Unsteady squeezing flow of magnetohydrodynamic carbon nanotube nanofluid in rotating channels with entropy generation and viscous dissipation

Abstract

The most favorable gift of modern science is nanofluid. The nanofluid can able to move freely through micro channels with the spreading of nanoparticles. Due to improved convection between the base liquid surfaces and nanoparticles, the nano suspensions express high thermal conductivity. Also, the benefits of suspending nanoparticles in base fluids are increased heat capacity, surface area, effective thermal conductivity, collision, and interaction among particles. This research aim to study squeezing flow of carbon nanotubes based on water (H₂O) in rotating channels. Entropy generation is evaluated and for this purpose, second law of thermodynamics is employed. The influences of thermal radiation, viscous dissipation, and applied magnetic field on nanofluid are taken into account. The flow of the nanofluid is considered in unsteady three dimensions. The transformed ordinary differential equations (ODEs) are solved by homotopy analysis method with the help of similarity variables. Results obtained for single and multi-wall carbon nanotubes are compared. Plots have been presented in order to examine how the velocities, temperature, and entropy profiles become affected by numerous physical parameters. Generally, the velocity profiles escalate when the upper plate of the channel moves toward the lower stretching one and reduces when the upper plate is moving away from the lower one. The velocity profile in y-direction escalates with the escalation in nanoparticle volume fraction and suction parameter while the rotation parameter bids dual behavior with the escalating values. The velocity profile in x-direction bids the oscillatory behavior with the enhancement in nanoparticle volume fraction, rotation parameter, and magnetic parameter. The physical properties of carbon nanotubes, thermo physical properties of carbon nanotubes and nanofluid of some base fluids, and thermal conductivity of carbon nanotubes with different volume fractions are shown through tables.

Introduction

Presently, the goal of many investigators is to examine the nanotechnology which was presented by Choi ¹ in 1995. He brings into being higher thermal conductivity of nanofluids than base fluids such as ethylene glycol and water. Nanofluid is one of the key parts of the nanotechnology which is potential heat transfer fluid. Carbon nanotube (CNT) nanofluids have many application in structural materials (textiles, body arms, concrete, polyethylene, sports equipment, bridges, flywheels, fire protections), electromagnetic (buckypaper, light bulb filament, magnets, solar cells, electromagnetic antenna), electroacoustic (loudspeaker), chemicals (air pollution filter, water filter, chemical nanowires, sensors), mechanicals (oscillator, waterproof), optical, electrical circuits, interconnects, transistors, cancer treatment, cardiac autonomic regulation, in drug delivery, platelet activation, tissues regeneration, super capacitors, thin film electronics, actuators, and so on. Due to these applications the researchers are busy to work on CNTs. Kang et al. ² revealed the precise results of Choi by testing experimentally. The preparation impacts of CNT nanofluid stability has been inspected by Ganji et al. ³ Haq et al. ⁴ reported that single-walled carbon nanotubes (SWCNTs) have higher Nusselt number and skin friction than multi-walled carbon nanotubes (MWCNTs) by considering water as a base fluid. Liu et al. ⁵ deliberated by studying ethylene glycol and synthetic engine oil in the existence of MWCNTs that ethylene glycol with CNTs have higher thermal conductivities than ethylene glycol suspension without CNTs. The theory of nanofluid paste over an exponentially stretching sheet has been presented by Nadeem and Lee. ⁶ With Reynolds’ model and Vogel’s model, Ellahi et al. ⁷ analytically probed the series solution of non-Newtonian nanofluids. Nadeem et al. ⁸ numerically studied the magnetohydrodynamic boundary layer flow of Maxwell fluid over a stretching sheet in the presence of nanoparticles. The other related studies to boundary layer flow can be seen in Nadeem and Haq.^9,10 Sheikholeslami ¹¹ numerically examined the nanofluid free convection in a porous enclosure under the impact of electric field. Shah et al. ¹² investigated the micropolar nanofluid flow of Casson fluid between two rotating parallel plates with hall effect. Nasir et al. ¹³ investigated the three-dimensional rotating flow of magnetohydrodynamic SWCNTs over a stretching sheet under the influence thermal radiation. Dawar et al. ¹⁴ examined the magnetohydrodynamic CNTs Casson nanofluid and radiative heat transfer in rotating channels. Sheikholeslami ¹⁵ analyzed the nanofluid flow magnetic field in a porous media considering magnetic field and Brownian motion. The other related studies of Sheikholeslami can be seen in the literature.^16–26

Entropy is a disorder of a system and surrounding. It occurs when heat transmission occurs, because some additional movements happen when it moves, for example, molecular vibration and friction, displacements of molecules, kinetic energy, and spin movements, due to which loss of useful heat occurs and thus heat cannot be transmitted fully into work. Chaos in a system and surrounding created due to these additional movements. This microscopic chaos results in macroscopic level occurs because of some unnecessary irreversibilities. For example, electric resistance, friction, mixing of fluids, unstained expansion, chemical reaction, inelastic deformation of solids, and unnecessary heat transmission in finite temperature difference. The entropy generation was originally formulated by Bejan ²⁷ over an unsteady stretching surface. Sarojamma et al. ²⁸ has examined the entropy generation on a thin film flow. Along with an inclined permeable surface, Soomro et al. ²⁹ recently examined numerically the entropy generation in magnetohydrodynamic (MHD) water-based CNTs. Mansour and Sahin ³⁰ examined the entropy generation rate in a laminar viscous flow in a circular flow and have deliberated that the entropy generation rate is high near the wall than that of the center of the pipe. Rashidi at al. ³¹ examined entropy generation over a permeable stretching sheet. The related study about entropy generation can be seen in the literature.^32–35 The flow of nanoparticles in a rotating system through entropy generation has been examined by Hayat et al. ³⁶ The flow of nanofluids with spherical heat source/sink through entropy generation has been examined by Nouri et al. ³⁷ Over a stretching surface, the flow of Jeffery nanofluids through entropy generation has been examined by Dalir et al. ³⁸ The unsteady squeezing flow of viscous fluid through entropy generation has been examined by Ahmed et al. ³⁹ The related studies about entropy generation can be seen in Rashidi et al.^40–42 Afridi et al. ⁴³ investigated the second law analysis of boundary layer flow with energy dissipation and variable fluid properties over an exponentially stretching sheet through entropy generation. They found that entropy generation can be minimizing by escalating viscosity parameter and reducing the operating temperature. Some important study of nanofluid with respect to their modern application can be seen in the literature.^44–51

The focal intention of this inspection is to examine the three-dimensional squeezing flow of CNTs nanofluids considering MHD and thermal radiation impacts through entropy generation. The influences of emerging parameters are deliberated through graphs. The formulated problem has been solved by homotopy analysis method (HAM).^52–58 The impacts of emerging parameters on velocities profiles, temperature profile, entropy generation, Bejan number, skin fraction coefficient, and local Nusselt number have been shown through graphs and discussed in detail as well. The formulation of the problem is presented in section “Problem statement.” In section “Entropy analysis and Bejan number,” the formulation of entropy generation and Bejan number is presented. In section “Solution by HAM,” the results and discussion of the graphs is presented. The analytical solution of the problem is presented in section “Results and discussion.” The central points of this investigation are presented in section “Conclusion.”

