Significance of radiant-energy and multiple slips on magnetohydrodynamic flow of single-walled carbon nanotube-water,titanium dioxide–water,multiwalled carbon nanotube–water,graphene oxide–water nanofluids

Abstract

The magnetohydrodynamic flow of nanofluid is studied in the present analysis by using two parallel, rotating, and stretchable disks with a porous medium. The thermal radiation effects along with velocity slips at the interface of fluid and disk are considered in this work. The water is taken as base fluid, and carbon nanotubes (CNTs), titanium dioxide (TiO ${}_2$ ), and graphene oxide (GO) are taken as nanoparticles. The corresponding equations are modeled in terms of partial differential equations and Von Karman similarity transformation approach is adopted. The resulting equations are solved by using a finite difference method-based ND Solver. The axial, radial, and tangential velocity profiles and temperature distribution are discussed with graphs and tables. Thermal radiation and convective boundary conditions are used in the heat transfer process. When the thermal Biot number of the lower disk rises, fluid temperature enhances, whereas, the fluid temperature falls with the rise in the thermal Biot number of the lower disk. It is observed that when the thermal Biot number of lower disk rises from 0.5 to 0.8, heat transfer at lower disk is increased by about 6.66% in hexagonal-shaped CNTs-based nanofluid and 6.66% in spherical shaped TiO ${}_2$ and GO-based nanofluid. The impact of physical parameters such as skin friction coefficient and Nusselt number are computed for governing parameters and discussed in detail.

Introduction

As per the current demand of the human population, it is necessary to enhance the thermophysical properties such as thermal diffusivity, viscosity, thermal conductivity, and convective heat transfer coefficients of some standard coolants (called base fluids) such as ethylene glycol, oil, water, etc. By adding some nanosized particles such as graphite, silver, aluminum, carbon nanotubes (CNTs), nickel, nitrides, metallic oxides (CuO, Al ${}_2$ O ${}_3$ ), etc., the thermal conductivity of these base fluids can be increased. The process of adding nanosized particles in the base fluid forms nanofluids. The nanofluids have numerous applications in industrial processes, viz. heat exchangers, combustors, axial blade compressors, microelectronic board circuits, nuclear reactors, transportation, fuel cells, gas turbines, electronics, biomedicine, food, etc. Choi et al. ¹ were the first who introduced nanofluids. Some related investigations on nanofluid flow are available in the literature.^2–12 CNTs are cylindrical molecules that consist of rolled-up sheets of single-layer carbon atoms. CNTs show unique properties such as high electrical conductivity, mechanical strength, and large current carrying capacity. Therefore CNTs exhibit diverse applications in the area of mechanical, optics, nano/microelectronics, biological sciences, etc. They are used in energy storage, molecular electronics, filtration processes, conductive adhesives, catalytic supporters, etc., as explained by Volder et al. ¹³ Single-walled and multi-walled carbon nanotubes (SWCNTs and MWCNTs) are two major classifications of CNTs. SWCNTs consist of a single graphene cylinder with a diameter of $<1$ nm, and MWCNTs possess several concentrically linked cylinders with a diameter of more than 100 nm. Heat transfer characteristics of nanofluids and their non-Newtonian behavior were numerically investigated by Kamali and Binesh. ¹⁴ Khan et al. ¹⁵ have used the Darcy–Forchheimer medium to analyze the entropy-optimized CNT-based nanomaterial flow between two disks, which can rotate at different rates and have different angular frequencies. The investigation of CNTs-based flow with entropy optimization has various practical and engineering applications. Imtiaz et al. ¹⁶ investigated the flow and heat transfer effects of water-based CNTs under convective boundary conditions.

Various metal oxides such as titanium oxide (TiO ${}_2$ ) and graphene oxide (GO) have unique properties due to their tensile strength, electrical conductivity, elasticity, stability, etc. TiO ${}_2$ nanoparticles have various uses in industries dealing with paper, pharmaceuticals, cosmetics, textile, plastics, food products, etc., as given by Waghmode et al. ¹⁷ Several other utilizations of TiO ${}_2$ are listed by Ziental et al. ¹⁸ Similarly, Feng et al. ¹⁹ explained multiple applications of GO in the medical field . It is an oxidized graphene derivative, easy to manufacture, and cheaper than graphene. Gul and Firdous ²⁰ analyze the flow of GO–H ${}_2$ O nanofluid between two rotating disks through an experimental study.

The flow and heat transfer process over stretchable and rotating disks has eminent applications in industries and research. Fluid flow and heat generation phenomenon over a rotating disk has plenty of uses in mechanical engineering fields such as turbine systems, centrifugal filtration, electric power generators, wastewater treatment, viscometry, hall accelerators, magnetohydrodynamic (MHD) accelerators, etc. Due to its tremendous applications in power generation systems, medical equipment, food processing technology, aerodynamics, etc., researchers have drawn much attention to this field. Initially, Karman and Angrew ²¹ have investigated the fluid flow characteristics due to rotating disks. With the help of some effective transformations, the governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs). After that these Von Karman transformations are used by many investigators to study various fluid flow problems. Stewartson ²² was the pioneer to study the fluid flow model due to two rotating disks. The fluid flow between two infinite spinning disks has been reviewed by Lance and Rogers. ²³ Later on, Mellor et al. ²⁴ explored the fluid flow between two disks. The heat transfer phenomenon due to fluid flow between two disks (rotating) has been studied by Arora and Stokes. ²⁵ MHD flow properties of fluid between a rotating disk and a stationary impermeable disk was explored by Kumar et al. ²⁶ Flow and heat transfer properties caused by two rotating and stretchable disks have been investigated by Turkyilmazoglu. ²⁷ Soong et al. ²⁸ examined the fluid flow properties between two co-axial spinning disks. The flow between two disks having stretching velocity was explained by Fang and Zhang. ²⁹ Flow in porous medium has grabbed the attention of researchers as it is encountered in many engineering applications, such as filtration, thermal insulation, oil flow, groundwater, and all types of heat exchangers. Some of the findings are.^30–32 Micropolar nanofluid flow in the porous regime was investigated by Kumar et al. ³³ Gorder et al. ³⁴ have analytically solved the coupled non-linear equations to examine the incompressible flow between two rotating disks. By considering the viscous dissipation and Ohmic heating, the flow over a disk in a porous regime was accomplished by Sibanda and Makinde. ³⁵ They have studied the relationship between skin friction and Eckert number. The MHD flow characteristics of nanofluid in a channel with a stretched surface were described by Shahmohamadi and Rashidi. ³⁶ Rauf et al. ³⁷ investigated the rotating disk model to analyze the flow in a porous medium with nanoparticles. Hayat et al. ³⁸ studied the peristaltic radiative flow of Sisko nanomaterials in the presence of ohmic heating, Hall current, dissipation, and radiation. For modeling the porous medium, modified Darcy’s law is used.

