of the physical constraints have been compared for the both types of single-wall carbon nanotubes and multiple-wall carbon nanotubes and discussed. The well-known BVPh 2.0 package of the optimal homotopy analysis method has been used to find the outcomes.

Introduction

The suspension of micro large size solid particle in the base fluids gives a remarkable change in their thermal, mechanical, optical, and electronic properties because the thermal efficiency of the solids is much more than that of fluids. The family of carbon plays an imperative role to growth the thermal conductivity of the base fluids. One of the well-known families of carbon called carbon nanotubes (CNTs) have been utilized in most of the recent research fields for the thermal and cooling applications. The CNTs are further divided into two classes, known as single-wall carbon nanotubes (SWCNTs) and multiple-wall carbon nanotubes (MWCNTs). SWCNTs are fashioned by a packaging layer of carbon with one atom thick layer, whereas the MWCNTs contain multiple rolled layers of carbon. The SWCNTs need catalysts for their fusion, whereas the MWCNTs are complex structured and can be fashioned without any catalyst.

Most of the work has done on the nanofluids as the large size nanoparticles do not give satisfactory results because of their stability problems. Most of the researches in early 1990 reported that CNT nanofluids have lot of importance in both academic and in industrial community and its applications in numerous fields like aerospace, electronics, optical, and energy conservation.^1,2 The CNTs have ultra-high thermal conductivity, that is, 2000–6000 W/mK—which is of a higher magnitude compared with copper and aluminum oxide nanofluid—that is why CNT is getting a great interest in the research area. The high thermal efficiency of CNTs reveals advanced thermal conductivity and heat flux as compared to pure fluid and other types of nanoparticles as discussed by Murshed and colleagues.^3,4 They examined that the carbon nanoparticles have great desperation behavior with most of the base fluids, which lead them to long-term stability. As mentioned before, the high thermal properties of CNT nanofluid employed many techniques for different types of CNT nanofluids. Furthermore, Kamali and Binesh ⁵ investigated the enhancement of heat transmission using CNT suspension. Most of the work is done on MWCNT.

Various mathematical models are used by the researchers to study the nanofluids, and these models are mostly dependent on the different initial and boundary conditions. Domairry and Aziz ⁶ have studied the viscous incompressible fluid flow between parallel disks. Heat transforms of motor oil within the sight of both sorts of CNTs between two concentric chambers is given in Haq et al. ⁷ It is depicted that the consideration of nanoparticles maximized the heat conductivity of liquid essentially for both turbulent and laminar flows. The results of the above problem were reviewed for SWCNT-motor oil and MWCNT-motor oil. Boundary layer stream of CNTs over an extended disk is examined by Gohar et al. ⁸ The homogeneous responses happening in the liquid are executed by first-order kinetics. The four types of nanofluids, for example, MWCNTs-H₂O, CuO-H₂O, SiO₂-H₂O, and CuO-C₂H₆O₂, are considered in Hwang et al. ⁹ Their thermal conductivity was estimated by a transient hot wire technique. The thermal conductivity improvement of water-based MWCNT nanofluid is expanded up to 11.3% at a volume fraction of 0.01. The efforts made by the researchers to define a comparative model for the improvement of the thermal conductivities of CNTs are given in Maxwell, ¹⁰ Jeffery, ¹¹ Davis, ¹² Lu and Lin, ¹³ and Hamilton and Crosser. ¹⁴

The above five models have been used for the augmentation of the thermal conductivity for spherical elements and not authenticate the CNTs space dispersal’ and therefore, the appropriate thermal conductivity model is required to fulfil this deficiency. Xue ¹⁵ suggested a model having a large axial ratio and repaying the CNTs space dispersal. The proposed thermal conductivity of the Xue model has been utilized for the CNT nanofluid. The study of the thin film flow over an extending surface has varieties of applications regarding the mechanical engineering and our daily life usages. The running of the thin drop on the widow of a car in a rainy weather, slipping on a wet bathroom, spraying, and painting are the common examples of the thin nanoliquid. The heat and mass transmission of the thin nanoliquid flow past an extending surface have been investigated by Khan et al. ¹⁶