Problem statement

Consider three-dimensional squeezing flow of CNTs nanofluids (SWCNT, MWCNT-nanoparticles) based on water $({\mathrm{H}}_2\mathrm{O})$ in a rotating channel with fixed and porous lower wall. The magnetic field $B_0$ is taken along y-direction and the nanofluid flow taken under the influence of thermal radiation. The lower wall of the channel is stretched along $y=0$ with velocity $U_w=\mathrm{ax}/(1-\alpha t)$ along x-axis and $v=-(V_0/(1-\alpha t))$ along y-axis. The channel height is varying with stretching out and shriveling from top to bottom of the walls. This height change follows from $v_h=\mathrm{dh}/\mathrm{dt}$ . For the stated problem, all the assumption and conditions are expressed in equation (9). The modeling of this problem is as follows ⁵²

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(1)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+\frac{2\beta }{1-\alpha t}w+\frac{1}{{\rho }_{\mathrm{nf}}}\frac{\partial p^{*}}{\partial x}}\\ -\frac{{\mu }_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2})+\frac{{\sigma }_{\mathrm{nf}}{B}_0^2}{{\rho }_{\mathrm{nf}}}\frac{u}{1-\alpha t}=0\hfill \end{array}$$

(2)

$$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+\frac{1}{{\rho }_{\mathrm{nf}}}\frac{\partial p^{*}}{\partial y}-\frac{{\mu }_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2})=0$$

(3)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}-\frac{2\beta }{1-\alpha t}w-\frac{{\mu }_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}(\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2})}\\ +\frac{{\sigma }_{\mathrm{nf}}{B}_0^2}{{\rho }_{\mathrm{nf}}}\frac{w}{1-\alpha t}=0\hfill \end{array}$$

(4)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial w}{\partial y}-\frac{{\mu }_{\mathrm{nf}}}{{\left(\rho C_p\right)}_{\mathrm{nf}}}(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2})}\\ -\frac{1}{{\left(\rho C_p\right)}_{\mathrm{nf}}}\frac{\partial \mathrm{Qr}}{\partial y}-\frac{{\mu }_{\mathrm{nf}}}{{\left(\rho C_p\right)}_{\mathrm{nf}}}\hfill \\ \{4{\left(\frac{\partial u}{\partial x}\right)}^2+(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y})+{\left(\frac{\partial w}{\partial x}\right)}^2+{\left(\frac{\partial w}{\partial y}\right)}^2\}\hfill \\ +\frac{{\sigma }_{\mathrm{nf}}{B}_0^2}{{\rho }_{\mathrm{nf}}}\frac{(u^2+w^2)}{1-\alpha t}=0\hfill \end{array}$$

(5)

Here $u,v,\mathrm{and}w$ are the components of velocity. Also $p^{*},{\sigma }_{\mathrm{nf}},{\mu }_{\mathrm{nf}},{\rho }_{\mathrm{nf}},(\rho C_p{)}_{\mathrm{nf}},T,\mathrm{Qr}$ indicate the modified pressure, electrical conductivity, dynamic viscosity, density, heat capacitance, temperature, and radiative heat flux, respectively (Figure 1).

Figure 1.

Geometrical representation of the problem.

The thermal flux is defined as follows

$$\mathrm{Qr}=-\frac{4{\sigma }^{*}\partial T^4}{3k^{*}\partial y}$$

(6)

where ${\sigma }^{*}$ is Stefan-Boltzman constant and $k^{*}$ is the absorption coefficient. By Taylor series expansion, $T^4$ in terms of $T_{\infty }$ is defined as follows

$$T^4\approx 4{\mathrm{TT}}_{\infty }^3-3T_{\infty }^3$$

(7)

Substituting equations (6) and (7) in equation (5), we obtain

$$\begin{array}{c}\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial w}{\partial y}-\frac{{\mu }_{\mathrm{nf}}}{{\left(\rho C_p\right)}_{\mathrm{nf}}}(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2})\hfill \\ -\frac{1}{{\left(\rho C_p\right)}_{\mathrm{nf}}}\frac{16{\sigma }^{*}}{3k^{*}}\frac{\partial }{\partial y}\left(T^3\frac{\partial T}{\partial y}\right)-\frac{{\mu }_{\mathrm{nf}}}{{\left(\rho C_p\right)}_{\mathrm{nf}}}\hfill \\ \{4{\left(\frac{\partial u}{\partial x}\right)}^2+(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y})+{\left(\frac{\partial w}{\partial x}\right)}^2+{\left(\frac{\partial w}{\partial y}\right)}^2\}\hfill \\ +\frac{{\sigma }_{\mathrm{nf}}{B}_0^2}{{\rho }_{\mathrm{nf}}}\frac{(u^2+w^2)}{1-\alpha t}=0\hfill \end{array}$$

(8)

The boundary conditions for the above stated problem are as follows

$$\begin{array}{c}u=U_w=\frac{\mathrm{ax}}{1-\alpha t},v=-\frac{V_0}{1-\alpha t},\hfill \\ w=0,T=T_w\mathrm{at}y=0\hfill \\ u=0,v=V_h=-\frac{\alpha }{2}{\left(\frac{\upsilon }{a(1-\alpha t)}\right)}^{1/2},\hfill \\ w=0,T=T_h\mathrm{as}y=h\left(t\right)\hfill \end{array}$$

(9)

The density of the nanofluid is as follows

$${\rho }_{\mathrm{nf}}=(1-\phi ){\rho }_f+\phi {\rho }_{\mathrm{CNT}}$$

(10)

The heat capacity of the nanofluid is as follows

$${\left(\rho C_p\right)}_{\mathrm{nf}}=(1-\phi ){\left(\rho C_p\right)}_f+\phi {\left(\rho C_p\right)}_{\mathrm{CNT}}$$

(11)

The dynamic viscosity of the nanofluid is as follows

$${\mu }_{\mathrm{nf}}=\frac{{\mu }_f}{{(1-\phi )}^{2.5}}$$

(12)

The thermal conductivity of the nanofluid is as follows

$$k_{\mathrm{nf}}=\frac{1-\phi +2\phi \frac{k_{\mathrm{CNT}}}{k_{\mathrm{CNT}}-k_f}\ln \frac{k_{\mathrm{CNT}}+k_f}{2k_f}}{1-\phi +2\phi \frac{k_f}{k_{\mathrm{CNT}}-k_f}\ln \frac{k_{\mathrm{CNT}}+k_f}{2k_f}}{k}_f$$

(13)

The electrical conductivity of the nanofluid is as follows

$${\sigma }_{\mathrm{nf}}=1+\frac{3(\frac{{\sigma }_{\mathrm{CNT}}}{{\sigma }_f}-1)\phi }{(\frac{{\sigma }_{\mathrm{CNT}}}{{\sigma }_f}+2)-(\frac{{\sigma }_{\mathrm{CNT}}}{{\sigma }_f}-1)\phi }{\sigma }_f$$

(14)

Here, ${\rho }_{\mathrm{CNT}},{\rho }_f,\phi ,k_f,\mathrm{and}{k}_{\mathrm{CNT}}$ are the density of CNTs, the fluid density, volume fraction of nanoparticles, the thermal conductivity of base fluid and CNTs, respectively.

In order to simplify equations (1)–(4), (8), and (9), the following definitions are given

$$\begin{array}{c}u=U_{w}f\prime \left(\lambda \right),v={\left(\frac{a\nu }{a(1-\alpha t)}\right)}^{1/2}f\left(\lambda \right),\hfill \\ w=U_{w}g\left(\lambda \right),\theta \left(\lambda \right)=\frac{T-T_w}{T_w-T_h},\lambda =\frac{y}{h\left(t\right)}\hfill \end{array}$$

(15)

Here $f\prime (\lambda ),f(\lambda ),g(\lambda ),\mathrm{and}\theta (\lambda )$ is the velocity function along x-axis, y-axis, rotational velocity function, and dimensionless temperature function, respectively, and $\lambda$ is the local similarity variable.