In the present study, we aim to investigate the flow and heat transfer characteristics of water-based nanofluids in the porous regime with SWCNTs, TiO ${}_2$ , MWCNTs, and GO as nanoparticles, within two rotating and stretchable disks. The convective boundary conditions, porous medium, velocity slip, and thermal radiation effects are taken into account. As CNTs and [TiO ${}_2$ , GO] possess hexagonal and spherical shapes, respectively, therefore, thermal conductivity expressions are considered separately. The Fourier law of heat conduction ³⁹ has been utilized to study the heat transfer properties. As per the authors’ concern, a comparative study of a flow model based on hexagonal and spherical shapes nanoparticles with velocity slip, convective boundary constraints, porous medium, and radiant energy is not described previously. It has various applications in rotor–stator spinning disk reactor, electrical and electronic industries, etc.

Mathematical formulation

The present model considers the radiative flow of steady, axisymmetric, electrically conducting nanofluid between two parallel, rotating, coaxial, and stretching disks. One disk is placed at $z=0$ , and the other is at $z=l$ , that is, separated by distance $l$ . Disks are rotated in the axial direction with angular velocities ${\mathrm{\Omega }}_1$ , ${\mathrm{\Omega }}_2$ and stretched along the radial direction with stretching rates $a_1$ , $a_2$ as presented in Figure 1. A uniform magnetic field of strength $B_0$ is applied along the axial direction ( $z$ -axis). The porous medium with porosity $k_0$ is considered between the disks. ⁴⁰ We have taken SWCNTs, MWCNTs, TiO ${}_2$ , and GO as nanoparticles and water as a base fluid. The velocity slip with convective boundary conditions is applied tothe disks. Let the surface of the lower disk is heated through convection at temperature $T_0$ , and the temperature of the upper disk is $T_1$ . The radiative heat flux ( $q_r$ ) is given by Rosseland approximations. ⁴¹

$$q_r=-\frac{4{\sigma }^{*}}{3k^{*}}\mathrm{\nabla }{T}^4$$

Here, ${\sigma }^{*}$ is the Stefan Boltzmann constant and $k^{*}$ is the mean absorption coefficient.

Figure 1.

Geometry of model.

Assuming that the temperature differences within the fluid flow are very small. So we can expand $T^4$ about $T_1$ in the form of the Taylor series by neglecting the higher order terms as:

$$T^4=4T_1^{3}T-3T_1^4$$

The thermophysical properties of magnetite and water are listed in Table 1. $u(r,\theta ,z),v(r,\theta ,z),w(r,\theta ,z)$ are the velocity components in $r$ , $\theta$ and $z$ directions, respectively, that is, $\mathbf{V}=[u(r,\theta ,z),v(r,\theta ,z),w(r,\theta ,z)]$ . The system of equations, governing the flow^15,16 are given by:

$$\mathrm{\nabla }\cdot \mathbf{V}=0$$

(1)

$$(\mathbf{V}\cdot \mathrm{\nabla })u=-\frac{1}{{\rho }_{nf}}\frac{\mathrm{\partial }p}{\mathrm{\partial }r}+{\nu }_{nf}{\mathrm{\nabla }}^{2}u-\frac{{\sigma }_{nf}}{{\rho }_{nf}}{B}_0^{2}u-{\nu }_{nf}\frac{u}{k_0}$$

(2)

$$(\mathbf{V}\cdot \mathrm{\nabla })v={\nu }_{nf}{\mathrm{\nabla }}^{2}v-\frac{{\sigma }_{nf}}{{\rho }_{nf}}{B}_0^{2}v-{\nu }_{nf}\frac{v}{k_0}$$

(3)

$$(\mathbf{V}\cdot \mathrm{\nabla })w=-\frac{1}{{\rho }_{nf}}\frac{\mathrm{\partial }p}{\mathrm{\partial }z}+{\nu }_{nf}{\mathrm{\nabla }}^{2}w-{\nu }_{nf}\frac{w}{k_0}$$

(4)

$$(\rho c_p{)}_{nf}(\mathbf{V}\cdot \mathrm{\nabla })T=k_{nf}{\mathrm{\nabla }}^{2}T-\mathrm{\nabla }\cdot q_r$$

(5)

The above equations in simplified form can be written as follows:

$$\begin{array}{r}\frac{\mathrm{\partial }u}{\mathrm{\partial }r}+\frac{\mathrm{\partial }w}{\mathrm{\partial }z}+\frac{u}{r}=0\end{array}$$

(6)

$$\begin{array}{rl}& u\frac{\mathrm{\partial }u}{\mathrm{\partial }r}+w\frac{\mathrm{\partial }u}{\mathrm{\partial }z}-\frac{v^2}{r}=-{\nu }_{nf}\frac{u}{k_0}\\ & +{\nu }_{nf}(-\frac{u}{r^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{r}^2}+\frac{1}{r}\frac{\mathrm{\partial }u}{\mathrm{\partial }r}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2})-\frac{1}{{\rho }_{nf}}\frac{\mathrm{\partial }p}{\mathrm{\partial }r}\\ & -\frac{{\sigma }_{nf}{B}_0^2}{{\rho }_{nf}}u\end{array}$$

(7)

$$\begin{array}{rl}& u\frac{\mathrm{\partial }v}{\mathrm{\partial }r}+w\frac{\mathrm{\partial }v}{\mathrm{\partial }z}+\frac{uv}{r}=-{\nu }_{nf}\frac{v}{k_0}\\ & +{\nu }_{nf}(-\frac{v}{r^2}+\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{r}^2}+\frac{1}{r}\frac{\mathrm{\partial }v}{\mathrm{\partial }r}+\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{z}^2})-\frac{{\sigma }_{nf}{B}_0^2}{{\rho }_{nf}}v\end{array}$$

