Analysis of Heat Transfer in Non-Coaxial Rotation of Newtonian Carbon Nanofluid Flow with Magnetohydrodynamics and Porosity Effects

Abstract

The study analyzed the heat transfer of water-based carbon nanotubes in non-coaxial rotation flow affected by magnetohydrodynamics and porosity. Two types of CNTs have been considered; single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). Partial differential equations are used to model the problem subjected to the initial and moving boundary conditions. Employing dimensionless variables transformed the system of equations into ordinary differential equations form. The resulting dimensionless equations are analytically solved for the closed form of temperature and velocity distributions. The obtained solutions are expressed in terms of a complementary function error. The impacts of the embedded parameters are graphically plotted in different graphs and are discussed in detail. The Nusselt number and skin friction are also evaluated. The temperature and velocity profiles have been determined to meet the initial and boundary conditions. An augment in the CNTs’ volume fraction increases both temperature and velocity of the nanofluid as well as enhances the rate of heat transport. SWCNTs provides high values of Nusselt number compared to MWCNTs. For verification, a comparison between the present solutions and a past study is conducted and achieved excellent agreement.

1. Introduction

The growing demand in manufacturing has led to a significant process of heat energy transfer in industry applications such as nuclear reactors, heat exchangers, radiators in automobiles, solar water heaters, refrigeration units and the electronic cooling devices. Enhancing the heating and cooling processes in industries will save energy, reduce the processing time, enhances thermal rate and increase the equipment’s lifespan. Sivashanmugam [1] found that nanofluid emergence has improved heat transfer capabilities for processes in industries. Choi and Eastman [2] established the nanofluid by synthesizing nanoparticles in the conventional base fluid. To be specific, nanofluid is created by suspending nano-sized particles with commonly less than 100 nm into the ordinary fluids such as ethylene glycol, propylene glycol, water and oils [3]. Various materials from different groups can be used as the nanoparticles such as Al₂O₃ and CuO from metalic oxide, Cu, Ag, Au from metals, SiC and TiC from carbide ceramics, as well as TiO₃ from semiconductors [4]. In addition, immersion of nanoparticles is a new way of enhancing thermal conductivity of ordinary fluids which directly improves their ability in heat transportation [5]. In line with nanofluid’s contribution in many crucial applications, a number of research has been carried out to discover the impacts of various nanofluid suspension on the flow features and heat transfer with several effects including Sulochana et al. [6] considering CuO-water and TiO-water, Sandeep and Reddy [7] using Cu-water, and Abbas and Magdy [8] choosing Al₂O₃-water as their nanofluid.

Magnetohydrodynamics (MHD) is known as the resultant effect due to mutual interaction of magnetic field and moving electrical conducting fluid. Their great applications such as power generation system, MHD energy conversion, pumps, motors, solar collectors have drawn significant attention of several researcher for MHD nanofluid in convective boundary layer flow [9]. Benos and Sarris [10] studied the impacts of MHD flow of nanofluid in a horizontal cavity. Hussanan et al. [11] analyzed the transportation of mass and heat for MHD nanofluid flow restricted to an accelerated plate in a porous media. In this study, water-based oxide and non-oxide had been considered as the nanofluids. Prasad et al. [12] performed similar work as [11] concerning the radiative flow of nanofluid over a vertical moving plate. Anwar et al. [13] conducted the MHD nanofluid flow in a porous material with heat source/sink and radiation effects. Cao et al. [14] analyzed the heat transfer and flow regimes for a Maxwell nanofluid under MHD effect. While, Ramzan et al. [15] investigated for a radiative Jeffery nanofluid and Khan et al. [16] carried out for a Casson nanofluid with Newtonian heating.

One of the greatest discoveries in material science history is carbon nanotubes (CNTs), which was discovered by a Japanese researcher in the beginning of the 1990s. Since the discovery, due to the unique electronic structural and mechanical characteristics, CNTs are found as valuable nanoparticles, especially in nanotechnology field. CNTs are great conductance which is highly sought in medical applications. They have been used as drug carriers and have benefited cancer therapy treatments [17]. The high thermal conductivity of CNTs has attracted significant attention from many researchers, including Xue [18], Khan et al. [19] and Saba et al. [20]. CNTs are hollow cylinders of carbon atoms in the forms of metals or semiconductors. CNTs are folded tubes of graphene sheet made up of hexagonal carbon rings, and their bundles are formed. CNTs are classified into two types with respectively differ in the graphene cylinder arrangement which are single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). SWCNTs has one layer [21], while MWCNTs consist of more than one graphene cylinder layers [22]. Khalid et al. [23] studied the characteristics of flow and heat transfer for CNTs nanofluid affected by MHD and porosity effects. Acharya et al. [24] discussed a comparative study on the properties of MWCNTs and SWCNTs suspended in water with the imposition of magnetic field. The CNTs nanofluid flow induced by a moving plate was investigated by Anuar et al. [25] and a prominent effect on heat transfer and skin friction by SWCNTs was observed. Ebaid et al. [26] analyzed convective boundary layer for CNTs nanofluid under magnetic field effect. The closed form solution was derived using Laplace transform method and the findings showed increasing magnetic strength and volume fraction of CNTs had deteriorated the rate of heat transport. Aman et al. [27] improved heat transfer for a Maxwell CNTs nanofluid moving over a vertical state plate with constant wall temperature. The investigation of velocity slip of carbon nanotubes flow with diffusion species was conducted by Hayat et al. [28]. Recently, the heat transmission analysis for water-based CNTs was discussed by Berrehal and Makinde [29], considering the flow over non-parallel plates and Ellahi et al. [30] considering flow past a truncated wavy cone.