By substituting equation (15) into equations (1)–(4), (8), and (9), the final obtained equations are as follows

$$\begin{array}{c}{f}^{\mathrm{iv}}-{(1-\phi )}^{2.5}{Z}_1\hfill \\ [(f\prime f″-\mathrm{ff}″\prime )+2\mathrm{\Omega }g\prime +\frac{S}{2}(3f″+\lambda f″\prime )]-Z_5\mathrm{Mf}″=0\hfill \end{array}$$

(16)

$$\begin{array}{c}g″+{(1-\phi )}^{2.5}{Z}_1\hfill \\ [(\mathrm{fg}\prime -\mathrm{gf}\prime )+2\mathrm{\Omega }f\prime -S(g+\frac{\lambda }{2}g\prime )]-Z_5\mathrm{Mg}=0\hfill \end{array}$$

(17)

$$\begin{array}{c}(1+\frac{R}{Z_4})\theta ″+\frac{Z_2\mathrm{Pr}}{Z_4{(1-\phi )}^{2.5}}(\theta f\prime -\frac{S}{2}\lambda \theta \prime )\hfill \\ +\frac{Z_2}{Z_4}\mathrm{Pr}\left\{\begin{array}{c}\mathrm{Ec}(4f^2+g^2)\hfill \\ \multicolumn{1}{c}{+E{c}_m({f″}^2+{g\prime }^2)}\end{array}\right\}-Z_{5}M({f\prime }^2+g^2)=0\hfill \end{array}$$

(18)

With the following boundary conditions

$$\begin{array}{c}f\left(0\right)=A,f\prime \left(0\right)=1,g\left(0\right)=0,\theta \left(0\right)=1\hfill \\ f\left(1\right)=\frac{S}{2},f\prime \left(1\right)=0,g\left(1\right)=0,\theta \left(1\right)=0\hfill \end{array}$$

(19)

In equations (16)–(19), the followings parameters and numbers are $\mathrm{\Omega }=\beta /a$ rotation parameter, $S=\alpha /a$ squeezing parameter, $R=(16{\sigma }^{*}{T}^3)/(3k^{*}{k}_f)$ radiation parameter, $M={\sigma }_f{B}_0^{2}h/{\rho }_f{\nu }_f$ magnetic parameter, $A=V_0/\mathrm{ah}$ suction parameter, $\mathrm{Pr}=(({\mu }_f(C_p{)}_f)/k)$ Prandtl number, $\mathrm{Ec}=v^2/(h^2(\mathrm{Cp}{)}_f(T_w-T_h))$ Eckert number, and $E{c}_m=U_w^2/((\mathrm{Cp}{)}_f(T_w-T_h))$ modified Eckert number. Also, $Z_1={\rho }_{\mathrm{nf}}=(1-\phi ){\rho }_f+\phi {\rho }_{\mathrm{CNT}},$ $Z_2=(\rho C_p{)}_{\mathrm{nf}}=(1-\phi )(\rho C_p{)}_f+\phi {(\rho C_p)}_{\mathrm{CNT}},$ $Z_3={\mu }_{\mathrm{nf}}=({\mu }_f/(1-\phi {)}^{2.5}),$ $Z_4=(k_{\mathrm{nf}}/k_f)=(1-\phi +2\phi (k_{\mathrm{CNT}}/(k_{\mathrm{CNT}}-k_f))\ln ((k_{\mathrm{CNT}}+k_f)/2k_f))/(1-\phi +2\phi (k_f/(k_{\mathrm{CNT}}-k_f))\ln ((k_{\mathrm{CNT}}+k_f)/2k_f)),$ and $Z_5={\sigma }_{\mathrm{nf}}=1+(3(({\sigma }_{\mathrm{CNT}}/{\sigma }_f)-1)\phi /(({\sigma }_{\mathrm{CNT}}/{\sigma }_f)+2)-(({\sigma }_{\mathrm{CNT}}/{\sigma }_f)-1)\phi ){\sigma }_f$ .

The skin fraction coefficient $(C{f}_x)$ and local Nusselt number $(N{u}_x)$ are defined as follows

$${\left(\mathrm{Cf}\right)}_{\mathrm{lower}}={\left(\frac{{\mu }_{\mathrm{nf}}\left(\frac{\partial u}{\partial y}\right)}{{\rho }_{\mathrm{nf}}{U}_0^2}\right)}_{y=0},{\left(\mathrm{Cf}\right)}_{\mathrm{upper}}={\left(\frac{{\mu }_{\mathrm{nf}}\left(\frac{\partial u}{\partial y}\right)}{{\rho }_{\mathrm{nf}}{U}_0^2}\right)}_{y=h\left(t\right)}$$

(20)

$$\begin{array}{c}{\left(\mathrm{Nu}\right)}_{\mathrm{lower}}={\left(\frac{X\left(\frac{\partial T}{\partial y}\right)}{k_{\mathrm{nf}}(T_w-T_h)}\right)}_{y=0}+\mathrm{Qr},\hfill \\ {\left(\mathrm{Nu}\right)}_{\mathrm{upper}}={\left(\frac{X\left(\frac{\partial T}{\partial y}\right)}{k_{\mathrm{nf}}(T_w-T_h)}\right)}_{y=h\left(t\right)}+\mathrm{Qr}\hfill \end{array}$$

(21)

After simplifying equations (20) and (21), $(C{f}_x)$ and $(N{u}_x)$ are defined as follows

$$\begin{array}{c}\multicolumn{1}{c}{{\left(\mathrm{Cf}{\mathrm{Re}}_x^{1/2}\right)}_{\mathrm{lower}}=\left(\frac{1}{Z_1{(1-\phi )}^{5/2}}\right)f″\left(0\right),{\left(\mathrm{Cf}{\mathrm{Re}}_x^{1/2}\right)}_{\mathrm{upper}}}\\ =\left(\frac{1}{Z_1{(1-\phi )}^{5/2}}\right)f″\left(1\right)\hfill \end{array}$$

(22)

$$\begin{array}{c}\multicolumn{1}{c}{{\left(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2}\right)}_{\mathrm{lower}}=-(Z_4+R)\theta \prime \left(0\right),{\left(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2}\right)}_{\mathrm{upper}}}\\ =-(Z_4+R)\theta \prime \left(1\right)\hfill \end{array}$$

(23)

where ${\mathrm{Re}}_x=(U_{w}X/{\nu }_f)$ is the local Reynolds number.