(8)

$$\begin{array}{rl}& u\frac{\mathrm{\partial }w}{\mathrm{\partial }r}+w\frac{\mathrm{\partial }w}{\mathrm{\partial }z}=-{\nu }_{nf}\frac{w}{k_0}\\ & +{\nu }_{nf}(\frac{1}{r}\frac{\mathrm{\partial }w}{\mathrm{\partial }r}+\frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{r}^2}+\frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{z}^2})-\frac{1}{{\rho }_{nf}}\frac{\mathrm{\partial }p}{\mathrm{\partial }z}\end{array}$$

(9)

$$\begin{array}{rl}& (\rho c_p{)}_{nf}(u\frac{\mathrm{\partial }T}{\mathrm{\partial }r}+w\frac{\mathrm{\partial }T}{\mathrm{\partial }z})=\\ & (k_{nf}+\frac{16{\sigma }^{*}}{3k^{*}}{T}_1^3)(\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{r}^2}+\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{z}^2}+\frac{1}{r}\frac{\mathrm{\partial }T}{\mathrm{\partial }r})\end{array}$$

(10)

The boundary conditions are as follows:

$$\begin{array}{rl}& u=r{a}_1+{\lambda }_1\frac{\mathrm{\partial }u}{\mathrm{\partial }z},v=r{\mathrm{\Omega }}_1+{\lambda }_2\frac{\mathrm{\partial }v}{\mathrm{\partial }z}\\ & k_{nf}\frac{\mathrm{\partial }T}{\mathrm{\partial }z}=-h_1(T_0-T),w=0,\mathrm{at}\mathrm{z}=0\\ & u=r{a}_2-{\lambda }_1\frac{\mathrm{\partial }u}{\mathrm{\partial }z},v=r{\mathrm{\Omega }}_2-{\lambda }_2\frac{\mathrm{\partial }v}{\mathrm{\partial }z}\\ & k_{nf}\frac{\mathrm{\partial }T}{\mathrm{\partial }z}=-h_2(T-T_1),w=0,\mathrm{at}\mathrm{z}=\mathrm{l}\end{array}$$

(11)

where $T$ represents the temperature of nanofluid, ${\lambda }_1$ and ${\lambda }_2$ are the velocity slip coefficients, ${\rho }_{nf}$ , ${\nu }_{nf}$ , ${\sigma }_{nf}$ , $k_{nf},$ and $(\rho c_p{)}_{nf}$ denote density, kinematic coefficient of viscosity, electrical conductivity, thermal conductivity, and heat capacitance of nanofluid, respectively. $h_1$ and $h_2$ denote convective heat transfer coefficients of lower and upper disks.

Table 1.

Important basic thermophysical properties of nanoparticles and water. ¹⁵

Properties	Unit	Water	SWCNT	MWCNT	TiO ${}_2$	GO
Heat capacitance	${\mathrm{Jkg}}^{-1}{\mathrm{K}}^{-1}$	4179	425	796	686.2	717
Electrical conductivity	$\mathrm{S}{\mathrm{m}}^{-1}$	0.05	$e^{-16}-e^8$	$e^6-e^7$	$2.6e^6$	$1.1e^{-5}$
Density	${\mathrm{kgm}}^{-3}$	997.1	2600	1600	4250	1800
Thermal conductivity	${\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$	0.613	6600	3000	8.9538	5000

SWCNT: single-walled carbon nanotube; MWCNT: multi-walled carbon nanotube; TiO ${}_2$ : titanium dioxide; GO: graphene oxide.

The various variables involved in nanofluid have been taken from the references^42,43 and are given below:

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

(12)

$$(\rho c_p{)}_{nf}=(\rho c_p{)}_s\chi +(\rho c_p{)}_f(1-\chi )$$

(13)

$${\rho }_{nf}={\rho }_s\chi +{\rho }_f(1-\chi )$$

(14)

$$\frac{k_{nf}}{k_f}=\frac{(1-\chi )+2\chi [k_s/(k_s-k_f)]\ln [({\mathrm{k}}_{\mathrm{s}}+{\mathrm{k}}_{\mathrm{f}})/2{\mathrm{k}}_{\mathrm{f}}]}{(1-\chi )+2\chi [{\mathrm{k}}_{\mathrm{f}}/({\mathrm{k}}_{\mathrm{s}}-{\mathrm{k}}_{\mathrm{f}})]\ln [({\mathrm{k}}_{\mathrm{s}}+{\mathrm{k}}_{\mathrm{f}})/2{\mathrm{k}}_{\mathrm{f}}]}$$

(15)

$$\frac{k_{nf}}{k_f}=\frac{k_s+2k_f-2\chi (k_f-k_s)}{k_s+2k_f+2\chi (k_f-k_s)}$$

(16)

$$\frac{{\sigma }_{nf}}{{\sigma }_f}=1+\frac{3[({\sigma }_s/{\sigma }_f)-1]\chi }{[({\sigma }_s/{\sigma }_f)+2]-[({\sigma }_s/{\sigma }_f)-1]\chi }$$

(17)

where the subscripts $f$ , $nf$ , and $s$ represent base fluid, nanofluid, nanoparticles, respectively, and $\chi$ represents nanoparticles volume fraction. Equations (12) and (13) denote the thermal conductivity for CNTs ⁴⁴ and TiO ${}_2$ /GO, ⁴⁵ respectively.

The von Karman transformation is used to convert nonlinear PDE into ODE ⁴⁶

$$\begin{array}{rl}& u=r{\mathrm{\Omega }}_1{f}^{\prime }(\zeta ),v=r{\mathrm{\Omega }}_{1}g(\zeta ),w=-2h{\mathrm{\Omega }}_{1}f(\zeta )\\ & p={\rho }_f{\nu }_f{\mathrm{\Omega }}_1(\frac{1}{2}\frac{r^2}{l^2}\epsilon +P(\zeta )),\zeta =\frac{z}{l},h(\zeta )=\frac{T-T_1}{T_0-T_1}\end{array}$$

(18)

By considering these transformations (15), continuity equation is identically satisfied and equations (7), (8), (9), and (7) can be written in a simplied form as

$$\begin{array}{rl}& f^{‴}-\frac{A_1}{A_2}\epsilon \\ & +A_1\mathrm{Re}(-{\mathrm{f}}^{\prime 2}+2\mathrm{f}{\mathrm{f}}^{″}-\frac{1}{\beta }{\mathrm{f}}^{\prime }+{\mathrm{g}}^2-\frac{\mathrm{M}{\mathrm{A}}_5}{{\mathrm{A}}_2}{\mathrm{f}}^{\prime })=0\end{array}$$