Aziz et al. ¹⁷ described the unsteady flow of the liquid film past a stretching surface with internal heating. Qasim et al. ¹⁸ conferred the time-dependent flow of a nanoliquid using Buongiorno’s model. Wang ¹⁹ has the innovator to describe the streaming of a thin layer over the surface of an extending cylinder. The nanoliquid spray over an extending tube for the thermal and cooling applications has been studied by Khan et al. ²⁰ Alshomrani and Gul ²¹ have extended the spray of a nanoliquid film over the slippery surface of an extending tube. Gul et al. ²² have examined $\gamma \mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-{\mathrm{H}}_2\mathrm{O}\mathrm{and}\gamma \mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-{\mathrm{C}}_2{\mathrm{H}}_6{\mathrm{O}}_2$ nanofluid spray over an extending cylinder. Gul et al. ²³ have studied the unsteady thin film CNT nanofluid flow on an upright cylinder. The numerical investigation of $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-{\mathrm{H}}_2\mathrm{O}$ nanofluid flow of two-phase model is presented by Abbassi et al. ²⁴ The turbulent behavior of free convection of $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-{\mathrm{H}}_2\mathrm{O}$ and $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{Cu}$ hybrid nanofluid is numerically investigated by Takabi and Shokouhmand. ²⁵ The effect of three-dimensional water–aluminum oxide nanofluid flow in a rectangular micro-channel with different nanoparticles is discussed by Akbari et al. ²⁶ The comparison of the two sorts of nanofluids has been studied using the steady case. Varieties of mathematical techniques have been utilized to find the outcomes of the proposed models.

Entropy generation in slip rotating flow is scrutinized by Rashidi et al. ²⁷ The numerical study of entropy generation on the magnetohydrodynamics slip flow past an extending cylinder under the effect of heat generation is comprehensively discussed by Jain and Bohra. ²⁸

The effect of the entropy on a Casson nanoliquid flow past a shrinking surface is nicely discussed by Qing et al. ²⁹ The study of entropy comprising nanoliquid flow through a porous plate is discussed by Bhatti et al. ³⁰ They observed that by increasing the thermophoresis parameters, the entropy generation enhances. Gul et al. ³¹ have inspected the entropy regime in the thin film streaming over an enlarging surface. The impact of the various entrenched constraints on entropy regime and Bejan number has been shown in their study.

The homotopy analysis method (HAM) is one of the analytic approximation methods for obtaining the solution of highly nonlinear differential equations. Liao ³² has introduced this technique for the first time in his PhD dissertation. Liao ³³ further modified this technique by introducing additional assisting parameters for the better convergence. These parameters are executed to overcome the convergence of the series solution. Recently, Liao ³⁴ has further improved this method for the fast convergence of the nonlinear problems by introducing the BVPh 2.0 package. This package is utilized for the range of the embedded parameters and square residual errors. Gul and Ferdous ³⁵ have used the OHAM-BVPh 2.0 package for the solution of the high nonlinear problems. This method is frequently used for the solution of nonlinear problems.^36,37

The objective of this work is to study the thin CNT nanoliquid flow past an extending cylinder. The modeling and simulation play a vigorous role in the field of Material Sciences and Engineering. It balances experimental testing and sometimes is very useful to detect those physical spectacles that cannot be specified experimentally. The mathematical tools enhance and support the spray materials through optimization. The main features of the current coating tactic and its computational outputs with uncertainty and optimization will play an imperative role in the field of coating for heat transmission. Also, from the existing literature, the spray over the stretching cylinder is steady while the recent study is unsteady and also the entropy generation and Bejan number are included.

Mathematical formulation

The CNT–water-based nanofluid spray over an unstable stretching cylinder with radius $a(t)=a\sqrt{1-\alpha t}$ has been considered for the enhancement of the heat transfer. Here, $t$ is the time and $\alpha$ is a positive constant. The $z\mathrm{and}r$ axes are chosen along the side of the cylinder and along the radial direction, respectively. The stretching velocity $W_w=(2c/(1-\alpha t))$ acts along the z-direction as mentioned in Gul et al. ²³ The constant $c>0$ is used for the stretching and $c<0$ is used for the shrinking of the surface.

The basic equations for the flow field and thermal boundary layer are

$$\frac{\partial \mathrm{ru}}{\partial r}+\frac{\partial \mathrm{rw}}{\partial z}=0$$

(1)

$${\rho }_{\mathrm{nf}}[\frac{\partial w}{\partial t}+\mathrm{u}\frac{\partial w}{\partial r}+\mathrm{w}\frac{\partial w}{\partial z}]={\mu }_{\mathrm{nf}}(\frac{{\partial }^{2}w}{\partial r^2}+\frac{1}{r}\frac{\partial w}{\partial r}+\frac{{\partial }^{2}w}{\partial z^2})$$

(2)

$${\rho }_{\mathrm{nf}}[\frac{\partial u}{\partial t}+\mathrm{u}\frac{\partial u}{\partial r}+\mathrm{w}\frac{\partial u}{\partial z}]=-\frac{\partial p}{\partial r}+{\mu }_{\mathrm{nf}}(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}-\frac{u}{r^2})$$

(3)

$${\left(\rho \mathrm{cp}\right)}_{\mathrm{nf}}[\frac{\partial T}{\partial t}+\mathrm{u}\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}]=k_{\mathrm{nf}}(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial z^2})$$

(4)