Entropy analysis and Bejan number

Entropy is a disorder of a system and surrounding. It occurs when heat transmission occurs, because some additional movements happen when it moves. The loss of useful heat occurs due to kinetic energy, spin movements, and so on and thus heat cannot be transmitted fully into work. Due to these additional movements, Chaos in a system and surrounding created. For the above stated problem, the local entropy generation rate is ⁵³

$$\begin{array}{c}\multicolumn{1}{c}{E_G=\frac{k_{\mathrm{nf}}}{T_w^2}(1+\frac{16{\sigma }^{*}{T}^3}{3k^{*}{k}_f})\{{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2\}}\\ +\frac{{\mu }_{\mathrm{nf}}}{T_w}\left\{\begin{array}{c}4{\left(\frac{\partial u}{\partial x}\right)}^2+(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y})\hfill \\ \multicolumn{1}{c}{+{\left(\frac{\partial w}{\partial x}\right)}^2+{\left(\frac{\partial w}{\partial y}\right)}^2}\end{array}\right\}+\frac{{\sigma }_{\mathrm{nf}}}{T_w}{B}_0^2(u^2+w^2)\hfill \end{array}$$

(24)

Executing equation (15), the above equation becomes

$$\begin{array}{c}\mathrm{Ns}={E_G}_0[(Z_4+R){\theta \prime }^2+\frac{\gamma \mathrm{Pr}{Z}_3^2}{Z_1{Z}_4}\{\mathrm{Ec}(4{f\prime }^2+g^2)+\frac{Z_{1}E{c}_m}{Z_3}(f\prime +{g\prime }^2)\}+\frac{Z_5\mathrm{ME}{c}_m}{Z_3}({f\prime }^2+g^2)]\hfill \end{array}$$

(25)

where $E_{G0},\gamma$ is the characteristics entropy generation and dimensionless temperature, respectively, whose expression are given by

$${E_G}_0=\frac{k_f{(T_w-T_h)}^2}{T_h^2{h}^2},g=\frac{T_h}{T_w-T_h}$$

(26)

The non-dimensional form of equation (25) is as follows

$$\mathrm{Ns}=\frac{E_G}{{E_G}_0}=\left[\begin{array}{c}(Z_4+R){\theta \prime }^2+\frac{\gamma \mathrm{Pr}{Z}_3^2}{Z_1{Z}_4}\{\mathrm{Ec}(4{f\prime }^2+g^2)+\frac{Z_{1}E{c}_m}{Z_3}(f\prime +{g\prime }^2)\}\hfill \\ \multicolumn{1}{c}{+\frac{Z_5\mathrm{ME}{c}_m}{Z_3}({f\prime }^2+g^2)}\end{array}\right]$$

(27)

The Bejan number Be is defined as follows

$$\mathrm{Be}=\frac{(Z_4+R){\theta \prime }^2}{\left[\begin{array}{c}(Z_4+R){\theta \prime }^2+\frac{\gamma \mathrm{Pr}{Z}_3^2}{Z_1{Z}_4}\{\mathrm{Ec}(4{f\prime }^2+g^2)+\frac{Z_{1}E{c}_m}{Z_3}(f\prime +{g\prime }^2)\}\hfill \\ \multicolumn{1}{c}{+\frac{Z_5\mathrm{ME}{c}_m}{Z_3}({f\prime }^2+g^2)}\end{array}\right]}$$

(28)

It is obvious from equation (28) that the Bejan number takes values from 0 to 1. $\mathrm{Be}=0$ leads to the case of irreversibility, due to magnetic field and fluid fraction dictate over the heat transfer irreversibility, while $\mathrm{Be}=1$ shows domination of irreversibility due to heat transfer on entropy generation. $\mathrm{Be}=1/2$ leads to the case where the irreversibility due to heat transfer is same as the sum of the irreversibility due to magnetic fields and fluid fraction.

Solution by HAM

To solve equations (16)–(18) with the boundary condition (19), we use HAM with the succeeding process.

The initial suppositions are chosen as follows

$$\begin{array}{c}\multicolumn{1}{c}{f_0(\lambda )=A+\lambda +(\frac{3}{2}S-2-3A){\lambda }^2}\\ +(1+2A-S){\lambda }^3,g_0(\lambda )=0,{\theta }_0(\lambda )=1-\lambda \hfill \end{array}$$

(29)

The $L_f,L_g,\mathrm{and}{L}_{\theta }$ are taken as follows

$$L_f(f)=f^{\mathrm{iv}},L_g(g)=g″,L_{\theta }(\theta )=\theta ″$$

(30)

which have the succeeding properties

$$\begin{array}{c}{L}_{\overline{F}}(r_1+r_2\lambda +r_3{\lambda }^2+r_4{\lambda }^3)=0,\hfill \\ L_{\overline{G}}(r_5+r_6\lambda )=0,L_{\overline{\varphi }}(r_7+r_8\lambda )=0\hfill \end{array}$$

(31)

where the general solution of the problem have $r_i(i=1-8)$ constants:

The resultant non-linear operators $N_f,N_g,\mathrm{and}{N}_{\theta }$ are indicated as follows

$$\begin{array}{c}{N}_f[f(\lambda ;\tau ),g(\lambda ;\tau )]=\frac{{\partial }^{4}f(\lambda ;\tau )}{\partial {\lambda }^4}-{(1-\phi )}^{2.5}{Z}_1\hfill \\ \left[\begin{array}{c}\{\frac{\partial f(\lambda ;\tau )}{\partial \lambda }\frac{{\partial }^{2}f(\lambda ;\tau )}{\partial {\lambda }^2}-f(\lambda ;\tau )\frac{{\partial }^{3}f(\lambda ;\tau )}{\partial {\lambda }^3}\}\hfill \\ \multicolumn{1}{c}{+2\mathrm{\Omega }\frac{\partial g(\lambda ;\tau )}{\partial \lambda }-\frac{S}{2}\{3\frac{\partial f^2(\lambda ;\tau )}{\partial {\lambda }^2}+\lambda \frac{{\partial }^{3}f(\lambda ;\tau )}{\partial {\lambda }^3}\}}\end{array}\right]-Z_{5}M\frac{\partial f^2(\lambda ;\tau )}{\partial {\lambda }^2}=0\hfill \end{array}$$

(32)

$$\begin{array}{c}{N}_g[f(\lambda ;\tau ),g(\lambda ;\tau )]=\frac{{\partial }^{2}g(\lambda ;\tau )}{\partial {\lambda }^2}+{(1-\phi )}^{2.5}{Z}_1\hfill \\ \left[\begin{array}{c}\{f(\lambda ;\tau )\frac{\partial g(\lambda ;\tau )}{\partial \lambda }-g(\lambda ;\tau )\frac{\partial f(\lambda ;\tau )}{\partial \lambda }\}\hfill \\ \multicolumn{1}{c}{+2\mathrm{\Omega }\frac{\partial f(\lambda ;\tau )}{\partial \lambda }-S\{g(\lambda ;\tau )+\frac{\lambda }{2}\frac{\partial g(\lambda ;\tau )}{\partial \lambda }\}}\end{array}\right]-Z_5\mathrm{Mg}(\lambda ;\tau )=0\hfill \end{array}$$

(33)

$$\begin{array}{c}{N}_{\theta }[f(\lambda ;\tau ),g(\lambda ;\tau ),\theta (\lambda ;\tau )]\hfill \\ =(1+\frac{R}{Z_4})\frac{{\partial }^2\theta (\lambda ;\tau )}{\partial {\lambda }^2}+\frac{Z_2}{Z_4}\mathrm{Pr}\left\{\begin{array}{c}\theta (\lambda ;\tau )\frac{\partial f(\lambda ;\tau )}{\partial \lambda }\hfill \\ \multicolumn{1}{c}{-\frac{S}{2}\lambda \frac{\partial \theta (\lambda ;\tau )}{\partial \lambda }}\end{array}\right\}\hfill \\ +\{\mathrm{Ec}(4{(f(\lambda ;\tau ))}^2+{(g(\lambda ;\tau ))}^2)+E{c}_m({\left(\frac{{\partial }^{2}f(\lambda ;\tau )}{\partial {\lambda }^2}\right)}^2+{\left(\frac{\partial g(\lambda ;\tau )}{\partial \lambda }\right)}^2)\}\hfill \\ -Z_5\mathrm{M}({\left(\frac{\partial f(\lambda ;\tau )}{\partial \lambda }\right)}^2+{(g(\lambda ;\tau ))}^2)\hfill \end{array}$$

(34)

The 0th-order problems from equations (16)–(18) are as follows

$$(1-\tau )L_f[f(\lambda ;\tau )-f_0(\lambda )]=\tau {\hslash }_f{N}_f[f(\lambda ;\tau ),g(\lambda ;\tau )]$$