(19)

$$g^{″}+A_1\mathrm{Re}(-2{\mathrm{f}}^{\prime }\mathrm{g}-\frac{1}{\beta }\mathrm{g}+2\mathrm{f}{\mathrm{g}}^{\prime }-\frac{\mathrm{M}{\mathrm{A}}_5}{{\mathrm{A}}_2}\mathrm{g})=0$$

(20)

$$P^{\prime }+2\frac{A_2}{A_1}{f}^{″}-\frac{2\mathrm{Re}{\mathrm{A}}_2}{\beta {\mathrm{A}}_1}f+4A_2\mathrm{Re}\mathrm{f}{\mathrm{f}}^{\prime }=0$$

(21)

$$\frac{1}{Pr}(A_4+Rd)h^{″}+2Re{A}_{3}f{h}^{\prime }=0$$

(22)

The corresponding boundary conditions are as follows:

$$\begin{array}{rl}& f(0)=0,g(0)=1+{\gamma }_2{g}^{\prime }(0),f^{\prime }(0)=B_1+{\gamma }_1{f}^{″}(0)\\ & f(1)=0,g(1)=\mathrm{\Omega }-{\gamma }_2{g}^{\prime }(1),f^{\prime }(1)=B_2-{\gamma }_1{f}^{″}(1)\\ & h^{\prime }(0)=-\frac{1}{A_4}{\delta }_1(1-h(0)),h^{\prime }(1)=-\frac{1}{A_4}{\delta }_{2}h(1),P(0)=0\end{array}$$

(23)

where $\mathrm{Re}={\mathrm{\Omega }}_1{\mathrm{l}}^2/{\nu }_{\mathrm{f}}$ , $Pr={\nu }_f(\rho c_p{)}_f/k_f$ , $M=B_0^2{\sigma }_f/{\rho }_f{\mathrm{\Omega }}_1$ , and $Rd=16{\sigma }^{*}{T}_1^3/3k_f{k}^{*}$ represent Reynolds number, Prandtl number, Hartman number, and radiation parameter, respectively. $\mathrm{\Omega }={\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1$ , $\beta ={\mathrm{\Omega }}_1{k}_0/{\nu }_{nf}$ , and $\lambda =\gamma {\mathrm{\Omega }}_1$ represent rotation parameter, porous permeability parameter, thermal relaxation parameter, respectively. $B_1=a_1/{\mathrm{\Omega }}_1$ and $B_2=a_2/{\mathrm{\Omega }}_1$ are ratios of stretching rates to angular velocities, known as scaled stretching parameters. ${\gamma }_1={\lambda }_1/l$ and ${\gamma }_2={\lambda }_2/l$ stand for the velocity slip parameters for lower and upper disks, respectively. ${\delta }_1=h_{1}l/k_f$ and ${\delta }_2=h_{2}l/k_f$ stand for thermal Biot numbers, $A_1=(1-\chi {)}^{2.5}[1-\chi +({\rho }_s/{\rho }_f)\chi ]$ , $A_2=1-\chi +({\rho }_s/{\rho }_f)\chi$ , $A_3=1-\chi +[(\rho c_p{)}_s/(\rho c_p{)}_f]\chi$ , $A_4=k_{nf}/k_f,$ and $A_5={\sigma }_{nf}/{\sigma }_f$ .

On differentiating equation (16) with respect to $\zeta$ , we get

$$f^{iv}+A_1\mathrm{Re}(2\mathrm{g}{\mathrm{g}}^{\prime }-\frac{1}{\beta }{\mathrm{f}}^{″}+2\mathrm{f}{\mathrm{f}}^{‴}-\frac{\mathrm{M}{\mathrm{A}}_5}{{\mathrm{A}}_2}{\mathrm{f}}^{″})=0$$

(24)

The pressure parameter $\epsilon$ from equation (16) is given by

$$\begin{array}{rl}\epsilon =& -A_2\mathrm{Re}(({\mathrm{f}}^{\prime }(0){)}^2+\frac{1}{\beta }{\mathrm{f}}^{\prime }(0)-(\mathrm{g}(0){)}^2-2\mathrm{f}(0){\mathrm{f}}^{″}(0)\\ & +\frac{M{A}_5}{A_2}{f}^{\prime }(0))+\frac{A_2}{A_1}{f}^{‴}(0)\end{array}$$

(25)

Pressure can be obtained by integrating equation (18) from $0$ to $\zeta$

$$P=-2\frac{A_2}{A_1}{f}^{\prime }+\frac{2Re{A}_2}{\beta A_1}{\int }_0^{\zeta }fd\zeta +2\frac{A_2}{A_1}{f}^{\prime }(0)-2Re{A}_2{f}^2$$

(26)

In the radial direction, shear stress ( ${\tau }_{zr}$ ) for lower disk is

$${\tau }_{zr}={\mu }_{nf}{\frac{\mathrm{\partial }u}{\mathrm{\partial }z}|}_{z=0}=\frac{{\mu }_{f}r{\mathrm{\Omega }}_1}{(1-\chi {)}^{2.5}l}{f}^{″}(0)$$

(27)

In the tangential direction, shear stress ( ${\tau }_{z\theta }$ ) for lower disk is

$${\tau }_{z\theta }={\mu }_{nf}{\frac{\mathrm{\partial }v}{\mathrm{\partial }z}|}_{z=0}=\frac{{\mu }_{f}r{\mathrm{\Omega }}_1}{(1-\chi {)}^{2.5}l}{g}^{\prime }(0)$$

(28)

Total shear stress is given by

$${\tau }_w=\sqrt{{\tau }_{zr}^2+{\tau }_{z\theta }^2}=\frac{{\mu }_{f}r{\mathrm{\Omega }}_1}{(1-\chi {)}^{2.5}l}[(f^{″}(0){)}^2+(g^{\prime }(0){)}^2{]}^{1/2}$$

(29)

The local skin friction coefficients $C_1$ and $C_2$ at the lower and upper disks are given by

$$C_1=\frac{{{\tau }_w|}_{z=0}}{{\rho }_f(r{\mathrm{\Omega }}_1{)}^2}=\frac{1}{{\mathrm{Re}}_{\mathrm{r}}(1-\chi {)}^{2.5}}[(f^{″}(0){)}^2+(g^{\prime }(0){)}^2{]}^{1/2}$$

(30)

$$C_2=\frac{{{\tau }_w|}_{z=h}}{{\rho }_f(r{\mathrm{\Omega }}_2{)}^2}=\frac{1}{{\mathrm{Re}}_{\mathrm{r}}(1-\chi {)}^{2.5}}[(f^{″}(1){)}^2+(g^{\prime }(1){)}^2{]}^{1/2}$$

(31)

where ${\mathrm{Re}}_{\mathrm{r}}=\mathrm{r}{\mathrm{\Omega }}_1\mathrm{h}/{\nu }_{\mathrm{f}}$ represents local Reynolds number.