Due to its broad range of uses such as car brake system, manufacturing of glass and plastic films, gas turbines, and medical equipment’s, numerous researchers have effectively studied heat transfer and fluid flow in a rotation system [31]. The impact of MHD and porosity on rotating nanofluid flow with double diffusion by using regular nanoparticles was discussed by Krishna and Chamkha [32]. More features of heat transfer affected by porosity and magnetic field for a rotating fluid flow were referred in Das et al. [33] and Krishna et al. [34, 35]. Kumam et al. [36] implemented CNTs in analyzing the flow behavior for a rotating nanofluid. The nanofluid was considered as an electrical conducting fluid moving in a channel under heat source/sink and radiation effects. More study on the heat propagation for a convective flow of nanofluid in a rotating system affected by CNTs with several effects and different geometries were presented by Imtiaz et al. [37], Mosayebidorcheh and Hatami [38] and Acharya et al. [39]. Interestingly, several researchers had recently concentrated their study on the non-coaxial rotation flow. Mixer machines in food processing industry, cooling pad of electronic devices and rotating propellers for aircraft have become great application to exemplify the non-coaxial rotating phenomenon in various industries. Mohamad et al. [40] presented the mathematical expression for heat transfer in non-coaxial rotation of viscous fluid flow. As the extension of the previous study, the heat and mass transfer effects (double diffusion) were considered by Mohamad et al. [41] and followed by Mohamad et al. [42] investigating porosity effect in double diffusion flow of MHD fluid. Ersoy [43] imposed a disk with non-torsional oscillation to study the convective non-coaxial rotating flow for a Newtonian fluid. Mohamad et al. [44, 45] worked on similar study considering the second grade fluid and Rafiq et al. [46] concerning the Casson fluid model. The time dependent flow of an incompressible fluid with MHD, chemical reaction and radiation effects under non-coaxial rotation was investigated by Rana et al. [47]. Subjecting to the same type of rotation, Mohamad et al. [48] studied the porosity and MHD consequences in mixed convection flow influenced by an accelerated disk. The study was improved by Noranuar et al. [49] including the effects of double diffusive flow. According to the review of non-coaxial rotation, it is clear that most of the study are subjected to the ordinary fluid. However, the study of non-coaxial rotation for nanofluid by using regular nanoparticles had been performed by Das et al. [50] and Ashlin and Mahanthesh [51] but then the study reporting the implementation of CNTs in non-coaxial rotation flow remains limited.

Inspiring from the above literature, new study is essential to explore more findings on non-coaxial rotation of CNTs nanofluid. Therefore, the investigation of MHD non-coaxial rotating flow of CNTs nanofluid due to free convection in a porous medium become the primary focus of the current study. Water base fluid is chosen to suspend nanoparticle of SWCNTs and MWCNTs. The exact solutions for velocity and temperature distributions are attained by solving the problem analytically using the Laplace transform method. The results are illustrated in several graphs and tables for further analysis of various embedded parameters.

2. Problem formulation

The incompressible time-dependent carbon nanofluid instigated by non-coaxial rotation past a vertical disk with an impulsive motion is considered as illustrated in Figure 1, where $x$and $z$are the Cartesian coordinates with $x$-axis is chosen as the upward direction and $z$-axis is the normal of it. The semi-finite space $z>0$is occupied by nanofluid that composed by constant kinematic viscosity ${\upsilon }_{\mathit{nf}}$of SWCNTs and MWCNTs suspended in water and acts as an electrically conducting fluid flowing through a porous medium. The disk is placed vertically along the $x$-axis with forward motion and a uniform transverse magnetic field of strength $B_0$is applied orthogonal to it. The plane $x=0$is considered as rotation axes for both disk and fluid. Initially, at $t=0$, the fluid and disk are retained at temperature $T_{\infty }$and rotate about $z^{\prime }$-axis with the same angular velocity $\mathrm{\Omega }$. After time $t>0$, the fluid remains rotating at $z^{\prime }$-axis while the disk begins to move with velocity $U_0$and rotates at $z$-axis. Both rotations have a uniform angular velocity $\mathrm{\Omega }$. The temperature of the disk raises to $T_w$and the distance between the two axes of rotation is equal to $\ell$. With above assumptions, the usual Boussinesq approximation is applied, and the nanofluid model proposed by Tiwari and Das [52] is used to represent the problem in the governing equations, express as

Figure 1.

Physical model of the problem.

$${\rho }_{n\mathrm{f}}\frac{\partial F}{\partial t}+\left({\rho }_{\mathit{nf}}\mathrm{\Omega }i+{\sigma }_{\mathit{nf}}{B_0}^2+\frac{{\mu }_{\mathit{nf}}}{k_1}\right)={\mu }_{\mathit{nf}}\frac{{\partial }^{2}F}{\partial z^2}+{\left(\rho {\beta }_T\right)}_{\mathit{nf}}{g}_x\left(T-T_{\infty }\right)+\left({\rho }_{\mathit{nf}}\mathrm{\Omega }i+{\sigma }_{\mathit{nf}}{B_0}^2+\frac{{\mu }_{\mathit{nf}}}{k_1}\right)\mathrm{\Omega }\ell ,$$E1

$${\left(\rho C_p\right)}_{\mathit{nf}}\frac{\partial T}{\partial t}=k_{\mathit{nf}}\frac{{\partial }^{2}T}{\partial z^2}.$$E2

The corresponding initial and boundary conditions are

$$F\left(z,0\right)=\mathrm{\Omega }\ell ;T\left(z,0\right)=T_{\infty };z>0,$$

$$F\left(0,t\right)=U_0;T\left(0,t\right)=T_w;t>0,$$E3

$$F\left(\infty ,t\right)=\mathrm{\Omega }\ell ;T\left(\infty ,t\right)=T_{\infty };t>0,$$

in which $F=f+\mathit{ig}$is the complex velocity; $f$and $g$are (real) primary and (imaginary) secondary velocities respectively, $T$is the temperature of nanofluid and $U_0$is the characteristic velocity. The following nanofluid constant for dynamic viscosity ${\mu }_{\mathit{nf}}$, density ${\rho }_{\mathit{nf}}$, heat capacitance ${\left({\mathit{\rho C}}_p\right)}_{\mathit{nf}}$, electrical conductivity ${\sigma }_{\mathit{nf}}$, thermal expansion coefficient ${\left({\beta }_T\right)}_{\mathit{nf}}$and thermal conductivity $k_{\mathit{nf}}$can be used as