(35)

$$(1-\tau )L_g[g(\lambda ;\tau )-g_0(\lambda )]=\tau {\hslash }_g{N}_g[f(\lambda ;\tau ),g(\lambda ;\tau )]$$

(36)

$$(1-\tau )L_{\theta }[\theta (\lambda ;\tau )-{\theta }_0(\lambda )]=\tau {\hslash }_{\theta }{N}_{\theta }[f(\lambda ;\tau ),g(\lambda ;\tau ),\theta (\lambda ;\tau )]$$

(37)

The equivalent boundary conditions are as follows

$$\begin{array}{c}{f(\lambda ;\tau )|}_{\lambda =0}=A,{\frac{\partial f(\lambda ;\tau )}{\partial \lambda }|}_{\lambda =0}=1,\hfill \\ {f(\lambda ;\tau )|}_{\lambda =1}=\frac{S}{2},{\frac{\partial f(\lambda ;\tau )}{\partial \lambda }|}_{\lambda =1}=0\hfill \\ {g(\lambda ;\tau )|}_{\lambda =0}=0,{g(\lambda ;\tau )|}_{\lambda =1}=0\hfill \\ {\theta (\lambda ;\tau )|}_{\lambda =0}=1,{\theta (\lambda ;\tau )|}_{\lambda =1}=0\hfill \end{array}$$

(38)

When $\tau =0\mathrm{and}\tau =1$ , we have

$$f(\lambda ;1)=f(\lambda ),g(\lambda ;1)=g(\lambda ),\mathrm{and}\theta (\lambda ;1)=\theta (\lambda )$$

(39)

Expanding $f(\lambda ;\tau ),g(\lambda ;\tau ),\mathrm{and}\theta (\lambda ;\tau )$ by Taylor’s series

$$\begin{array}{c}f(\lambda ;\tau )=f_0(\lambda )+\sum _{q=1}^{\infty }{f}_q(\lambda ){\tau }^q\hfill \\ g(\lambda ;\tau )=g_0(\lambda )+\sum _{q=1}^{\infty }{g}_q(\lambda ){\tau }^q\hfill \\ \theta (\lambda ;\tau )={\theta }_0(\lambda )+\sum _{q=1}^{\infty }{\theta }_q(\lambda ){\tau }^q\hfill \end{array}$$

(40)

where

$$\begin{array}{c}{f}_q(\lambda )={\frac{1}{q!}\frac{\partial f(\lambda ;\tau )}{\partial \lambda }|}_{\tau =0},g_q(\lambda )={\frac{1}{q!}\frac{\partial g(\lambda ;\tau )}{\partial \lambda }|}_{\tau =0},\hfill \\ \mathrm{and}{\theta }_q(\lambda )={\frac{1}{q!}\frac{\partial \theta (\lambda ;\tau )}{\partial \lambda }|}_{\tau =0}\hfill \end{array}$$

(41)

The secondary constraints ${\hslash }_f,{\hslash }_g,\mathrm{and}{\hslash }_{\theta }$ are selected such that the series (40) converges at $\tau =1$ , changing $\tau =1$ in equation (40), we get

$$\begin{array}{c}f(\lambda )=f_0(\lambda )+\sum _{q=1}^{\infty }{f}_q(\lambda )\hfill \\ g(\lambda )=g_0(\lambda )+\sum _{q=1}^{\infty }{g}_q(\lambda )\hfill \\ \theta (\lambda )={\theta }_0(\lambda )+\sum _{q=1}^{\infty }{\theta }_q(\lambda )\hfill \end{array}$$

(42)

The $q\mathrm{th}$ -order problem satisfies the following

$$\begin{array}{c}{L}_f[f_q(\lambda )-{\chi }_q{f}_{q-1}(\lambda )]={\hslash }_f{U}_q^f(\lambda ),\hfill \\ L_g[g_q(\lambda )-{\chi }_q{g}_{q-1}(\lambda )]={\hslash }_g{U}_q^g(\lambda ),\hfill \\ L_{\theta }[{\theta }_q(\lambda )-{\chi }_q{\theta }_{q-1}(\lambda )]={\hslash }_{\theta }{U}_q^{\theta }(\lambda ).\hfill \end{array}$$

(43)

The equivalent boundary conditions are as follows

$$\begin{array}{c}{f}_q(0)={f\prime }_q(0)=f_q(1)={f\prime }_q(1)=0\hfill \\ g_q(0)=g_q(1)=0\hfill \\ {\theta }_q(0)={\theta }_q(1)=0\hfill \end{array}$$

(44)

Here

$$\begin{array}{c}{U}_q^f(\lambda )=f_{q-1}^{\mathrm{iv}}-{(1-\phi )}^{2.5}{Z}_1\hfill \\ [\{\sum _{k=0}^{q-1}{f\prime }_{q-1-k}{f″}_k-\sum _{k=0}^{q-1}{f}_{q-1-k}{f″\prime }_k\}\hfill \\ +2\mathrm{\Omega }{g\prime }_{q-1}-\frac{S}{2}\{3{f″}_{q-1}+\lambda {f″\prime }_{q-1}\}]-Z_5{\mathrm{Mf}}_{q-1}^{″}\hfill \end{array}$$

(45)

$$\begin{array}{c}{U}_q^g(\lambda )=g_{q-1}^{″}+{(1-\phi )}^{2.5}{Z}_1\hfill \\ [\{\sum _{k=0}^{q-1}{f}_{q-1-k}{g\prime }_k-\sum _{k=0}^{q-1}{g}_{q-1-k}{f\prime }_k\}\hfill \\ +2\mathrm{\Omega }{f″}_{q-1}-S\{g_{q-1}+\frac{\lambda }{2}{g\prime }_{q-1}\}\hfill \end{array}]-Z_{5}M{g}_{q-1}$$

(46)

$$\begin{array}{c}\multicolumn{1}{c}{U_q^{\theta }(\lambda )=(1+\frac{R}{Z_4}){\theta ″}_{q-1}+\frac{Z_2}{Z_4}\mathrm{Pr}\{\sum _{k=0}^{q-1}{\theta }_{q-1-k}{f\prime }_k-\frac{S}{2}\lambda {\theta \prime }_{q-1}\}}\\ +\frac{Z_2}{Z_4}\mathrm{Pr}\{\mathrm{Ec}(4\sum _{k=0}^{q-1}{f}_{q-1-k}{f}_k+\sum _{k=0}^{q-1}{g}_{q-1-k}{g}_k)+E{c}_m(\sum _{k=0}^{q-1}{f″}_{q-1-k}{f″}_k+\sum _{k=0}^{q-1}{g\prime }_{q-1-k}{g\prime }_k)\}\hfill \\ -Z_{5}M(\sum _{k=0}^{q-1}{f\prime }_{q-1-k}{f\prime }_k+\sum _{k=0}^{q-1}{g}_{q-1-k}{g}_k)\hfill \end{array}$$

(47)

where

$${\chi }_q=\{\begin{array}{c}0,\mathrm{if}\tau \le 1\hfill \\ \multicolumn{1}{c}{1,\mathrm{if}\tau >1}\end{array}$$

Results and discussion

In this segment, we have briefly studied the influence of emerging parameters and non-dimensional numbers on linear and angular velocity functions $(f(\lambda ),f\prime (\lambda ),g(\lambda ))$ , temperature function $(\theta (\lambda ))$ , entropy generation $(\mathrm{Ns})$ , Bejan number $(\mathrm{Be})$ , skin fraction coefficient $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2})$ and local Nusselt number $(\mathrm{Nu}{\mathrm{Re}}_x^{1/2})$ for both SWCNTs and MWCNTs based on water $({\mathrm{H}}_2\mathrm{O})$ . These emerging parameters and non-dimensional numbers are nanoparticle volume fraction $(\phi )$ , rotation parameter $(\mathrm{\Omega })$ , squeeze parameter $(S)$ , suction parameter $(A)$ , radiation parameter $(R)$ , magnetic parameter $(M)$ , Eckert number $(\mathrm{Ec})$ , and modified Eckert number $(E{c}_m)$ , respectively.