The local Nusselt number for both lower and upper disks are given by

$$\begin{array}{rl}& N{u}_1={\frac{lq}{k_f(-T_1+T_0)}|}_{z=0},\\ & N{u}_2={\frac{lq}{k_f(-T_1+T_0)}|}_{z=l}\end{array}$$

(32)

where $q$ is the sum of wall heat flux ( $q_w$ ) and radiative heat flux ( $q_r$ ), which can be written as

$${q_w|}_{z=0}=-k_{nf}{\frac{\mathrm{\partial }T}{\mathrm{\partial }z}|}_{z=0},{q_w|}_{z=l}=-k_{nf}{\frac{\mathrm{\partial }T}{\mathrm{\partial }z}|}_{z=l}$$

(33)

$${q_r|}_{z=0}=-\frac{16{\sigma }^{*}}{3k^{*}}{T}_1^3{\frac{\mathrm{\partial }T}{\mathrm{\partial }z}|}_{z=0},{q_w|}_{z=l}=-\frac{16{\sigma }^{*}}{3k^{*}}{T}_1^3{\frac{\mathrm{\partial }T}{\mathrm{\partial }z}|}_{z=l}$$

(34)

From equations (29), (30), and (31), the resulting expression for Nusselt number is

$$N{u}_1=-(A_4+Rd)h^{\prime }(0),N{u}_2=-(A_4+Rd)h^{\prime }(1)$$

(35)

Solution methodology

Equations (17), (19), and (21) associated with boundary conditions (20) are solved using ND-Solve command of MATHEMATICA software.^47,48 This software is based on the finite difference method to solve differential equations. Using the ND-Solve command, the numerical solutions of the differential equations are obtained in the discretized domain by changing the values of various involved parameters.

Validation of results

Considering the model as per the work of Imtiaz et al., ¹⁶ that is, neglecting the porous media, magnetic field, and the velocity slip coefficients in the boundary conditions, the values of skin friction coefficient are matched at the lower and upper end of the disk for different parameters involved. Validation is performed using CNTs as nanoparticles. Table 2 compares our numerical findings with the literature. Excellent matching of results is observed.

Table 2.

Validation table for the comparison of skin friction coefficient for various involved parameters.

				SWCNT				MWCNT
				Imtiaz et al. 16		Present result		Imtiaz et al. 16		Present result
$\chi$	$B_1$	$B_2$	$\mathrm{\Omega }$	${\mathrm{Re}}_{\mathrm{r}}{\mathrm{C}}_1$	${\mathrm{Re}}_{\mathrm{r}}{\mathrm{C}}_2$	${\mathrm{Re}}_{\mathrm{r}}{\mathrm{C}}_1$	${\mathrm{Re}}_{\mathrm{r}}{\mathrm{C}}_2$	${\mathrm{Re}}_{\mathrm{r}}{\mathrm{C}}_1$	${\mathrm{Re}}_{\mathrm{r}}{\mathrm{C}}_2$	${\mathrm{Re}}_{\mathrm{r}}{\mathrm{C}}_1$	${\mathrm{Re}}_{\mathrm{r}}{\mathrm{C}}_2$
0.1	0.5	0.5	0.1	4.0535	4.1084	4.0534	4.1082	4.0554	4.1055	4.0553	4.1055
0.2	0.5	0.5	0.1	5.4459	5.5084	5.4454	5.5082	5.4498	5.5028	5.4494	5.5027
0.1	0.1	0.5	0.1	2.1194	3.1319	2.1193	3.1315	2.1234	3.1285	2.1236	3.1285
0.1	0.4	0.5	0.1	3.5519	3.8622	3.5516	3.8622	3.5543	3.8592	3.5541	3.8590
0.1	0.5	0.1	0.1	3.0886	2.1811	3.0885	2.1813	3.0890	2.1798	3.0890	2.1796
0.1	0.5	0.5	0.3	3.9883	4.0397	3.9880	4.0396	3.9901	4.0370	3.9903	4.0370

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

Results and discussion

In this section, the effects of various involved parameters on the axial, tangential, and radial velocity distributions and temperature profiles are demonstrated through graphs and tables. Also, tables are created to show the effect of various parameters on Nusselt number and skin friction coefficient. All graphs are shown using dotted and plane lines. Plane lines represent the SWCNTs and TiO ${}_2$ based nanofluids while dotted lines are used for MWCNTs and GO-based nanofluids. The default values of various parameters used in this work are $\mathrm{Re}=0.9,\mathrm{Pr}=6.2,{\mathrm{B}}_1=0.4,{\mathrm{B}}_2=0.9,\chi =0.2,{\gamma }_1=0.4,{\gamma }_2=0.7,\beta =0.5,\mathrm{\Omega }=0.3$ , $\lambda =0.2$ , ${\delta }_1=0.4,{\delta }_2=0.5,\mathrm{Rd}=0.3$ , until otherwise stated.

Axial velocity variation

Figures 2 to 4 are plotted to examine the impact of Re, ${\gamma }_1$ and $B_1$ on the axial velocity profile. Figure 2 presents the variation of ${\gamma }_1$ on axial velocity. As the value of ${\gamma }_1$ rises, the axial velocity of the fluid reduces near the lower disk. After the point, $\zeta =0.35$ (nearly), the reverse trend in fluid velocity is seen. This is due to the fact that ${\gamma }_1$ relates to the slip velocity of the lower disk, and it is assumed in the slip condition that the fluid adjacent to the surface of the lower disk experience greater resistance, which leads to a decrease in the velocity profile. Figure 3 depicts that axial velocity increases with the Reynolds number rising from 2 to 10. This is due to the fact that the rotation of the lower disk enhances due to a rise in inertial effects. It is also visible here that the graphs for SWCNTs and MWCNTs are almost overlapped but TiO ${}_2$ nanoparticles boost the axial velocity more than GO nanoparticles due to the significant variation of thermal conductivities between them. Figure 4 illustrates the effect of stretching parameter $B_1$ on axial velocity. There is a rise in the axial velocity profile as the values of $B_1$ increase. This can be attributed to the fact that the stretching rate of the lower disk increases with the increase in the values of the stretching parameter $B_1$ .