$$\begin{array}{cc}{\mu }_{\mathit{nf}}& =\frac{{\mu }_f}{{\left(1-\varphi \right)}^{2.5}},{\rho }_{\mathit{nf}}=\left(1-\varphi \right){\rho }_f+\varphi {\rho }_{\mathit{\text{CNTs}}},\\ {\left(\mathrm{\rho }{\mathrm{C}}_{\mathrm{p}}\right)}_{\mathrm{nf}}& =\left(1-\mathrm{\varphi }\right){\left(\mathrm{\rho }{\mathrm{C}}_{\mathrm{p}}\right)}_{\mathrm{f}}+\mathrm{\varphi }{\left(\mathrm{\rho }{\mathrm{C}}_{\mathrm{p}}\right)}_{\text{CNTs}},\\ \frac{{\mathrm{\sigma }}_{\mathrm{nf}}}{{\mathrm{\sigma }}_{\mathrm{f}}}& =1+\frac{3\left(\frac{{\mathrm{\sigma }}_{\text{CNTs}}}{{\mathrm{\sigma }}_{\mathrm{f}}}-1\right)\mathrm{\varphi }}{\left(\frac{{\mathrm{\sigma }}_{\text{CNTs}}}{{\mathrm{\sigma }}_{\mathrm{f}}}+2\right)-\mathrm{\varphi }\left(\frac{{\mathrm{\sigma }}_{\text{CNTs}}}{{\mathrm{\sigma }}_{\mathrm{f}}}-1\right)},\\ {\left({\mathrm{\beta }}_{\mathrm{T}}\right)}_{\mathrm{nf}}& =\frac{\left(1-\mathrm{\varphi }\right){\left(\mathrm{\rho }{\mathrm{\beta }}_{\mathrm{T}}\right)}_{\mathrm{f}}+\mathrm{\varphi }{\left(\mathrm{\rho }{\mathrm{\beta }}_{\mathrm{T}}\right)}_{\text{CNTs}}}{{\mathrm{\rho }}_{\mathrm{nf}}},\\ \frac{{\mathrm{k}}_{\mathrm{nf}}}{{\mathrm{k}}_{\mathrm{f}}}& =\frac{1-\mathrm{\varphi }+2\mathrm{\varphi }\frac{{\mathrm{k}}_{\text{CNTs}}}{{\mathrm{k}}_{\text{CNTs}}-{\mathrm{k}}_{\mathrm{f}}}ln\left(\frac{{\mathrm{k}}_{\text{CNTs}}+{\mathrm{k}}_{\mathrm{f}}}{2{\mathrm{k}}_{\mathrm{f}}}\right)}{1-\mathrm{\varphi }+2\mathrm{\varphi }\frac{{\mathrm{k}}_{\mathrm{f}}}{{\mathrm{k}}_{\text{CNTs}}-{\mathrm{k}}_{\mathrm{f}}}ln\left(\frac{{\mathrm{k}}_{\text{CNTs}}+{\mathrm{k}}_{\mathrm{f}}}{2{\mathrm{k}}_{\mathrm{f}}}\right)},\end{array}$$E4

where the subscripts $f$is for fluid and $\mathit{\text{CNTs}}$is for carbon nanotubes. Meanwhile, $\varphi$is the solid volume fraction of nanofluid. The constants in Eq. (4) are used based on the thermophysical features in Table 1.

Material	Properties
	$\rho \left(\mathit{Kg}{m}^{-3}\right)$	$C_p\left(\mathit{JK}{g}^{-1}{K}^{-1}\right)$	$k\left(W{m}^{-1}{K}^{-1}\right)$	$\beta \times {10}^{-5}\left(K^{-1}\right)$	$\sigma \left(S{m}^{-1}\right)$
Water	997.1	4179	0.613	$21$	0.05
SWCNTs	2600	425	6600	$27$	${10}^6-{10}^7$
MWCNTs	1600	796	3000	$44$	$1.9\times {10}^{-4}$

Table 1.

Thermophysical features of water, SWCNTs, and MWCNTs.

Introducing following dimensionless variables

$$F^{\ast }=\frac{F}{\mathrm{\Omega }\ell }-1,z^{\ast }=\sqrt{\frac{\mathrm{\Omega }}{\upsilon }}z,t^{\ast }=\mathrm{\Omega t},T^{\ast }=\frac{T-T_{\infty }}{T_w-T_{\infty }}.$$E5

Using Eqs. (4) and (5), the governing equations in Eqs. (1)–(3) reduce to (excluding the * notation to simplify the equations)

$$\frac{\partial F}{\partial t}+d_{1}F=\frac{1}{{\varphi }_1}\frac{{\partial }^{2}F}{\partial z^2}+{\varphi }_3\mathit{GrT},$$E6

$$\frac{\partial T}{\partial t}=\frac{1}{a_1}\frac{{\partial }^{2}T}{\partial z^2}$$E7

and the conditions take the form

$$\begin{array}{cc}\mathrm{F}\left(\mathrm{z},0\right)& =0,\mathrm{T}\left(\mathrm{z},0\right)=0;\mathrm{z}>0,\\ \mathrm{F}\left(0,\mathrm{t}\right)& =\mathrm{U}-1,\mathrm{T}\left(0,\mathrm{t}\right)=1;\mathrm{t}>0,\\ F\left(\infty ,t\right)& =0,T\left(\infty ,t\right)=0;t>0,\end{array}$$E8

where

$$\begin{array}{cc}{d}_1& =\left(\mathrm{i}+M^2{\varphi }_2+\frac{1}{{\varphi }_{1}K}\right),a_1=\frac{\mathit{Pr}{\varphi }_4}{\lambda },M=\frac{{\sigma }_f{B_0}^2}{\Omega {\rho }_f},\frac{1}{K}=\frac{{\upsilon }_f}{k_1\Omega },\\ Pr& =\frac{{\upsilon }_f{\left(\rho C_p\right)}_f}{k_f},\mathit{Gr}=\frac{g_x{{\beta }_T}_f\left(T_w-T_{\infty }\right)}{{\Omega }^2\ell },U=\frac{U_0}{\Omega \ell }.\end{array}$$E9