Velocity functions $(f(\lambda ),f\prime (\lambda ),$ and $g(\lambda ))$

Figures 2–16 are plotted to see the influence of emerging parameters and non-dimensional numbers on linear and angular velocity functions $(f(\lambda ),f\prime (\lambda ),\mathrm{and}g(\lambda ))$ for both SWCNTs and MWCNTs based on water $({\mathrm{H}}_2\mathrm{O})$ . The impacts of nanoparticle volume fraction with increasing amount on $(f(\lambda ),f\prime (\lambda ),\mathrm{and}g(\lambda ))$ are shown in Figures 2–4. Figure 2 indicates the impact of $(\phi )$ on $f(\lambda )$ . From here, we see that the increasing values of $(\phi )$ shows enhancement in velocity function $f(\lambda )$ in y-direction for both SWCNTs and MWCNTs. This leads us to the claim that $f(\lambda )$ permanently has a superior value at $(S>0)$ that of $(S<0)$ . This is due to the fact that the density of SWCNTs is greater than those of MWCNTs. Figure 3 is plotted to see the impact of $(\phi )$ on $f\prime (\lambda )$ . In the interval $0\le \lambda \le 4$ , the increasing rate of nanoparticle in velocity function along x-direction is declining, while in the interval $4<\lambda \le 1$ the variation in nanoparticle volume fraction along x-direction shows escalation. It should be illustrious that when $S>0$ , the variations in velocity function along x-direction are higher. When $S<0$ , then this leads to the enhancement in rotational velocity function $g(\lambda )$ . This result is shown in Figure 4. It is also observed that $g(\lambda )$ has bigger value for the case of MWCNTs than those of SWCNTs. It should be noteworthy that $g(\lambda )$ has larger value at $S>0$ than that of $S<0$ . The impacts of rotation parameter $(\mathrm{\Omega })$ on $(f(\lambda ),f\prime (\lambda ))$ $\mathrm{and}g(\lambda )$ are shown in Figures 5–7. It can be understood from Figure 5 that $(\mathrm{\Omega })$ gives dual behavior on $f(\lambda )$ . It can also be seen that the escalation in $(\mathrm{\Omega })$ shows reducing behavior in the interval $0\le \lambda \le 0.6$ for both cases of SWCNTs and MWCNTs, while in the interval $0.6<\lambda \le 1$ shows escalating behavior with the escalating values of rotation parameter. It is significant to mention that values of $f(\lambda )$ for MWCNTs is greater than those of SWCNTs. Also the velocity at $S>0$ is higher than that of $S<0$ . Figure 6 displays the impact of rotation parameter on $f\prime (\lambda )$ . From here, we see that $f\prime (\lambda )$ expresses an oscillatory behavior comparative to the variation in rotation parameter. At closed distances to the upper and lower wall of the channel, the increase in the rotation parameter, in the intervals $0\le \lambda \le 0.3$ and $0.8\le \lambda \le 1.0$ has decreased $f\prime (\lambda )$ in both cases of SWCNTs and MWCNTs. But in mean position of the channel $0.3<\lambda <0.8$ , the escalating behavior is observed. Figure 7 shows the impact of rotation parameter on $g(\lambda )$ . From here, we see that the variation in $(\mathrm{\Omega })$ has reverse relationship with $g(\lambda )$ . We see from here that for the case of MWCNTs, the velocity function $g(\lambda )$ is always higher than those of SWCNTs, and when $S>0$ the velocity enhances more quickly compared to when $S<0$ . The impacts of squeezing parameter $(S)$ on $(f(\lambda ),f\prime (\lambda ),\mathrm{and}g(\lambda ))$ are shown in Figures 8–10. Figure 8 depicts the impact of S on $f(\lambda )$ . From here, we see that the increase in the squeeze parameter $(S)$ enhances the velocity function in y-direction. Also, the velocity increases when the channel is contracted $(S>0)$ and the velocity reduces when the channel is stretched out $(S<0)$ . Similar impacts can be seen in Figures 9 and 10. In the dominant area of the channel, there is a reverse flow that produced a noteworthy decline in the velocity, while at the upper and lower walls of the channel, this reverse slowly vanishes. Figures 11–13 are schemed to see the impact of suction parameter $(A)$ on $(f(\lambda ),f\prime (\lambda ),\mathrm{and}g(\lambda ))$ . In both $(S>0)$ and $(S<0)$ cases, the increasing suction parameter enhances the velocity in y-direction, which is shown in Figure 11. Due to the fluid suction in the area near to the lower wall of the channel, a significant enhancement in velocity is observed. It should be noted that the velocity is enhanced by MWCNTs more than those of SWCNTs. Figure 12 indicates the relationship between the distribution rate of nanofluid velocity $f\prime (\lambda )$ and $(A)$ . From here, we see the reverse relationship between $(A)$ and $f\prime (\lambda )$ . The rise in $(A)$ results decline in $f\prime (\lambda )$ . It is interesting to mention that the reverse flow is greater when $(S<0)$ as compared to $(S>0)$ . In addition $f\prime (\lambda )$ in the case of MWCNTs is larger than those of SWCNTs. The same result is plotted for the rotational velocity function $g(\lambda )$ in Figure 13. With the difference that $g(\lambda )$ in the case of SWCNTs has been reduced more those of MWCNTs. Figures 14–16 are plotted to see the impact of magnetic parameter $(M)$ on velocity functions $(f(\lambda ),f\prime (\lambda ),\mathrm{and}g(\lambda ))$ . According to Lorentz force theory, the opposing of the flow is due to the existence of $(M)$ in electrically conducting fluid. This opposing force inclines to reduce the velocity of the flow. Figures 14 and 16 show reducing behavior in velocity functions $(f(\lambda )\mathrm{and}g(\lambda ))$ , while in Figure 15, the velocity function $f\prime (\lambda )$ shows reduction behavior in the interval $0.4\le \lambda \le 1.0$ . This behavior is due to the stretching of the lower plate. In Figure 14, it interesting to mention that the reverse flow is greater when $(S>0)$ as compared to $(S<0)$ . This behavior is due to the opposing force of $(M)$ . Also, we see from the figures that for the case of MWCNTs, the velocity functions $(f(\lambda ),f\prime (\lambda ),\mathrm{and}g(\lambda ))$ are always higher than those of SWCNTs.

Figure 2.

Influence of $\phi$ on $f(\lambda )$ .

Figure 3.

Influence of $\phi$ on $f\prime (\lambda )$ .

Figure 4.

Influence of $\phi$ on $g(\lambda )$ .

Figure 5.

Outcome of $\mathrm{\Omega }$ on $f(\lambda )$ .

Figure 6.

Influence of $\mathrm{\Omega }$ on $f\prime (\lambda )$ .

Figure 7.

Influence of $\mathrm{\Omega }$ on $g(\lambda )$ .

Figure 8.

Influence of S on $f(\lambda )$ .

Figure 9.

Influence of S on $f\prime (\lambda )$ .

Figure 10.

Influence of S on $g(\lambda )$ .

Figure 11.

Influence of A on $f(\lambda )$ .

Figure 12.

Influence of A on $f\prime (\lambda )$ .

Figure 13.

Influence A on $g(\lambda )$ .

Figure 14.

Influence of M on $f(\lambda )$ .

Figure 15.

Influence of M on $f\prime (\lambda )$ .

Figure 16.

Influence of M on $g(\lambda )$ .

Temperature function $(\theta (\lambda ))$

Figures 17–22 are plotted to see the influence of emerging parameters and non-dimensional numbers on temperature function $(\theta (\lambda ))$ for both SWCNTs and MWCNTs based on water $({\mathrm{H}}_2\mathrm{O})$ . Figure 17 shows the relationship between $(\phi )$ and $\theta (\lambda )$ . From here, we see that the variation in $\theta (\lambda )$ is reducing with the escalating values of $(\phi )$ . The case where the channel is contracted or the case where the channel is stretched out, this result is identical for both cases of SWCNTs and MWCNTs. Figure 18 shows the impact of $(S)$ on $\theta (\lambda )$ . From here, we see that growing $(S)$ in both cases $(S>0)$ and $(S<0)$ increases the nanofluid temperature for both SWCNTs and MWCNTs. Figure 19 shows the impact of M on $\theta (\lambda )$ . From here, we see the increasing behavior in $\theta (\lambda )$ with the increasing values of magnetic parameter. Figure 20 indicates the impact of $\left(R\right)$ on $\theta (\lambda )$ . The temperature function $\theta (\lambda )$ reduces with escalation in $(R)$ for both cases $(S>0)$ and $(S<0)$ . This behavior physically understands as the enhancement in heat transfer rate and the decline in $\theta (\lambda )$ caused by enhancement in $(R)$ . It should be noted that $\theta (\lambda )$ is more declined for the case of SWCNTs than those of MWCNTs when $S>0$ , but this result is opposite when $S<0$ . There is a direct relationship between $\theta (\lambda )$ and $(\mathrm{Ec})$ , and $(E{c}_m)$ . The influence of $(\mathrm{Ec})$ and $(E{c}_m)$ on $\theta (\lambda )$ for both $(S>0)$ and $(S<0)$ cases are shown in Figures 21 and 22, respectively. This direct relationship physically represents the relationship between the velocity and $(\mathrm{Ec})$ , and hence the kinetic energy. Escalating Eckert number escalates the intermolecular collision and this escalates the temperature function. In addition, the temperature function for MWCNTs increases more than those of SWCNTs in the case of $(S>0)$ , it is due to the fact that MWCNTs has less amount of thermal conductivity than the SWCNTs which results the reverse impact obtained in the case of $(S<0)$ .

Figure 17.

Influence of $\phi$ on $\theta (\lambda )$ .

Figure 18.

Influence of S on $\theta (\lambda )$ .

Figure 19.

Influence of M on $\theta (\lambda )$ .

Figure 20.

Influence of R on $\theta (\lambda )$ .

Figure 21.

Influence of Ec on $\theta (\lambda )$ .

Figure 22.

Influence of $E{c}_m$ on $\theta (\lambda )$ .

Entropy analysis (Ns) and Bejan number (Be)

Figures 23–32 are plotted to see the impact of emerging parameters and non-dimensional numbers on entropy generation rate $(\mathrm{Ns})$ and Bejan number $(\mathrm{Be})$ for both SWCNTs and MWCNTs. Figure 23 is plotted to see the impact of $(R)$ on $(\mathrm{Ns})$ for both cases of SWCNTs and MWCNTs. From Figure 23, we observed that as the radiative parameter $(R)$ becomes higher entropy generation rate across the channel escalates. This is due to the enhancement in emission of $(R)$ . The impact of $(R)$ on $(\mathrm{Be})$ is depicted in Figure 24. The rise in $(R)$ results the enhancement in $(\mathrm{Be})$ for both SWCNTs and MWCNTs. Figure 25 illustrates the influence of $(\mathrm{Ec})$ on $(\mathrm{Ns})$ . The escalating values of $(\mathrm{Ec})$ shows dual behavior in $(\mathrm{Ns})$ . In the region $(0\le \lambda \le 0.5)$ , it shows increasing behavior, while in the region $(0.5<\lambda \le 1.0)$ , it shows decreasing behavior. Figure 26 depicts the escalating behavior in $(\mathrm{Be})$ with the escalation in $(\mathrm{Ec})$ . Figure 27 depicts the impact of $(M)$ on $(\mathrm{Ns})$ . From here, we see that $(M)$ gives dual behavior with the escalation in $(\mathrm{Ns})$ . While from Figure 28 we have seen that the escalation in $(M)$ gives decreasing behavior in $(\mathrm{Be})$ . Figure 29 illustrates the impact of $(E{c}_m)$ on $(\mathrm{Ns})$ . The escalating $(E{c}_m)$ gives dual behavior in $(\mathrm{Ns})$ . In the region $(0\le \lambda \le 0.3)$ , the modified Eckert number gives increasing behavior, while in the region $(0.3<\lambda \le 1.0)$ , the modified Eckert number gives reducing behavior. The Influence of $(E{c}_m)$ on $(\mathrm{Be})$ is shown in Figure 30. We have seen the increasing behavior in $(\mathrm{Be})$ with the escalation in $(E{c}_m)$ . Figure 31 illustrates the impact of $(\mathrm{Pr})$ on $(\mathrm{Ns})$ . From here, we have seen decreasing behavior in the region $(0\le \lambda \le 0.39)$ , while in the region $(0.39<\lambda \le 1.0)$ , we have seen the escalating behavior in $(\mathrm{Ns})$ with the escalation in $(\mathrm{Pr})$ . The escalating $(\mathrm{Pr})$ gives decreasing behavior in $(\mathrm{Be})$ , which is shown in Figure 32.

Figure 23.

Influence of R on Ns.

Figure 24.

Influence of R on Be.

Figure 25.

Influence of Ec on Ns.

Figure 26.

Influence of Ec on Be.

Figure 27.

Influence of M on Ns.

Figure 28.

Influence of M on Be.

Figure 29.

Influence of $E{c}_m$ on Ns.

Figure 30.

Influence of $E{c}_m$ on Be.

Figure 31.

Influence of $\mathrm{Pr}$ on Ns.

Figure 32.

Influence of $\mathrm{Pr}$ on Be.

Skin fraction (Cf Re_x^1/2) and local Nusselt number (Nu Re_x^1/2)

Figures 33–42 are plotted to see the impact of emerging parameters and non-dimensional numbers on skin fraction coefficient $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2})$ and local Nusselt number $(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2})$ for both SWCNTs and MWCNTs. Figures 33 and 34 illustrate the influence of $(\mathrm{\Omega })$ on $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2})$ at both lower and upper walls of the channel. The surface drag force reduces with the escalation in $(\mathrm{\Omega })$ at the lower wall of the channel in both $(S>0)$ and $(S<0)$ cases, which is shown in Figure 33. In addition, when the upper and lower walls are separated further more from each other, $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2})$ reduces abundance more. Also in the case of MWCNTs, the skin fraction coefficient has reduced more than those of SWCNTs. While reverse impact can be seen at the upper wall of the channel, which is shown in Figure 34. Figures 35–38 depict the impact of squeezing parameter $(S)$ on $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2})$ and $(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2})$ at both upper and lower walls of the channel. Generally, escalating squeeze parameter, the $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2})$ increase at the lower wall of the channel and reduces at the upper wall of the channel. In addition, at the lower wall of the channel, $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2})$ has reduced in the case of SWCNTs than in that of MWCNTs. Also, there is a direct relationship between $(S)$ and $(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2})$ at the lower wall of the channel, while at the upper wall of the channel there is a reverse relationship between $\left(S\right)$ and $(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2})$ . Figures 39 and 40 depict the impact of $(M)$ on $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2})$ at both upper and lower walls of the channel. It can be seen from both figures that the increase in $(M)$ increases $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2})$ and reduces $(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2})$ at both lower and upper walls of the channel in both $(S>0)$ and $(S<0)$ cases. In addition, $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2})$ has increased in the case of MWCNTs than those of SWCNTs, while $(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2})$ reduced in the case of SWCNTs than in that of MWCNTs. Figures 41 and 42 depict to see the effect of $(R)$ on $(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2})$ at both lower and upper walls of the channel. When the upper wall moves to the lower wall of the channel, the escalating $(R)$ escalates $(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2})$ . It is obvious from the figure that escalating $(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2})$ in the case of SWCNTs is more than those of MWCNTs.

Figure 33.

Influence of $\mathrm{\Omega }$ on $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2}{)}_{\mathrm{lower}}$ .

Figure 34.

Influence of $\mathrm{\Omega }$ on $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2}{)}_{\mathrm{upper}}$ .

Figure 35.

Influence of S on $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2}{)}_{\mathrm{lower}}$ .

Figure 36.

Influence of S on $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2}{)}_{\mathrm{upper}}$ .

Figure 37.

Influence of S on $(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2}{)}_{\mathrm{lower}}$ .

Figure 38.

Influence of S on $(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2}{)}_{\mathrm{upper}}$ .

Figure 39.

Influence of M on $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2}{)}_{\mathrm{lower}}$ .

Figure 40.

Influence of M on $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2}{)}_{\mathrm{upper}}$ .

Figure 41.

Influence of R on $(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2}{)}_{\mathrm{lower}}$ .

Figure 42.

Influence of R on $(\mathrm{Nu}{\mathrm{Re}}_x^{-1/2}{)}_{\mathrm{upper}}$ .

Tables 1–3 are schemed to study the Physical properties of CNTs, thermo physical properties CNTs and nanofluids of some base fluids, and thermal conductivity ( $k_{\mathrm{nf}}$ ) of CNTs with different volume fraction $(\phi )$ , respectively.

Table 1.

Physical properties of CNTs.

Materials	SWCNT	MWCNT
Thermal conductivity, $k_{\mathrm{nf}}$ (W/mK)	3000	3000
Electrical conductivity, ${\phi }_{\mathrm{nf}}$ (S/m)	${10}^6-{10}^7$	${10}^6-{10}^7$
Tensile strength (GPa)	150	150
Young’s modulus (GPa)	1054	1200

CNTs: carbon nanotubes; SWCNT: single-walled carbon nanotube; MWCNT: multi-walled carbon nanotube.

Table 2.

The thermo physical properties of CNTs and nanofluids of some base fluids.

Physical properties		Specific heat, $c_p(\mathrm{J}/\mathrm{K})$	Density, $\rho (\mathrm{kg}/{\mathrm{m}}^3)$	Thermal conductivity, $k(\mathrm{W}/\mathrm{mK})$
Base fluid	Water	$4.197\times {10}^3$	$9.97\times {10}^2$	$6.13\times {10}^{-1}$
	Kerosene (lamp) oil	$2.090\times {10}^3$	$7.83\times {10}^2$	$1.45\times {10}^{-1}$
	Engine oil	$1.910\times {10}^3$	$8.84\times {10}^2$	$1.44\times {10}^{-1}$
Nanofluids	SWCNT	$4.25\times {10}^2$	$2.6\times {10}^3$	$6.6\times {10}^3$
	MWCNT	$7.96\times {10}^2$	$1.6\times {10}^3$	$3\times {10}^3$

CNTs: carbon nanotubes; SWCNT: single-walled carbon nanotube; MWCNT: multi-walled carbon nanotube.

Table 3.

Thermal conductivity $(k_{\mathrm{nf}})$ of CNTs with different volume fractions (ϕ).

$\phi$	$k_{\mathrm{nf}}$ for SWCNT	$k_{\mathrm{nf}}$ for MWCNT
0	$1.45\times {10}^{-1}$	$1.45\times {10}^{-1}$
0.01	$1.74\times {10}^{-1}$	$1.72\times {10}^{-1}$
0.02	$2.04\times {10}^{-1}$	$2\times {10}^{-1}$
0.03	$2.35\times {10}^{-1}$	$2.28\times {10}^{-1}$
0.04	$2.66\times 10^{-1}$	$2.57\times 10^{-1}$

CNTs: carbon nanotubes; SWCNT: single-walled carbon nanotube; MWCNT: multi-walled carbon nanotube.

Conclusion

Entropy generation investigation in three-dimensional squeezing flow of CNTs based on water $({\mathrm{H}}_2\mathrm{O})$ has been done. The influences of thermal radiation, viscous dissipation, and applied magnetic field on nanofluid are taken into account. SWCNTs and MWCNTs are used in this model. The impacts of embedded of parameters are shown graphically. The concluding remarks of this investigation are listed below:

• The velocity profile $f(\lambda )$ in y-direction escalates with the escalation in $(\phi )$ and $(A)$ , and $(\mathrm{\Omega })$ gives dual behavior with the escalating values, while with the escalation in $(M)$ , the velocity function gives decreasing behavior.

• The velocity profile $f\prime (\lambda )$ in x-direction reduces with the escalation in $(A)$ and gives the oscillatory behavior with the enhancement in $(\phi )$ , $(\mathrm{\Omega })$ , and $(M)$ .

• The velocity profile $g(\lambda )$ escalates with the escalation in $(\phi )$ and reduces with the escalation in $(\mathrm{\Omega })$ , $(A)$ , and $(M)$ .

• Generally, the velocity profiles escalate with the decrease in $(S>0)$ and reduce with escalation in $(S<0)$ .

• The temperature function $\theta (\lambda )$ escalates with the escalation in $(S)$ , $(\mathrm{Ec})$ , and $(E{c}_m)$ and reduces with the escalation in $(\phi )$ and $(R)$ .

• The entropy generation $(\mathrm{Ns})$ escalates with the escalation in $(R)$ , while $(\mathrm{Ec})$ , $(E{c}_m)$ , $(M)$ , and $(\mathrm{Pr})$ have dual behavior on $(\mathrm{Ns})$ .

• The Bejan number $(\mathrm{Be})$ escalates with the escalation in $(R)$ , $(\mathrm{Ec})$ , and $(E{c}_m)$ , while declines with $(M)$ and $(\mathrm{Pr})$ .

• The skin fraction coefficient $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2}{)}_{\mathrm{lower}}$ escalates with the escalation in $(S)$ , $(M)$ , and $(R)$ , while reduces with the escalation in $(\mathrm{\Omega })$ .

• The skin fraction coefficient $(\mathrm{Cf}{\mathrm{Re}}_x^{1/2}{)}_{\mathrm{upper}}$ enhances with the enhancement in $(\mathrm{\Omega })$ while reduces with $(S)$ and $(M)$ .

• The local Nusselt number $(\mathrm{Nu}{\mathrm{Re}}_x^{1/2}{)}_{\mathrm{lower}}$ escalates with the enhancement in $(R)$ , while reduces with $(S)$ .

• The local Nusselt number $(\mathrm{Nu}{\mathrm{Re}}_x^{1/2}{)}_{\mathrm{upper}}$ increases with the escalation in $(R)$ , while reduces with $(S)$ .