Figure 2.

Impact of ${\gamma }_1$ on axial velocity profile: (a) SWCNT and MWCNT and (b) TiO ${}_2$ and GO. SWCNT: single-walled carbon nanotube; MWCNT: multi-walled carbon nanotube; TiO ${}_2$ ; titanium dioxide; GO: graphene oxide.

Figure 3.

Impact of $Re$ on axial velocity profile (a) SWCNT and MWCNT and (b) TiO ${}_2$ and GO. SWCNT: single-walled carbon nanotube; MWCNT: multi-walled carbon nanotube; TiO ${}_2$ ; titanium dioxide; GO: graphene oxide.

Figure 4.

Impact of $B_1$ on axial velocity profile (a) SWCNT and MWCNT and (b) TiO ${}_2$ and GO. SWCNT: single-walled carbon nanotube; MWCNT: multi-walled carbon nanotube; TiO ${}_2$ ; titanium dioxide; GO: graphene oxide.

Radial velocity variation

The influence of parameters ${\gamma }_1$ , Re and $B_2$ on the radial velocity profile are shown in Supplemental Figures 5 to 7. The impact of Re is depicted by Supplemental Figure 5. The Reynolds number (Re) is defined as the ratio of inertial force to viscous force; therefore, $Re$ tends to increase radial velocity near the lower disc, but a reverse behavior is captured as approaches to the upper disk. Also, it can be visualized from the graphs that TiO ${}_2$ -based nanofluid has higher radial velocity than GO-based nanofluid. From Supplemental Figure 6, it is clear that enhancement in ${\gamma }_1$ causes a reduction in the radial velocity near both the disks. This parameter causes enhancement in radial velocity near the mid-section of the flow domain. $\zeta =0.45$ is the critical point at which the radial velocity function obtained its minima. It is evident from Supplemental Figure 7 that parameter $B_2$ tends to reduce radial velocity near the lower disk and accelerate near the upper disk. This trend of increasing radial velocity near the upper disk is captured due to the increased centripetal force near the same disk. A similar kind of behavior is observed for all types of nanoparticles.

Tangential velocity variation

Supplemental Figures 8 to 12 reflect the change in tangential velocity due to parameters Re, ${\gamma }_2$ , and $M$ . These figures indicate that tangential velocity is a decreasing function of Re, M, and ${\gamma }_2$ . It is observed that the rise in the values of Re and M causes the decrement in the tangential velocity of the fluid. The decreasing behavior of tangential velocity for $M$ is justified due to retarding nature of Lorentz force (Supplemental Figure 10). Since increasing Re directly reflects the viscous force reduction, tangential velocity reduces significantly (Supplemental Figure 8). Supplemental Figure 9 shows the influence of ${\gamma }_2$ on tangential velocity distribution. Inciting the values of ${\gamma }_2$ means velocity slip at the upper disk’s surface rises, which reduces the tangential velocity magnitude at that end. GO nanoparticles have a more tangential velocity as compared to all other nanoparticles. Supplemental Figure 11 presents the effect of rotation parameter ( $\mathrm{\Omega }$ ) on the tangential velocity profile. The case $\mathrm{\Omega }<0$ signifies that both the disks are rotating in the opposite direction. The value $\mathrm{\Omega }=0$ implies that the rotation rate of the upper disk is zero. The situation $\mathrm{\Omega }>0$ indicates the same rotating direction of both disks. The graph depicts that the magnitude of tangential velocity suddenly rises near the point $\zeta =0.5$ when $\mathrm{\Omega }>0$ . A fall in magnitude of tangential velocity is observed when $\mathrm{\Omega }<0$ . TiO ${}_2$ - and GO-based nanofluids show more variation in the tangential velocity profile than SWCNTs- and MWCNTs-based nanofluids.

Supplemental Figure 12 shows the influence of nanoparticles concentration on the tangential velocity profile. It can be visualized from the graph that tangential velocity is a decreasing function of nanoparticles concentration. The rise in the volumetric fraction of nanoparticles intensifies the density of the fluid particles.

Temperature variation

The temperature distribution for parameters Rd, $\chi$ , ${\delta }_1$ , and ${\delta }_2$ are displayed in Supplemental Figures 13 to 16. The impact of thermal Biot number ${\delta }_1$ is shown in Supplemental Figure 13. With the rise in ${\delta }_1$ , the convection at the lower disk enhances which causes an increase in the temperature profile. On the other hand, an increase in ${\delta }_2$ remarks the more convection at the upper disk. So the fluid temperature reduces (see Supplemental Figure 14). Supplementary Figure 15 shows the impact of radiation parameter Rd on temperature. There is a slight decrease in temperature with the rise in Rd because it is inversely related to the mean absorption coefficient. The temperature of MWCNT-based fluid is higher than SWCNT-based fluid; similarly, TiO ${}_2$ -based fluid shows a higher temperature than GO nanoparticles-based fluid. Supplemental Figure 16 shows the variation of temperature with the volume fraction of nanoparticles. It can be observed from the graphs that inciting the volume fraction of nanoparticles causes a decrement in temperature near the lower end of the disk but nearly before $\zeta =0.8$ , that is, close to the upper disk, a reverse trend is observed. Temperature variation of TiO ${}_2$ and GO nanoparticles-based fluid are more visible than that of SWCNT- and MWCNT-based fluid. Supplemental Figure 17 compares the temperature and tangential velocity profiles for differently chosen nanoparticles. At the lower end of the disk, SWCNT-based fluid experiences the lowest temperature while TiO ${}_2$ -based fluid has the greatest temperature enhancement. SWCNT’s highest thermal conductivity and lowest heat capacitance account for this occurrence. GO-based fluid has a higher tangential velocity than TiO ${}_2$ -based fluid because GO has a lower density than TiO ${}_2$ nanoparticles.