At this point, $d_1$and $a_1$are constant parameters, $M$is the magnetic parameter (magnetic field), $K$is the porosity parameter, $Pr$is Prandtl number, $\mathit{Gr}$is Grashof number and $U$is the amplitude of disk. Besides that, the other constant parameters are

$$\begin{array}{cc}\mathrm{\lambda }& =\frac{{\mathrm{k}}_{\mathrm{nf}}}{{\mathrm{k}}_{\mathrm{f}}},{\mathrm{\varphi }}_1={\left(1-\mathrm{\varphi }\right)}^{2.5}\left(\left(1-\mathrm{\varphi }\right)+\frac{\mathrm{\varphi }{\mathrm{\rho }}_{\text{CNTs}}}{{\mathrm{\rho }}_{\mathrm{f}}}\right),\\ {\mathrm{\varphi }}_2& =\left(1+\frac{3\left(\frac{{\mathrm{\sigma }}_{\text{CNTs}}}{{\mathrm{\sigma }}_{\mathrm{f}}}-1\right)\mathrm{\varphi }}{\left(\frac{{\mathrm{\sigma }}_{\text{CNTs}}}{{\mathrm{\sigma }}_{\mathrm{f}}}+2\right)-\mathrm{\varphi }\left(\frac{{\mathrm{\sigma }}_{\text{CNTs}}}{{\mathrm{\sigma }}_{\mathrm{f}}}-1\right)}\right)\frac{1}{\left(\left(1-\mathrm{\varphi }\right)+\frac{\mathrm{\varphi }{\mathrm{\rho }}_{\text{CNTs}}}{{\mathrm{\rho }}_{\mathrm{f}}}\right)},\\ {\mathrm{\varphi }}_3& =\frac{\left(1-\mathrm{\varphi }\right)+\frac{\mathrm{\varphi }{\left(\mathrm{\rho \beta }\right)}_{\text{CNTs}}}{{\left(\mathrm{\rho \beta }\right)}_{\mathrm{f}}}}{\left(1-\mathrm{\varphi }\right)+\frac{\mathrm{\varphi }{\mathrm{\rho }}_{\text{CNTs}}}{{\mathrm{\rho }}_{\mathrm{f}}}},{\mathrm{\varphi }}_4=\left(1-\mathrm{\varphi }\right)+\frac{\mathrm{\varphi }{\left(\mathrm{\rho }{\mathrm{C}}_{\mathrm{p}}\right)}_{\text{CNTs}}}{{\left(\mathrm{\rho }{\mathrm{C}}_{\mathrm{p}}\right)}_{\mathrm{f}}}.\end{array}$$E10

3. Exact solution

Next, the system of equations in Eqs. (6)–(8) after applying Laplace transform yield to the following form

$$\frac{d^2}{\mathrm{d}{z}^2}\overline{F}\left(z,q\right)-\left({\varphi }_{1}q+d_2\right)\overline{F}\left(z,q\right)=-d_3\mathit{Gr}\overline{T}\left(z,q\right),$$E11

$$\overline{F}\left(0,q\right)=\left(U-1\right)\frac{1}{q},\overline{F}\left(\infty ,q\right)=0,$$E12

$$\frac{d^2}{d{z}^2}\overline{T}\left(z,q\right)-\left(a_{1}q\right)\overline{T}\left(z,q\right)=0,$$E13

$$\overline{T}\left(0,q\right)=\frac{1}{q},\overline{T}\left(\infty ,q\right)=0.$$E14

Then, Eqs. (11) and (13) are solved by using the boundary conditions, Eqs. (12) and (14). After taking some manipulations on the resultant solutions, the following Laplace solutions form

$$\overline{F}\left(z,q\right)={\overline{F}}_1\left(z,q\right)-{\overline{F}}_2\left(z,q\right)-{\overline{F}}_3\left(z,q\right)+{\overline{F}}_4\left(z,q\right)+{\overline{F}}_5\left(z,q\right)-{\overline{F}}_6\left(z,q\right),$$E15

$$\overline{T}\left(z,q\right)=\frac{1}{q}\mathit{exp}\left(-z\sqrt{a_{1}q}\right),$$E16

where

$$\begin{array}{cc}{\overline{\mathrm{F}}}_1\left(\mathrm{z},\mathrm{q}\right)& =\frac{\mathrm{U}}{\mathrm{q}}exp\left(-\mathrm{z}\sqrt{{\mathrm{\varphi }}_1\mathrm{q}+{\mathrm{d}}_2}\right),{\overline{\mathrm{F}}}_2\left(\mathrm{z},\mathrm{q}\right)=\frac{1}{\mathrm{q}}exp\left(-\mathrm{z}\sqrt{{\mathrm{\varphi }}_1\mathrm{q}+{\mathrm{d}}_2}\right),\\ {\overline{\mathrm{F}}}_3\left(\mathrm{z},\mathrm{q}\right)& =\frac{{\mathrm{a}}_4}{\mathrm{q}}exp\left(-\mathrm{z}\sqrt{{\mathrm{\varphi }}_1\mathrm{q}+{\mathrm{d}}_2}\right),{\overline{\mathrm{F}}}_4\left(\mathrm{z},\mathrm{q}\right)=\frac{{\mathrm{a}}_4}{\mathrm{q}-{\mathrm{a}}_3}exp\left(-\mathrm{z}\sqrt{{\mathrm{\varphi }}_1\mathrm{q}+{\mathrm{d}}_2}\right),\\ {\overline{\mathrm{F}}}_5\left(\mathrm{z},\mathrm{q}\right)& =\frac{{\mathrm{a}}_4}{\mathrm{q}}exp\left(-\mathrm{z}\sqrt{{\mathrm{a}}_1\mathrm{q}}\right),{\overline{\mathrm{F}}}_6\left(\mathrm{z},\mathrm{q}\right)=\frac{{\mathrm{a}}_4}{\mathrm{q}-{\mathrm{a}}_3}exp\left(-\mathrm{z}\sqrt{{\mathrm{a}}_1\mathrm{q}}\right)\end{array}$$E17

are defined, respectively. The exact solutions for the temperature and velocity are finally generated by utilizing the inverse Laplace transform on Eqs. (15) and (16). Hence, it results

$$F\left(z,t\right)=F_1\left(z,t\right)-F_2\left(z,t\right)-F_3\left(z,t\right)+F_4\left(z,t\right)+F_5\left(z,t\right)-F_6\left(z,t\right)$$E18

$$T\left(z,t\right)=\mathit{erfc}\left(\frac{z}{2}\sqrt{\frac{a_1}{t}}\right)$$E19