The velocity mechanisms $u(r,z),w(r,z)$ are along the $(r,z)$ directions consistently. p stands for the pressure term. T stands for the temperature field. Moreover, $(\mathrm{Cp}{)}_{\mathrm{nf}},{\rho }_{\mathrm{nf}},{\kappa }_{\mathrm{nf}},{\mu }_{\mathrm{nf}}$ are the heat capacitance, density, thermal conductivity, and viscosity of the nanofluid. The physical conditions of the above-stated problem are

$$u=U_w,w=W_w,T=T_w,\mathrm{at}r=a\left(t\right)$$

(5)

$$\mu \frac{\partial w}{\partial r}=\frac{\partial T}{\partial r}=0,u=\frac{\mathrm{db}}{\mathrm{dt}},\mathrm{at}r=b$$

(6)

where $a(t)=a\sqrt{(1-\alpha t)}$ is the unsteady radius of the tube and $b(t)$ is the radius of the nanoliquid film such that $b(t)-a(t)>0$ . $U_w=(\mathrm{ca}/\sqrt{\eta (1-\alpha t)})$ is the suction/injection velocity, and the thickness of the liquid film is denoted by $\delta .$ The temperature of the fluid at the surface of the cylinder is denoted by $T_w=T_{a(t)}$ . The temperature at the liquid film surface is taken as $T=T_{b(t)}$ . Here, $T_w=T_{b(t)}-T_{\mathrm{ref}}(c{z}^2/v_f(1-\alpha t))$ satisfies^20–22 for the steady case $(t=0)$ . The kinematic viscosity of the base solvent is denoted by ${\nu }_f$ , and $T_{\mathrm{ref}}$ is known as the reference temperature such that $0\le T_{\mathrm{ref}}\le T_{b(t)}$ .

The dimensionless liquid film thickness is defined as

$$\beta ={\left(\frac{b}{a\sqrt{(1-\alpha t)}}\right)}^2,\frac{\mathrm{db}}{\mathrm{dt}}=\frac{-a\alpha }{2}\sqrt{\frac{\beta }{(1-\alpha t)}}$$

(7)

The thermophysical constraints of the CNTs are

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\rho }_{\mathrm{nf}}}{{\rho }_f}=(1-\phi )+\phi \frac{{\rho }_{\mathrm{CNT}}}{{\rho }_f},{\mu }_{\mathrm{nf}}={\mu }_f{(1-\phi )}^{-2.5},}\\ \multicolumn{1}{c}{\frac{{\left(\rho C_p\right)}_{\mathrm{nf}}}{{\left(\rho C_p\right)}_f}=(1-\phi )+\frac{{\left(\rho C_p\right)}_{\mathrm{CNT}}}{{\left(\rho C_p\right)}_f}\phi }\\ \multicolumn{1}{c}{\frac{k_{\mathrm{nf}}}{k_f}=\frac{1-\phi +2\phi \left(\frac{k_{\mathrm{CNT}}}{k_{\mathrm{CNT}}-k_f}\right)\ln \left(\frac{k_{\mathrm{CNT}}+k_f}{2k_f}\right)}{1-\phi +2\phi \left(\frac{k_f}{k_{\mathrm{CNT}}-k_f}\right)\ln \left(\frac{k_{\mathrm{CNT}}+k_f}{2k_f}\right)},}\\ \multicolumn{1}{c}{\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}=\left(\frac{1+3(\sigma -1)\phi }{(\sigma +2)-(\sigma -1)\phi }\right),{\alpha }_{\mathrm{nf}}=\frac{{\mu }_{\mathrm{nf}}}{{\left(\rho c_p\right)}_{\mathrm{nf}}}}\end{array}$$

(8)

where $\phi$ is the nanoparticle volume fraction; ${\kappa }_{\mathrm{nf}}$ is the thermal conductivity of the nanoliquid; ${\mu }_{\mathrm{nf}}$ is the viscosity of the nanoliquid; ${\rho }_{\mathrm{CNT}}$ is the density of the CNTs; and ${\mu }_f,(\rho C_p{)}_f,{\rho }_f,\mathrm{and}{\kappa }_f$ are the viscosity, specific heat density, and thermal conductivity of the base fluid.

Introducing the similarity transformations to alter the basic governing equations into the dimensionless form and satisfying^20–22 for $t=0$

$$\begin{array}{c}\multicolumn{1}{c}{\eta ={\left(\frac{r}{a\sqrt{(1-\alpha t)}}\right)}^2,w=\frac{2\mathrm{cz}}{(1-\alpha t)}\frac{\mathrm{df}\left(\eta \right)}{d\eta },}\\ \multicolumn{1}{c}{u=-\frac{\mathrm{ca}}{\sqrt{\eta (1-\alpha t)}}f\left(\eta \right),\Theta \left(\eta \right)=\frac{T-T_{b(t)}}{T_{a(t)}-T_{b(t)}}}\end{array}$$

(9)