Skin friction coefficient and Nusselt number of nanofluid

Skin friction coefficient for SWCNT and MWCNT nanofluids

Table 3 depicts the numerical values of skin friction coefficient for SWCNT- and MWCNT-based nanofluids at both ends of the disk for various involved parameters such as volume fraction of nanoparticles ( $\chi$ ), stretching parameters ( $B_1$ and $B_2$ ), velocity slip parameters ( ${\gamma }_1$ and ${\gamma }_2$ ), rotation parameter ( $\mathrm{\Omega }$ ), magnetic field parameter ( $M$ ), and porous permeability parameter ( $\beta$ ). The parameters that are kept constant in this table are also mentioned. Skin friction arises due to internal friction between the fluid layers. It can be seen from the table that the increasing $\chi$ , $B_1$ , and $B_2$ cause an increase in skin friction coefficient at both ends of the disk. In contrast, velocity slip parameters ${\gamma }_1$ and ${\gamma }_2$ tend to reduce the skin friction coefficient. The increase in rotation parameter ( $\mathrm{\Omega }$ ) leads to a decreased skin friction coefficient at the lower end, whereas an increase is at the upper disk for both SWCNT- and MWCNT-based fluid. Hartman number ( $M$ ) and porous permeability parameter ( $\beta$ ) cause increment and decrement in skin friction coefficient, respectively.

Table 3.

Numerical values of skin friction coefficient for SWCNT and MWCNT at lower and upper disk when $Re=0.9,Pr=6.2,\lambda =0.2,{\delta }_1=0.4,{\delta }_2=0.5,Rd=0.3$ .

								SWCNT		MWCNT
$\chi$	$B_1$	$B_2$	${\gamma }_1$	${\gamma }_2$	$\mathrm{\Omega }$	$M$	$\beta$	$R{e}_r{C}_1$	$R{e}_r{C}_2$	$R{e}_r{C}_1$	$R{e}_r{C}_2$
0.1	0.4	0.9	0.2	0.5	0.1	0.5	0.5	2.08342	2.89491	2.07554	2.88863
0.2								2.77692	3.87097	2.76034	3.85856
0.3								3.824269	5.38135	3.81647	5.36329
0.1	0.5							2.33683	2.96963	2.32906	2.96394
	0.7							2.86328	3.11806	2.85502	3.11362
	0.4	1.0						2.14486	3.18235	2.13807	3.17981
		1.2						2.2689	3.75876	2.26433	3.74865
		0.9	0.4					1.52511	1.92569	1.51234	1.9236
			0.6					1.31855	1.45162	1.30287	1.45146
			0.2	0.7				2.01281	2.88766	2.00791	2.8811
				0.9				1.96709	2.88381	1.96405	2.87713
				0.5	0.3			2.05491	2.89715	2.04618	2.88971
					0.5			2.02859	2.89975	2.01912	2.89660
					0.1	0.7		2.09701	2.90004	2.0895	2.89366
						0.9		2.10998	2.90537	2.10282	2.89893
						0.5	0.7	2.04664	2.88344	2.04118	2.87844
							0.9	2.02417	2.87818	2.02036	2.87380

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

Skin friction coefficient for TiO ${}_2$ and GO nanofluids

Table 4 is designed to predict the variation of skin friction coefficient for TiO ${}_2$ - and GO-based nanofluids for various parameters, as stated in the table. It can be seen from the table that on increasing the values of $\chi$ , $B_1$ , and $B_2$ , there is an increase in skin friction coefficient at both ends of the disk. The velocity slip parameters ${\gamma }_1$ and ${\gamma }_2$ cause a reduction in skin friction coefficient. The skin friction coefficient decreases at the lower end of the disk with the enhancement of the rotation parameter ( $\mathrm{\Omega }$ ), whereas it increases at the upper end for both TiO ${}_2$ and GO nanoparticles.

Table 4.

Numerical values of skin friction coefficient for TiO ${}_2$ and GO at lower and upper disk when $Re=0.9,Pr=6.2,\lambda =0.2,{\delta }_1=0.4,{\delta }_2=0.5,Rd=0.3$ .

								TiO ${}_2$		GO
$\chi$	$B_1$	$B_2$	${\gamma }_1$	${\gamma }_2$	$\mathrm{\Omega }$	$M$	$\beta$	$R{e}_r{C}_1$	$R{e}_r{C}_2$	$R{e}_r{C}_1$	$R_r{C}_2$
0.1	0.4	0.9	0.2	0.5	0.1	0.5	0.5	2.09602	2.90529	2.06415	2.88562
0.2								2.80301	3.89166	2.73367	3.85288
0.3								3.93512	5.43081	3.76908	5.35584
0.1	0.5							2.34929	2.97905	2.31828	2.96132
	0.7							2.90592	3.13475	2.84477	3.11173
	0.4	1.0						2.15574	3.19475	2.12727	3.17099
		1.2						2.3020	3.79653	2.25465	3.74329
		0.9	0.4					1.54528	1.92939	1.49714	1.92346
			0.6					1.39132	1.45227	1.28525	1.4528
			0.2	0.7				2.02053	2.89841	2.00006	2.87737
				0.9				1.98563	2.89033	1.95846	2.827298
				0.5	0.3			2.06888	2.89939	2.03408	2.87537
					0.5			2.04376	2.91073	2.00654	2.88031
					0.1	0.7		2.10902	2.91053	2.07355	2.88867
						0.9		2.12144	2.91595	2.08267	2,89184
						0.5	0.7	2.05551	2.8917	2.02774	2.87603
							0.9	2.03045	2.8854	2.00564	2.87198

TiO ${}_2$ : titanium dioxide; GO: graphene oxide.

Nusselt number for SWCNT and MWCNT nanofluids

Table 5 compares the values of Nusselt number for various parameters such as nanoparticles volume fraction ( $\chi$ ), radiation parameter ( $Rd$ ), thermal Biot numbers ( ${\delta }_1$ and ${\delta }_2$ ), porous permeability parameter ( $\beta$ ), and rotation parameter ( $\mathrm{\Omega }$ ). The Nusselt number determines the heat transfer rate. It can be visualized from the table that the Nusselt number falls when the volume fraction of nanoparticles increases. The rate of heat transfer enhances with the rise in $Rd$ . Higher the values of ${\delta }_1$ and ${\delta }_2$ , the heat transfer rate also increases. Nusselt number is a decreasing function of $\beta$ and $\mathrm{\Omega }$ at the lower end of the disk and an increasing function at the upper end.

Table 5.

Numerical values of Nusselt number for SWCNT and MWCNT at lower and upper disk when $Re=0.9,Pr=6.2,B_1=0.4,B_2=0.9,\lambda =0.2,{\gamma }_1=0.2,{\gamma }_2=0.5,M=0.5$ .