with

$$\begin{array}{cc}{\mathrm{F}}_1\left(\mathrm{z},\mathrm{t}\right)& =\frac{\mathrm{U}}{2}exp\left(\mathrm{z}\sqrt{{\mathrm{\varphi }}_1{\mathrm{d}}_4}\right)erfc\left(\frac{\mathrm{z}}{2}\sqrt{\frac{{\mathrm{\varphi }}_1}{\mathrm{t}}}+\sqrt{{\mathrm{d}}_4\mathrm{t}}\right)\\ & +\frac{\mathrm{U}}{2}exp\left(-\mathrm{z}\sqrt{{\mathrm{\varphi }}_1{\mathrm{d}}_4}\right)erfc\left(\frac{\mathrm{z}}{2}\sqrt{\frac{{\mathrm{\varphi }}_1}{\mathrm{t}}}-\sqrt{{\mathrm{d}}_4\mathrm{t}}\right),\\ {\mathrm{F}}_2\left(\mathrm{z},\mathrm{t}\right)& =\frac{1}{2}exp\left(\mathrm{z}\sqrt{{\mathrm{\varphi }}_1{\mathrm{d}}_4}\right)erfc\left(\frac{\mathrm{z}}{2}\sqrt{\frac{{\mathrm{\varphi }}_1}{\mathrm{t}}}+\sqrt{{\mathrm{d}}_4\mathrm{t}}\right)\\ & +\frac{1}{2}exp\left(-\mathrm{z}\sqrt{{\mathrm{\varphi }}_1{\mathrm{d}}_4}\right)erfc\left(\frac{\mathrm{z}}{2}\sqrt{\frac{{\mathrm{\varphi }}_1}{\mathrm{t}}}-\sqrt{{\mathrm{d}}_4\mathrm{t}}\right),\\ {\mathrm{F}}_3\left(\mathrm{z},\mathrm{t}\right)& =\frac{{\mathrm{a}}_4}{2}exp\left(\mathrm{z}\sqrt{{\mathrm{\varphi }}_1{\mathrm{d}}_4}\right)erfc\left(\frac{\mathrm{z}}{2}\sqrt{\frac{{\mathrm{\varphi }}_1}{\mathrm{t}}}+\sqrt{{\mathrm{d}}_4\mathrm{t}}\right)\\ & +\frac{{\mathrm{a}}_4}{2}exp\left(-\mathrm{z}\sqrt{{\mathrm{\varphi }}_1{\mathrm{d}}_4}\right)erfc\left(\frac{\mathrm{z}}{2}\sqrt{\frac{{\mathrm{\varphi }}_1}{\mathrm{t}}}-\sqrt{{\mathrm{d}}_4\mathrm{t}}\right),\\ {\mathrm{F}}_4\left(\mathrm{z},\mathrm{t}\right)& =\frac{{\mathrm{a}}_4}{2}exp\left({\mathrm{a}}_3\mathrm{t}+\mathrm{z}\sqrt{{\mathrm{\varphi }}_1\left({\mathrm{a}}_3+{\mathrm{d}}_4\right)}\right)erfc\left(\frac{\mathrm{z}}{2}\sqrt{\frac{{\mathrm{\varphi }}_1}{\mathrm{t}}}+\sqrt{\left({\mathrm{a}}_3+{\mathrm{d}}_4\right)\mathrm{t}}\right)\\ & +\frac{{\mathrm{a}}_4}{2}exp\left({\mathrm{a}}_3\mathrm{t}-\mathrm{z}\sqrt{{\mathrm{\varphi }}_1\left({\mathrm{a}}_3+{\mathrm{d}}_4\right)}\right)erfc\left(\frac{\mathrm{z}}{2}\sqrt{\frac{{\mathrm{\varphi }}_1}{\mathrm{t}}}-\sqrt{\left({\mathrm{a}}_3+{\mathrm{d}}_4\right)\mathrm{t}}\right),\\ {\mathrm{F}}_5\left(\mathrm{z},\mathrm{t}\right)& ={\mathrm{a}}_{4}erfc\left(\frac{\mathrm{z}}{2}\sqrt{\frac{{\mathrm{a}}_1}{\mathrm{t}}}\right),\\ {\mathrm{F}}_6\left(\mathrm{z},\mathrm{t}\right)& =\frac{{\mathrm{a}}_4}{2}exp\left({\mathrm{a}}_3\mathrm{t}+\mathrm{z}\sqrt{{\mathrm{a}}_1{\mathrm{a}}_3}\right)erfc\left(\frac{\mathrm{z}}{2}\sqrt{\frac{{\mathrm{a}}_1}{\mathrm{t}}}+\sqrt{{\mathrm{a}}_3\mathrm{t}}\right)\\ & +\frac{{\mathrm{a}}_4}{2}exp\left({\mathrm{a}}_3\mathrm{t}-\mathrm{z}\sqrt{{\mathrm{a}}_1{\mathrm{a}}_3}\right)erfc\left(\frac{\mathrm{z}}{2}\sqrt{\frac{{\mathrm{a}}_1}{\mathrm{t}}}-\sqrt{{\mathrm{a}}_3\mathrm{t}}\right),\end{array}$$E20

where

$${\mathrm{d}}_2={\mathrm{\varphi }}_1{\mathrm{d}}_1,{\mathrm{d}}_3={\mathrm{\varphi }}_1{\mathrm{\varphi }}_3,{\mathrm{d}}_4=\frac{{\mathrm{d}}_2}{{\mathrm{\varphi }}_1},{\mathrm{a}}_2={\mathrm{a}}_1-{\mathrm{\varphi }}_1,{\mathrm{a}}_3=\frac{{\mathrm{d}}_2}{{\mathrm{a}}_2},{\mathrm{a}}_4=\frac{{\mathrm{d}}_3\mathrm{Gr}}{{\mathrm{a}}_2{\mathrm{a}}_3}.$$E21

4. Physical quantities

In this study, the skin friction $\tau \left(t\right)$and Nusselt number $\mathit{Nu}$for the flow of Newtonian nanofluid in non-coaxal rotation are also analyzed. Their dimensional form is expressed as

$$\tau \left(t\right)=-{\left.{\mu }_{\mathit{nf}}\frac{\partial F}{\partial z}\right|}_{z=0}$$E22

$$\mathit{Nu}={\left.-k_{\mathit{nf}}\frac{\partial T}{\partial z}\right|}_{z=0}$$E23

Incorporating Eqs. (22) and (23) with the nanofluid model Eq. (4), dimensionless variables Eq. (5) and solutions Eqs. (18) and (19), the following dimensionless skin friction and Nusselt number form as

$$\begin{array}{cc}\mathrm{\tau }\left(\mathrm{t}\right)& ={\left.-\frac{1}{{\left(1-\mathrm{\varphi }\right)}^{2.5}}\frac{\mathrm{\partial F}}{\mathrm{\partial z}}\right|}_{\mathrm{z}=0},\\ & =-\frac{1}{{\left(1-\mathrm{\varphi }\right)}^{2.5}}\left({\mathrm{\tau }}_1\left(\mathrm{t}\right)-{\mathrm{\tau }}_2\left(\mathrm{t}\right)-{\mathrm{\tau }}_3\left(\mathrm{t}\right)+{\mathrm{\tau }}_4\left(\mathrm{t}\right)-{\mathrm{\tau }}_5\left(\mathrm{t}\right)+{\mathrm{\tau }}_6\left(\mathrm{t}\right)\right),\end{array}$$E24

$$\mathit{Nu}={\left.-\frac{k_{\mathit{nf}}}{k_f}\frac{\partial T}{\partial z}\right|}_{z=0}=\lambda \sqrt{\frac{a_1}{\mathit{\pi t}}},$$E25

where

$$\begin{array}{cc}{\mathrm{\tau }}_1\left(\mathrm{t}\right)& =\mathrm{U}\sqrt{{\mathrm{\varphi }}_1{\mathrm{d}}_4}erfc\left(\sqrt{{\mathrm{d}}_4\mathrm{t}}\right)-\mathrm{U}\sqrt{{\mathrm{\varphi }}_1{\mathrm{d}}_4}-\frac{\mathrm{U}}{2}\sqrt{\frac{{\mathrm{\varphi }}_1}{\mathrm{\pi t}}}exp\left(-{\mathrm{d}}_4\mathrm{t}\right),\\ {\mathrm{\tau }}_2\left(\mathrm{t}\right)& =\sqrt{{\mathrm{\varphi }}_1{\mathrm{d}}_4}erfc\left(\sqrt{{\mathrm{d}}_4\mathrm{t}}\right)-\sqrt{{\mathrm{\varphi }}_1{\mathrm{d}}_4}-\sqrt{\frac{{\mathrm{\varphi }}_1}{\mathrm{\pi t}}}exp\left(-{\mathrm{d}}_4\mathrm{t}\right),\\ {\mathrm{\tau }}_3\left(\mathrm{t}\right)& ={\mathrm{a}}_4\sqrt{{\mathrm{\varphi }}_1{\mathrm{d}}_4}erfc\left(\sqrt{{\mathrm{d}}_4\mathrm{t}}\right)-{\mathrm{a}}_4\sqrt{{\mathrm{\varphi }}_1{\mathrm{d}}_4}-{\mathrm{a}}_4\sqrt{\frac{{\mathrm{\varphi }}_1}{\mathrm{\pi t}}}exp\left(-{\mathrm{d}}_4\mathrm{t}\right),\\ {\mathrm{\tau }}_4\left(\mathrm{t}\right)& ={\mathrm{a}}_4\sqrt{{\mathrm{\varphi }}_1\left({\mathrm{a}}_3+{\mathrm{d}}_4\right)}exp\left({\mathrm{a}}_3\mathrm{t}\right)erfc\left(\sqrt{\left({\mathrm{a}}_3+{\mathrm{d}}_4\right)\mathrm{t}}\right)-{\mathrm{a}}_4\sqrt{\frac{{\mathrm{\varphi }}_1}{\mathrm{\pi t}}}exp\left(-{\mathrm{d}}_4\mathrm{t}\right)\\ & -{\mathrm{a}}_4\sqrt{{\mathrm{\varphi }}_1\left({\mathrm{a}}_3+{\mathrm{d}}_4\right)}exp\left({\mathrm{a}}_3\mathrm{t}\right),\\ {\mathrm{\tau }}_5\left(\mathrm{t}\right)& =-{\mathrm{a}}_4\sqrt{\frac{{\mathrm{a}}_1}{\mathrm{\pi t}}},\\ {\tau }_6\left(t\right)& =a_4\sqrt{a_1{a}_3}\mathit{exp}\left(a_{3}t\right)\mathit{erfc}\left(\sqrt{a_{3}t}\right)-a_4\sqrt{a_1{a}_3}\mathit{exp}\left(a_{3}t\right)-a_4\sqrt{\frac{a_1}{\mathit{\pi t}}},\end{array}$$E26

with ${\tau }^{\ast }=\raisebox{1ex}{$\tau \sqrt{{\nu }_f}$}\!\left/ \!\raisebox{-1ex}{${\mu }_f{\mathrm{\Omega }}^{\frac{3}{2}}\ell $}\right.$.

5. Analysis of results

The dimensionless differential equations of non-coaxial rotating nanofluid flow with associated boundary and initial conditions are analytically solved using the method of Laplace transform to obtain the closed form solutions of heat transfer. Further analysis for the role of dimensionless time $t$, Grashof number $\mathit{Gr}$, volume fraction of nanoparticles $\varphi$, porosity parameter $K$, magnetic field parameter $M$and amplitude of disk $U$on velocity and temperature distributions as well as Nusselt number and skin friction are presented in figures and tables. The profiles are plotted with the physical value of parameters as $Pr=6.2,\mathit{Gr}=0.5,M=0.2,K=2.0,\varphi =0.02,U=2.0$and $t=0.2$. The values are same unless for the investigated parameter of the profile. Since the rotating nanofluid is part of the problem, the results are discussed by presenting the graph of velocity profile in real and imaginary parts, specifically describes the primary $f$and secondary $g$velocities. The velocity profiles are demonstrated in Figures 2–7 and the temperature profiles are illustrated in Figures 8 and 9. From these profiles, it is found that all the obtained results satisfy both boundary and initial conditions. SWCNTs and MWCNTs have an identical nature of fluid flow and heat transfer.

Figure 2.

Profile of$f$and$g$for varied values of$t.$

Figure 3.

Profile of$f$and$g$for varied values of$\mathit{Gr}.$

Figure 4.

Profile of$f$and$g$for varied values of$M$.

Figure 5.

Profile of$f$and$g$for varied values of$K$.

Figure 6.

Profile of$f$and$g$for varied values of$\varphi$.

Figure 7.

Profile of$f$and$g$for varied values of$U$.

Figure 8.

Profile of$T\left(z,t\right)$for varied values of$t$.

Figure 9.

Profile of$T\left(z,t\right)$for varied values of$\varphi$.

Figure 2 depicts the plotting of $f$and $g$profiles with varying $t$values. Overall, the velocity of both SWCNTs and MWCNTs rises over time. As $t$increases, the buoyancy force becomes more effective and functions as an external source of energy to the flow, causing the velocity of fluid to increase. Figure 3 illustrates the variation of $f$and $g$profiles for SWCNTs and MWCNTs cases under the effect of $\mathit{Gr}$. It is essential to note that $\mathit{Gr}$is an approximation of the buoyancy force to the viscous force exerting on the flow. Hence, an increase of $\mathit{Gr}$suggests to the domination of buoyancy force and reduces the viscosity of fluid. Thus, growing $\mathit{Gr}$leads to an augment of fluid velocity.

On the other hand, Figure 4 discloses the nature of fluid flow in response to $M$. For both SWCNTs and MWCNTs cases, the figure suggests that amplifying $M$decreases $f$and $g$profiles. This impact is owing to the fact that a greater $M$value increases the frictional forces acting on the fluid, commonly known as the Lorentz force. Consequently, the fluid encounters substantial resistance along the flow and its velocity decreases. Next, the contribution of $K$in SWCNTs and MWCNTs nanofluids for both $f$and $g$profiles are displayed in Figure 5. It suggests that $K$value increases linearly with the velocities for both SWCNTs and MWCNTs. Noting that porosity is also greatly affected by the permeability of a medium, where it determines the ability of a medium to enable the fluid to flow through it. Then, the increasing values of $K$cause the medium to be more permeable and the fluid can easily pass through the medium. Therefore, it increases both $f$and $g$profiles.

Figure 6 reveals the consequences of $\varphi$on $f$and $g$profiles in the cases of SWCNTs and MWCNTs. It shows that increasing $\varphi$values result in the increment of $f$profiles and fluctuating trend of $g$profiles. This suggests significant advantages of non-coaxial rotation in CNTs, especially in industrial and medical applications. In line with a general finding, an analysis proceeding in cancer treatment has reported that the CNTs with higher velocity have been used to reach the tumor’s site. Besides, referring to Figure 7, it is noticed that ascending $U$also has a positive impact on velocity profiles for both CNTs suspensions, where the velocity ascends linearly with the values of $U$. As $U$increases, this proposes to the creation of external sources, which are used to enhance the thrust force acting in the fluid flow. Thus, the velocity fluid elevates with increasing $U$.

Furthermore, the temperature profiles $T\left(z,t\right)$under the impacts of $t$and $\varphi$are displayed graphically in Figures 8 and 9. It reveals that increment of $t$and $\varphi$contributes to a rise in nanofluid temperature for both types of CNTs case and followed by the magnification of thermal boundary layer. Physically, the addition of sufficient $\varphi$of CNTs can improve nanofluid’s thermal conductivity. The more CNTs being inserted, the higher the thermal conductivity, which unsurprisingly improves the ability of fluid to conduct heat. Therefore, a growth of temperature profile is exhibited for increasing $\varphi$. The comparison of physical behavior for SWCNTs and MWCNTs are clearer when referring to the zooming box of each graph. Overall, Figures 2–7 reveal that the velocity profile of MWCNTs case is more significant compared to the velocity of SWCNTs. This behavior is agreed to the thermophysical features in Table 1, where MWCNTs have low density, which also being a key factor for the increase of velocity profiles. Meanwhile, from Figures 8 and 9, SWCNTs have provided a prominent effect on temperature profiles as it is affected by a high thermal conductivity property.

Tables 2 and 3 show the results of skin friction (${\tau }_p$and ${\tau }_s$) and Nusselt number $\mathit{Nu}$for various parameters on both cases SWCNTs and MWCNTs. According to Table 2, it shows that both ${\tau }_p$and ${\tau }_s$of SWCNTs and MWCNTs rise when the strength of $M$higher. These effects cause the surface to produce high friction drag due to the maximization of wall shear stress. On the contrary, as $\mathit{Gr},K$and $t$increase, both suspension of SWCNTs and MWCNTs report a diminution in ${\tau }_p$and ${\tau }_s$. This shows that augmentation of $\mathit{Gr},K$and $t$have reduced the friction between fluid and surfaces which lead the velocity to increase. Meanwhile, as $\varphi$and $U$increase, both suspension of SWCNTs and MWCNTs report a growth of ${\tau }_p$and a diminution in ${\tau }_s$. From Table 3, it shows that $\mathit{Nu}$for both CNTs cases decrease as the values of $t$increase. However, when involving high $\varphi$, both SWCNTs and MWCNTs have large $\mathit{Nu}$which also implies to have a great of heat transfer rate. This effect is also directly affected by the reduction of nanofluid heat capacitance as $\varphi$increases. Overall, for Table 3, it is found that SWCNTs case have high value of $\mathit{Nu}$compared to MWCNTs, due to its reduction of heat capacitance. This effect also signifies for a better heat transfer process that can be used in several engineering and industrial system.

$t$	$\mathit{Gr}$	$M$	$K$	$\varphi$	$U$	SWCNTs		MWCNTs
						${\tau }_p$	${\tau }_s$	${\tau }_p$	${\tau }_s$
0.2	0.5	0.2	2	0.02	2	1.3811	−0.2550	1.3691	−0.2523
0.4	0.5	0.2	2	0.02	2	1.0318	−0.3492	1.0236	−0.3455
0.2	5	0.2	2	0.02	2	0.6276	−0.2705	0.6195	−0.2676
0.2	0.5	3	2	0.02	2	3.2171	−0.1596	3.0871	−0.1620
0.2	0.5	0.2	3	0.02	2	1.3377	−0.2578	1.3252	−0.2552
0.2	0.5	0.2	2	0.12	2	1.6820	−0.3138	1.6016	−0.2966
0.2	0.5	0.2	2	0.02	3	2.8459	−0.5082	2.8214	−0.5030

Table 2.

Values of primary ${\tau }_p$and secondary ${\tau }_s$skin friction for SWCNTs and MWCNTs.

The significance of bold emphasis used in Table 2 is for the comparison of the effects for varied values of the particular parameters. For each parameter, the changes of skin friction values are compared among the bold values of parameters.

$t$	$\varphi$	${Nu}$
		SWCNTs	MWCNTs
0.2	0.02	3.6238	3.5818
0.4	0.02	2.5624	2.5327
0.2	0.12	5.4840	5.3185

Table 3.

Values of Nusselt number $\mathit{Nu}$for SWCNTs and MWCNTs.

The significance of bold emphasis used in Table 3 is for the comparison of the effects for varied values of the particular parameters. For each parameter, the changes of Nusselt number values are compared among the bold values of parameters.

The accuracy of the obtained solution is verified by comparing solution in Eq. (18) with the solution obtained by Mohamad et al. [40] in Eq. (53). The comparison is conducted by letting magnetic parameter and nanoparticle volume fraction $M=\varphi =0$, and porosity parameter $K\to \infty$in the present solution for both types of CNTs and letting phase angle $\omega =0$and amplitude of disk oscillation $U=2$in the published work. This comparison shows that $f$and $g$profiles for both present and previous works are identical to each other as clearly presented in Figures 10 and 11, which thus proves that the accuracy of obtained solution is verified. Meanwhile, another verification is also carried out to verify the validity of present solution by comparing the values of velocity profiles from the present work with the numerical values solved by numerical Gaver-Stehfest algorithm [53, 54]. Tables 4 and 5 observe that the results of $f$and $g$profiles from the exact solution in Eq. (18) and the results from numerical solution are in excellent agreement.

Figure 10.

Comparison of$f$profiles from present results inEq. (18)with the published work by Mohamad et al. [40] in Eq. (53).

Figure 11.

Comparison of$g$profiles from present results inEq. (18)with the published work by Mohamad et al. [40] in Eq. (53).

$z$	Exact Eq. (18)		Numerical Laplace Eq. (15)
	SWCNTs	MWCNTs	SWCNTs	MWCNTs
0	1.0000	1.0000	1.0000	1.0000
0.5	0.4165	0.4206	0.4165	0.4206
1.0	0.1089	0.1122	0.1089	0.1121
1.5	0.0171	0.0182	0.0172	0.0182
2.0	0.0016	0.0017	0.0015	0.0017

Table 4.

Comparison of exact and numerical solution of $f$profiles for SWCNTs and MWCNTs with $t=0.2,\mathit{Gr}=0.5,M=0.2,K=2,\varphi =0.02,U=2,\mathit{Pr}=6.2$.

$z$	Exact Eq. (18)		Numerical Laplace Eq. (15)
	SWCNTs	MWCNTs	SWCNTs	SWCNTs
0	0.0000	0.0000	0.0000	0.0000
0.5	0.0366	0.0367	0.0366	0.0367
1.0	0.0146	0.0149	0.0146	0.0150
1.5	0.0027	0.0029	0.0027	0.0029
2.0	0.0003	0.0003	0.0003	0.0003

Table 5.

Comparison of exact and numerical solution of $g$profiles for SWCNTs and MWCNTs with $t=0.2,\mathit{Gr}=0.5,M=0.2,K=2,\varphi =0.02,U=2,\mathit{Pr}=6.2$.

6. Summary with conclusion

The unsteady non-coaxial rotation of water-CNTs nanofluid flow in a porous medium with MHD effect is analytically solved for the exact solutions by applying the Laplace transform method. The temperature and velocity profiles with various values of parameter for the immersion of SWCNTs and MWCNTs are plotted graphically and analyzed for their effects. From the discussion, significant findings emerge:

1. Both primary and secondary velocities for SWCNTs and MWCNTs suspension increase as the values of $t,\mathit{Gr},K,$and $U$increase while decrease as the values of $M$increase.

2. The insertion of higher $\mathrm{\varphi }$of SWCNTs and MWCNTs increases the primary velocity profiles while for secondary velocity profiles, fluctuating trend is reported for both cases.

3. The temperature of nanofluid increases when $\mathrm{\varphi }$and $\mathrm{t}$increase for both SWCNTs and MWCNTs cases.

4. MWCNTs have higher primary and secondary velocity profiles compared to SWCNTs because of their low-density property

5. SWCNTs have higher temperature profile than MWCNTs owing to their high thermal conductivity property.

6. The increasing values of $\mathrm{t},\mathrm{Gr}$and $\mathrm{K}$decrease both primary and secondary skin friction for both types of CNTs while the increase of $\mathrm{M}$gives opposite effect on both skin friction.

7. Nusselt number for both CNTs cases reduce as $\mathrm{t}$increases and amplify as $\mathrm{\varphi }$increases.

8. The findings in present work are in accordance to findings in Mohamad et al. [40] and numerical values obtained by numerical Gaver-Stehfest algorithm.

Acknowledgments

The authors would like to acknowledge the Ministry of Higher Education Malaysia and Research Management Centre-UTM, Universiti Teknologi Malaysia (UTM) for the financial support through vote number 17 J98, FRGS/1/2019/STG06/UTM/02/22 and 08G33.

Conflict of interest

The authors declare that they have no conflicts of interest to report regarding the present study.

Nomenclature

βT	Thermal expansion coefficient
Cp	Specific heat
ρ	Density
σ	Electrical conductivity
μ	Dynamic viscosity
gx	Acceleration due to gravity
k	Thermal conductivity
T	Temperature of nanofluid
T∞	Free stream temperature
Tw	Wall temperature
B0	Magnetic field
k1	Permeability
U0	Characteristic of velocity
Nu	Nusselt number
τ	Skin friction
τp	Primary skin friction
τs	Secondary skin friction
F	Complex velocity
f	Primary velocity
g	Secondary velocity
ϕ	Volume fraction nanoparticles
Ω	Angular velocity
t	Time
i	Imaginary unit
Pr	Prandtl number
Gr	Grashof number
K	Porosity
CNTs	Carbon nanotubes
nf	Nanofluid
f	Fluid