The similarity variables defined in equation (15) have been used in equations (7)–(12). The continuity equation satisfied automatically and the rest of the equations are altered as

$$\begin{array}{c}\multicolumn{1}{c}{\frac{1}{\left({(1-\phi )}^{2.5}((1-\phi )+\phi \frac{{\rho }_{\mathrm{CNT}}}{{\rho }_f})\right)}(\eta \frac{d^{3}f}{d{\eta }^3}+\frac{d^{2}f}{d{\eta }^2})}\\ \multicolumn{1}{c}{+\mathrm{Re}[f\frac{d^{2}f}{d{\eta }^2}-{\left(\frac{\mathrm{df}}{d\eta }\right)}^2-\frac{S}{2}(\frac{\mathrm{df}}{d\eta }+\eta \frac{d^{2}f}{d{\eta }^2})]=0,}\end{array}$$

(10)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\frac{k_{\mathrm{nf}}}{k_f}}{\mathrm{Pr}((1-\phi )+\phi \frac{{\left(\rho c_p\right)}_{\mathrm{CNT}}}{{\left(\rho c_p\right)}_f})}(2\eta \frac{d^2\Theta }{d{\eta }^2}+\frac{d\Theta }{d\eta })}\\ \multicolumn{1}{c}{+\mathrm{Re}[f\frac{d\Theta }{d\eta }-2\frac{\mathrm{df}}{d\eta }\Theta -\frac{S}{2}(\Theta +\eta \frac{d\Theta }{d\eta })]=0.}\end{array}$$

(11)

The nanofluid constraint $k_{\mathrm{nf}}/k_f$ has been used as defined by Xue. ¹⁵ The transformed boundary conditions satisfy^20–22 for the steady flow $S=0$ and are defined as

$$f\left(1\right)=\frac{\mathrm{df}\left(1\right)}{d\eta }=\Theta \left(1\right)=1,\mathrm{at}\eta =1$$

(12)

$$\frac{d^{2}f\left(\beta \right)}{d{\eta }^2}=\frac{d\Theta \left(\beta \right)}{d\eta }=0,f\left(\beta \right)=\frac{S\beta }{2}$$

(13)

The Reynolds number, Prandtl number, and unsteady parameter are defined as follows

$$\mathrm{Re}=\frac{\mathrm{ca}{\left(t\right)}^2}{2v_f},\mathrm{Pr}=\frac{{\left(\mu C_p\right)}_f}{k_f},S=\frac{\alpha }{c}$$

(14)

The pressure distribution term is evaluated from equation (2) as following

$$\begin{array}{c}\multicolumn{1}{c}{\frac{p-p_b}{c{\mu }_f}=-\frac{\mathrm{Re}}{\eta }((1-\phi )+\phi \frac{{\rho }_{\mathrm{CNT}}}{{\rho }_f}){(1-\phi )}^{2.5}{f}^2}\\ \multicolumn{1}{c}{-(\eta S\mathrm{Re}((1-\phi )+\phi \frac{{\rho }_{\mathrm{CNT}}}{{\rho }_f}){(1-\phi )}^{2.5}+2)\frac{\mathrm{df}}{d\eta }}\end{array}$$

(15)

The shear stress at the outer surface is zero and defined as

$$\frac{d^{2}f\left(\beta \right)}{d{\eta }^2}=0$$

(16)

The shear stress at the surface of the cylinder is taken as

$${\tau }_w=\frac{{\left(\rho \upsilon \right)}_{\mathrm{nf}}4\mathrm{cz}}{a\left(t\right)}\frac{d^{2}f\left(1\right)}{d{\eta }^2}=\frac{{\mu }_{\mathrm{nf}}4\mathrm{cz}}{a\left(t\right)}\frac{d^{2}f\left(1\right)}{d{\eta }^2}$$

(17)

The deposit velocity V is a function of the nanofluid thickness $\beta$ and is defined as

$$-V=\frac{-\mathrm{ca}}{\sqrt{\eta (1-\alpha t)}}\frac{\mathrm{df}\left(\beta \right)}{d\eta }$$

(18)

The mass flux $m_1$ is defined as

$$m_1=\frac{2\pi b\left(t\right)\mathrm{ca}\left(t\right)}{\sqrt{\eta (1-\alpha t)}}\frac{\mathrm{df}\left(\beta \right)}{d\eta }=V2\pi b\left(t\right)$$

(19)

The normalized mass flux $m_2$ is defined as

$$m_2=\frac{m_1}{2\pi c{a}^2\left(t\right)}=\frac{m_1}{4\pi {\upsilon }_f\mathrm{Re}}=f\left(\beta \right)$$

(20)

The significant physical constraint of the drag force and rate of the heat transfer are as

$$C_f=\frac{2{\upsilon }_{\mathrm{nf}}}{W_w}{\left(\frac{\partial w}{\partial r}\right)}_{r=a(t)},\mathrm{Nu}=-\frac{a{k}_{\mathrm{nf}}}{\mathrm{\Delta }T}{\left(\frac{\partial T}{\partial r}\right)}_{r=a(t)}$$

(21)

The non-dimensional form of the above constraints are

$$\left(\frac{z\mathrm{Re}}{a\left(t\right)}\right)C_f={(1-\phi )}^{-2.5}\frac{d^{2}f\left(1\right)}{d{\eta }^2},\mathrm{Nu}=-2\frac{k_{\mathrm{nf}}}{k_f}\frac{d\Theta \left(1\right)}{d\eta }$$

(22)

Entropy analysis

$$S″\prime =\frac{k_{\mathrm{nf}}}{T_b^2}{\left(\frac{\partial T}{\partial r}\right)}^2+\frac{{\mu }_{\mathrm{nf}}}{T_b}{\left(\frac{\partial w}{\partial r}\right)}^2$$

(23)

according to Jain and Bohra. ²⁸

After the nondimensionalization, the above equation becomes

$$N_s=\frac{S″\prime }{{S″\prime }_0}=\frac{1}{X}[\eta {(\Theta \prime )}^2+\frac{\mathrm{Br}}{\mathrm{\Omega }}\eta {(f″)}^2]$$

(24)

according to Jain and Bohra. ²⁸ Here, $S″{\prime }_0=(4k_f(T_w-T_b{)}^2)/T_{\mathrm{ref}}^2{L}^2$ represents the reference volumetric entropy generation, $X=(a/L{)}^2$ is the dimensionless parameter, $\mathrm{\Omega }=(T_w-T_b)/T_{\mathrm{ref}}$ is temperature in dimensionless form, and Brinkman is denoted by $\mathrm{Br}=(\mu w_e^2(z))/k(T_w-T_b)$

$$N_S=N_H+N_f$$

(25)

where $N_H$ represents the entropy generation due to heat transfer and $N_f$ represents the fluid friction

$$N_H=\frac{1}{X}\eta \Theta {\prime }^2,N_f=\frac{1}{X}\left(\frac{\mathrm{Br}}{\mathrm{\Omega }}\right){(f\prime )}^2$$

(26)

The total and heat transfer entropy generation has many applications in optimization and engineering design problem. The Bejan number $\mathrm{Be}$ is defined as the ratio of $N_H$ to $N_S$ ²⁸

$$\mathrm{Be}=\frac{N_H}{N_S}$$

(27)

OHAM solution

In this section, the nonlinear flow modeled in equations (10)–(13) has been attempted analytically through OHAM. For this purpose, the BVPh 2.0 package as mentioned in previous studies^33–38 is utilized to obtain the tenth-order approximations for which the residual error is minimized. The initial trials play an important role in the OHAM solution and are calculated as

$$\begin{array}{c}\multicolumn{1}{c}{f_0\left(\eta \right)=\frac{\beta (S-2)}{4{(1-\beta )}^3}}\\ \multicolumn{1}{c}{[{\eta }^3-3\beta {\eta }^2-(3-6\beta )\eta +(2-3\beta )]+\eta ,{\Theta }_0\left(\eta \right)=1}\end{array}$$

(28)

The linear operators ${\mathrm{L}}_f$ and ${\mathrm{L}}_{\Theta }$ are selected as

$${\mathrm{L}}_f=\frac{{\partial }^4}{\partial {\eta }^4}\mathrm{and}{\mathrm{L}}_{\Theta }=\frac{{\partial }^2}{\partial {\eta }^2}$$

(29)

Consequently, the general outputs of ${\mathrm{L}}_f$ and ${\mathrm{L}}_{\Theta }$ are

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{L}}_f[{\mathrm{\Phi }}_1+\left({\mathrm{\Phi }}_2\right)\eta +\left({\mathrm{\Phi }}_3\right){\eta }^2+\left({\mathrm{\Phi }}_4\right){\eta }^3]=0\mathrm{and}}\\ \multicolumn{1}{c}{{\mathrm{L}}_{\Theta }[{\mathrm{\Phi }}_5+\left({\mathrm{\Phi }}_6\right)\eta +\left({\mathrm{\Phi }}_7\right){\eta }^2]=0}\end{array}$$

(30)

Equations (10)–(13) are confirmed as

$${\epsilon }_m^f=\frac{1}{k+1}\sum _{j=1}^k{\left[{\aleph }_f{\left(\sum _{i=1}^{m}f\left(\eta \right)\right)}_{\xi =j\delta \xi }\right]}^2$$

(31)

$${\epsilon }_m^{\Theta }=\frac{1}{k+1}\sum _{j=1}^k{\left[{\aleph }_{\Theta }{(\sum _{i=1}^{m}f\left(\eta \right),\sum _{i=1}^m\Theta \left(\eta \right))}_{\xi =j\delta \xi }\right]}^2$$

(32)

The sum of the total and square residual error is defined as^33–38

$${\epsilon }_m^t={\epsilon }_m^f+{\epsilon }_m^{\Theta }$$

(33)

The total residual error ${\epsilon }_m^t$ in the form of auxiliary parameters has been revealed in Figures 2 and 3 for the velocity pitch and temperature. The numerical outputs of the optimal convergence controlling constraints $h_f=-0.5768\mathrm{and}{h}_{\Theta }=-1.6521$ have been achieved in the case of the SWCNTs–water while $h_f=-0.5801\mathrm{and}{h}_{\Theta }=-2.0121$ have been obtained in the case of MWCNTs–water. To inculcate the accuracy of the obtained result, one can find the required range of the embedding constraints elaborated in the problem. Due to this importance, the range of the physical constraints involved in the modeled problem has been calculated and presented in Figures 4 and 5.

Results and discussion

The spray analysis of the CNT nanofluids over an unstable and stretching cylinder has been investigated. The velocity, temperature, and pressure profiles for both SWCNTs and MWCNTs suspended in water as a base fluid are investigated. The modeled nonlinear differential equations have been solved through OHAM. The BVPh 2.0 package of the OHAM technique is applied for the solution of equations (15)–(18) and (20). The solution for velocity pitch, temperature field, and pressure profile are accomplished using the thermal conductivity model. ¹⁵

The stepwise detail discussion of the OHAM method convergence, velocity field, temperature profile, pressure distribution and spray rate are as follows:

Residual error sketches

In OHAM technique, the optimal values of the auxiliary parameters $h_f$ and $h_{\Theta }$ play a vigorous role in the convergence. The BVPh 2.0 package as mentioned in previous studies^9,33–38 is utilized to obtain the tenth-order approximations for which the residual error is minimized. The geometry of the problem is presented in Figure 1.

Figure 1.

The geometry of the proposed problem.

The numerical values of the optimal convergence controlling constraints $h_f=-0.5768,h_{\Theta }=-1.6521$ have been archived in case of the SWCNTs–water while $h_f=-0.5801,h_{\Theta }=-1.9121$ have been obtained in the case of MWCNTs–water.

Velocity profile

The striking features in the sprayed pattern on the exterior surface of an extending cylinder are the thickness of the nanoliquid film $\beta$ and the nanoparticle concentration $\phi$ for both sorts of the CNTs. From Figure 2, it is obvious that the velocity expressions decline with larger thickness parameter $\beta$ . The increasing thickness enhances the mass of the nanofluid, and therefore, the fluid flow is declining. In other words, the motion of the thinning liquid is quicker as compared to the thick liquid. Therefore, the increasing value of $\beta$ (thickening parameter) reduces the fluid motion. The enhancement of the spray rate and radial velocity $f\prime (\eta )$ is appropriated with the thinning liquid film, and this effect is clearer in the SWCNTs due to its strong physical properties as compared to the MWCNTs.

Figure 2.

Impact of $\beta$ on the velocity field.

The nanoparticle volume fraction $\phi$ has an important role in the thermal and cooling applications, inserting the MWCNTs and SWCNTs for different base fluids. The larger nanoparticle volume fraction $\phi$ increases the velocity field as shown in Figure 3.

Figure 3.

Impact of $\phi$ on the velocity field.

According to the thermal conductivity model as given in equation (8), it is very clear that the values of $\phi =0.01$ to $\phi =0.04$ enhance the thermal efficiency of the base liquid and reduce the viscous forces to stop the fluid motion. Therefore, the larger quantity of $\phi$ enhances the radial velocity, and this effect is comparatively strong using the SWCNTs.

The significances of the Reynolds number Re and unsteady parameter S are shown in Figures 4 and 5, respectively. The inertial forces (resistive forces) are in the direct contact with the Reynolds number Re and the increasing value of Re, enhancing the inertial forces which cause the fluid motion. Therefore, stronger Reynolds number Re declines the velocity of the both sorts of CNTs. The thickness of the momentum boundary layer upsurges for greater values of S (unsteadiness parameter), which indicates the decline in the velocity field, and this effect is almost much closed for both sorts of CNTs. In fact, the rising value of the unsteadiness parameter S enhances the thickness of the momentum boundary layer, and as a result, the velocity field declines.

Figure 4.

Impact of Re on the velocity field.

Figure 5.

Impact of S on the velocity field.

Temperature profile

The CNTs play a vital role due to its imperative thermophysical properties to enhance the thermal efficiencies of the nanofluid during spray phenomenon. The impact of the embedded constraints on the temperature field has been compared for the SWCNTs and MWCNTs. The temperature distribution decelerates for greater values of $\beta$ , which is evident in Figure 6. Physically, the mass of the fluid rises with the greater quantity of the thick liquid, which exhausts the temperature profile. The heat can easily penetrate in the thin liquids as compared to the thick liquids. Therefore, the larger thickness parameter $\beta$ decelerates the temperature field, and this effect is much closed for both sorts of CNTs. The value of $\beta =1.32,1.34,1.36$ shows the thickness of the thin layer near the cylinder surface having radius 1.

Figure 6.

Impact of $\beta$ on the temperature field.

The thermal conductivity model as described by Xue ¹⁵ has been used for the CNT nanofluid, and the impact of the nanoparticle volume fraction has been calculated and revealed in Figure 7. According to Xue model, the greater particle volume fraction $0<\phi <5\mathrm{\%}$ upsurges the temperature distribution. The thermal efficiency increases for both sorts of CNTs, and the available literature authenticates that this effect is more prominent up to 5% increase in $\phi .$

Figure 7.

Impact of $\phi$ on the temperature field.

Figure 8 indicates the effects of the Prandtl number Pr on temperature field $\Theta (\eta )$ . The excited cooling is attained with the larger values of the Prandtl number, and this influence is faster for the CNTs due to its extraordinary thermophysical properties. In fact, the larger value of the Prandtl number increases the momentum diffusivity and the fluids with lowest Prandtl number are more capable for the heat transfer rate. Therefore, the larger Prandtl number reduces the temperature field. Furthermore, the increasing values of the Prandtl number reduce the thermal boundary layer thickness, which shows that effective cooling is attained rapidly for the CNT nanofluid with high viscous dissipation.

Figure 8.

Impact of Pr on the temperature field.

Figure 9 shows the effect of the Reynolds number and implies that the inertial forces (resistive forces) in the Reynolds number alter the viscous forces. Due to the strong inertial forces, the molecules and atoms of the nanoliquid are tightly bound and too much energy is required to collapse them. The decline effect due to the larger Reynolds number is comparatively rapid in the SWCNTs.

Figure 9.

Impact of Re on the temperature field.

Pressure distribution

Figure 10 specifies that pressure distribution is enhanced for the larger values of the thickness constraint $\beta$ . Physically, more pressure is required for the thick boundary layer as compared to the thinning liquids during spray phenomenon. The pressure distribution boosts up with the larger thickness parameter $\beta$ and extra power is essential to stun the stress spawned due to the greater thickness of the nanoliquid film. This effect is observed relatively strong in the SWCNTs.

Figure 10.

Impact of the $\beta$ on the pressure distribution.

Figure 11 depicted that greater Reynolds number values decline the pressure term. Due to the inertial forces, the pressure inevitably goes to the lowest rate in the broad way of motion. The particles of the nanoliquid are tightly packed and more pressure is required for the fluid motion.

Figure 11.

Impact of the Re on the pressure distribution.

Entropy and Bejan number

Figure 12 illustrates the influence of the Reynolds number Re on Bejan number Be. The larger Re enhances the Bejan number. The rate of Be cuts to the surface of the tube, and after the critical point, it starts retardation. The cumulative values of $\mathrm{Br}/\mathrm{\Omega }$ decreasing the Bejan number $\mathrm{Be}$ is shown in Figure 13. In fact, the heat source vanishes from the above-mentioned ratio, and as a result, the Bejan number decline with the increasing value of $\mathrm{Br}/\mathrm{\Omega }$ .

Figure 12.

Impact of the Re on Bejan number.

Figure 13.

Graph of $\mathrm{Br}/\mathrm{\Omega }$ (Bejan number).

The impact of the effect of Re and S on the entropy regime has been displayed in Figures 14 and 15, respectively. The larger values of the Reynolds number decline the entropy regime, and this effect changes after the critical point. Physically, the heat source is responsible to reduce the entropy generation for the cumulative values of Re. The effect of the unsteadiness S on entropy regime has been depicted in Figure 15. The rising values of S decline the entropy generation near the cylinder surface, and this effect changes after the point of inflection.

Figure 14.

Impact of the Re on the entropy generation.

Figure 15.

Impact of the S (unsteady parameter) on the entropy generation.

Skin friction and Nusselt number

Table 1 indicates the physical possessions of the base solvent water and of the two sorts of CNTs. The sum of the squared residual errors for the SWCNTs and MWCNTs is presented in Table 2. The influence of the unsteadiness parameter S and Reynolds number Re on the local skin friction is scrutinized in Table 3. The larger quantities of the unsteady parameter S and Reynolds number Re rises the skin friction coefficient $f″(1)$ . In fact, the larger values of these parameters enhance the resistance force, and consequently the skin friction upsurges. The influence of the unsteadiness S and Reynolds number $\mathrm{Re}$ verses the heat transfer rate (Nusselt number) is scrutinized in Table 4.

Table 1.

The thermophysical properties of CNTs and the base fluid water.

Physical properties	Density, $\rho$	Thermal conductivity, k	Specific heat, $c_p$
Base fluid water	$0.613$	$0.613$	$4197$
Nanoparticles
SWCNT	$2600$	$6600$	$425$
MWCNT	$1600$	$3000$	$796$

CNT: carbon nanotubes; SWCNT: single-wall carbon nanotubes; MWCNT: multiple-wall carbon nanotubes.

Table 2.

Individual averaged squared residual errors for SWCNTs/MWCNTs–water when $\beta =1.1,\mathrm{Pr}=6.7,\mathrm{Rd}=\mathrm{M}=\mathrm{S}=\mathrm{Re}=\mathrm{Gr}=0.1,\phi =0.01$ .

$m$	${\epsilon }_m^f\mathrm{SWCNTs}$	${\epsilon }_m^f\mathrm{MWCNTs}$	${\epsilon }_m^{\Theta }\mathrm{SWCNTs}$	${\epsilon }_m^{\Theta }\mathrm{MWCNTs}$
2	$4.86\times {10}^{-2}$	$3.21\times {10}^{-2}$	$3.81\times {10}^{-3}$	$2.72\times {10}^{-3}$
6	$4.21\times {10}^{-4}$	$3.53\times {10}^{-4}$	$4.85\times {10}^{-5}$	$3.24\times {10}^{-5}$
10	$6.18\times {10}^{-6}$	$5.89\times {10}^{-6}$	$2.03\times {10}^{-7}$	$1.55\times {10}^{-7}$
12	$8.71\times {10}^{-7}$	$7.71\times {10}^{-7}$	$7.75\times {10}^{-8}$	$6.61\times {10}^{-8}$

SWCNT: single-wall carbon nanotubes; MWCNT: multiple-wall carbon nanotubes.

Table 3.

Exhibits the numerical values for skin friction coefficient for different physical parameters when $\beta =1.1,\mathrm{Pr}=10.7,\mathrm{Re}=0.1,\phi =0.01$ .

$S$	$\mathrm{Re}$	$\begin{array}{c}f″(1)\hfill \\ \mathrm{MWCNTs}\hfill \end{array}$	$\begin{array}{c}f″(1)\hfill \\ \mathrm{SWCNTs}\hfill \end{array}$
0.7	0.8	0.110882	0.112002
0.8		0.115651	0.120112
0.9		0.120394	0.129546
	0.9	0.121308	0.130145
	1.0	0.131821	0.139887

SWCNT: single-wall carbon nanotubes; MWCNT: multiple-wall carbon nanotubes.

Table 4.

Exhibits the numerical values of local Nusselt number for different physical parameters when $\beta =1.1,\mathrm{Re}=0.1,\phi =0.01$ .

$S$	$\mathrm{Pr}$	$\begin{array}{c}-\theta \prime (1)\hfill \\ \mathrm{MWCNTs}\hfill \end{array}$	$\begin{array}{c}-\theta \prime (1)\hfill \\ \mathrm{SWCNTs}\hfill \end{array}$
0.7	0.7	0.703065	0.71462
0.8		0.712599	0.723478
0.9		0.721968	0.731021
	0.8	0.811450	0.82941
	0.9	0.921894	0.932464

SWCNT: single-wall carbon nanotubes; MWCNT: multiple-wall carbon nanotubes.

The larger parameters S and Re enhance the heat transfer rate or cooling effect. The motive is obvious that the thermal boundary layer enhances with larger values of these parameters as mentioned previously, and as a result, the cooling effect upsurges. The effects of these parameters are more effective in the case of SWCNTs.

Conclusion

Water has been used as a base solvent for the diffusion of the CNTs to perform nanofluid. The two sorts—SWCNTs and MWCNTs—of the CNTs have been used for the spray pattern over the stretching surface of the cylinder. The thermal efficiency outcomes have been observed operating the Xue model. Xue ¹⁵ inspected that the previous proposed models of nanofluid are operational only for the sphere-shaped or elliptic-shaped revolving nature materials through the smaller axial part. These ideas do not describe the space scattering features of the thermal conductivities of the CNTs. To overcome this problem, Xue ¹⁵ have prolonged Maxwell’s concept in view of revolving elliptic-shaped nanotubes with very huge axial part and rewarding the impacts of space scattering on CNTs.

The solution of the modeled nonlinear differential equations has been obtained through OHAM. The absolute total residual error has been calculated for the velocity and temperature profiles, respectively. The range of the embedded parameters has also been obtained for the stable convergence of the OHAM method. The parallel and normalized spray rates have also been calculated for the uniformity of spray phenomenon. The pressure distribution is also necessary for the uniform spray coating and has been observed in this research.

The obtained outputs are deliberated as follow:

• It is observed that the increasing thickness parameter $\beta$ reduces the flow motion and also the more pressure is required for the spray pattern. Also, this effect is more efficient in the case of the SWCNTs as compared to the MWCNTs.

• It is observed that the larger quantity of the nanofluid volume fraction $\phi$ reduces intermolecular forces, and as a result, the flow motion is boosted up.

• The greater values of the Reynolds number decay the entropy regime, and this effect varies after the point of inflection.

• The Bejan number declines with the increasing values of $\mathrm{Br}/\mathrm{\Omega }$ .

• The impact of the different parameters against the skin friction and Nusselt number in the case of SWCNTs and MWCNTs have been calculated numerically.