						SWCNT		MWCNT
$\chi$	$Rd$	${\delta }_1$	${\delta }_2$	$\beta$	$\mathrm{\Omega }$	$N{u}_1$	$N{u}_2$	$N{u}_1$	$N{u}_2$
0.1	0.5	0.3	0.2	0.5	0.1	0.128782	0.139439	0.129247	0.140699
0.2						0.125103	1.30694	0.125428	0.131594
0.3						0.123395	0.126722	0.123628	0.127353
0.1	1					0.148585	0.159253	0.150267	0.161732
	1.5					0.16839	0.179066	0.171289	0.182764
	0.5	0.5				0.150891	0.163377	0.151287	0.164691
		0.8				0.167019	0.180841	0.167338	0.182164
		0.3	0.4			0.18331	0.198479	0.183943	0.200241
			0.6			0.213434	0.231096	0.214512	0.233127
			0.2	0.7		0.128697	0.139497	0.129163	0.140757
				0.9		0.12865	0.139529	0.129116	0.140789
				0.5	0.3	0.128769	0.139447	0.129235	0.140707
					0.5	0.128745	0.139464	0.12921	0.140724

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

Nusselt number for TiO ${}_2$ and GO nanofluids

For TiO ${}_2$ - and GO-based nanofluids, Table 6 presents the variation in Nusselt number. It can be visualized from the table that the Nusselt number shows a decreasing trend when the volume fraction of nanoparticles increases. The heat transfer rate rises with the values of $Rd$ at both ends of the disk. The rate of heat transfer enhances with the rise in convective heat transfer coefficients, that is, ${\delta }_1$ and ${\delta }_2$ . Nusselt number is an increasing function of $\beta$ and $\mathrm{\Omega }$ at the upper end of the disk and decreasing function at the lower end.

Table 6.

Numerical values of Nusselt number for TiO ${}_2$ and GO at lower and upper disk when $Re=0.9,Pr=6.2,B_1=0.4,B_2=0.9,\lambda =0.2,{\gamma }_1=0.2,{\gamma }_2=0.5,M=0.5$ .

						TiO ${}_2$		GO
$\chi$	$Rd$	${\delta }_1$	${\delta }_2$	$\beta$	$\mathrm{\Omega }$	$N{u}_1$	$N{u}_2$	$N{u}_1$	$N{u}_2$
0.1	0.5	0.3	0.2	0.5	0.1	0.137507	0.159435	0.136484	0.156644
0.2						0.132718	0.148223	0.131178	0.143827
0.3						0.128757	0.139242	0.126872	0.133973
0.1	1					0.177149	0.199144	0.173493	0.193707
	1.5					0.216807	0.238843	0.210513	0.230763
	0.5	0.5				0.158947	0.184294	0.158055	0.181401
		0.8				0.174228	0.202012	0.173477	0.199102
		0.3	0.4			0.194562	0.225589	0.193236	0.221779
			0.6			0.225791	0.261798	0.22433	0.257466
			0.2	0.7		0.137317	0.15957	0.136336	0.156748
				0.9		0.137214	0.159644	0.136256	0.156806
				0.5	0.3	0.137480	0.159453	0.136461	0.156659
					0.5	0.137426	0.159491	0.136416	0.156691

TiO ${}_2$ : titanium dioxide; GO: graphene oxide.

Conclusion

The present work deals with the flow of nanofluid having nanoparticles (SWCNT, MWCNT, TiO ${}_2$ , and GO) by considering velocity slip conditions at the surface of two rotating and stretchable disks. Some of the highlighted results are stated as follows:

At the lower end of the disk, SWCNT-based fluid experiences the lowest temperature while TiO ${}_2$ -based fluid has the greatest temperature enhancement. SWCNT’s highest thermal conductivity and lowest heat capacitance account for this occurrence.

GO-based fluid has a higher tangential velocity than TiO ${}_2$ -based fluid because GO has a lower density than TiO ${}_2$ nanoparticles.

All of the nanoparticles under consideration have nearly identical axial and radial velocity patterns.

A significant improvement in the heat transport is captured with the rise in parameters $Rd$ , ${\delta }_1$ , and ${\delta }_2$ for considered nanoparticles. Increasing ${\delta }_2$ from 0.4 to 0.6, heat transfer at lower disk is improved by 16.66% in hexagonal-shaped CNTsbased nanofluid and about 15.78% in spherical-shaped TiO ${}_2$ - and GO-based nanofluid.

Varying rotation parameter from 0.3 to 0.5, skin friction at lower disk is decreased by 1.46% for CNTs-based nanofluid and this decrement is 0.97% for TiO ${}_2$ - and GO-based nanofluid.

TiO ${}_2$ -based fluid exhibits more elevation in heat transfer rate than other nanoparticles because of its spherical form, but SWCNTs exhibit the least heat transfer rate at the lower and upper disks.

At both disks, GO-based fluid has the lowest skin friction coefficient while TiO ${}_2$ -based fluid has the highest.

Supplemental Material

sj-docx-1-pie-10.1177_09544089221145930 - Supplemental material for Significance of radiant-energy and multiple slips on magnetohydrodynamic flow of single-walled carbon nanotube-water, titanium dioxide–water, multiwalled carbon nanotube–water, graphene oxide–water nanofluids

Supplemental material, sj-docx-1-pie-10.1177_09544089221145930 for Significance of radiant-energy and multiple slips on magnetohydrodynamic flow of single-walled carbon nanotube-water, titanium dioxide–water, multiwalled carbon nanotube–water, graphene oxide–water nanofluids by Tanvi Singla, Sapna Sharma and Bhuvaneshvar Kumar in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Supplemental Material

sj-zip-2-pie-10.1177_09544089221145930 - Supplemental material for Significance of radiant-energy and multiple slips on magnetohydrodynamic flow of single-walled carbon nanotube-water, titanium dioxide–water, multiwalled carbon nanotube–water, graphene oxide–water nanofluids

Supplemental material, sj-zip-2-pie-10.1177_09544089221145930 for Significance of radiant-energy and multiple slips on magnetohydrodynamic flow of single-walled carbon nanotube-water, titanium dioxide–water, multiwalled carbon nanotube–water, graphene oxide–water nanofluids by Tanvi Singla, Sapna Sharma and Bhuvaneshvar Kumar in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering