A Thin Film Flow of Nanofluid Comprising Carbon Nanotubes Influenced by Cattaneo-Christov Heat Flux and Entropy Generation

Abstract

This study aims to scrutinize the thin film flow of a nanofluid comprising of carbon nanotubes (CNTs), single and multi-walled i.e., (SWCNTs and MWCNTs), with Cattaneo-Christov heat flux and entropy generation. The time-dependent flow is supported by thermal radiation, variable source/sink, and magneto hydrodynamics past a linearly stretched surface. The obtained system of equations is addressed by the numerical approach bvp4c of the MATLAB software. The presented results are validated by comparing them to an already conducted study and an excellent synchronization in both results is achieved. The repercussions of the arising parameters on the involved profiles are portrayed via graphical illustrations and numerically erected tables. It is seen that the axial velocity decreases as the value of film thickness parameter increases. It is further noticed that for both types of CNTs, the velocity and temperature distributions increase as the solid volume fraction escalates.

1. Introduction

The flow and heat transfer phenomenon in thin fluid film past stretched surfaces has promising applications including continuous casting, extrusion of plastic sheets, drawing of polymer surfaces, foodstuff processing, annealing and tinning of copper wires, and cooling of metallic plates [1]. The maintenance of the extrudes’ surface is vital in the extrusion process smooth surface with minimum friction and enough strength is necessary for the coating procedure. Additionally, all this highly rely on the flow and heat transfer properties of the thin film over stretched surfaces. Because of this, the analysis in such cases is quite essential. Wang’s [2] pioneering work by deliberating the hydrodynamics of time-dependent thin fluid film flow past a stretching sheet invited researchers to work in this attractive industry-oriented theme. Andersson et al. [3] further developed Wang’s idea for heat transfer analysis. This case is further presented in a more generalized form by Chung and Andersson [4]. The solution to the same problem is discussed analytically by Wang [5]. The thin film flow is later analyzed in various scenarios like magnetic impact [6,7], thermo-capillary impacts [8], and non-Newtonian fluids [9,10,11,12].

The above-mentioned studies on thin liquid films are limited to Newtonian and non-Newtonian fluids in the absence of nanofluids. In recent years, the subject of nanofluids, owing to their amazing characteristic of high thermal conductivity, has gained much attention of researchers and scientists. The seminal work by Choi and Eastman [13] introducing “nanofluids” has revolutionized the heat transfer processes. A nanofluid is an amalgamation of the solid metallic particles called “Nanoparticles” with a size of 1–100 nm and ordinary liquids. Nanofluids are the finest coolants with amazing applications including microelectronics, optical manufacturing and transportation [14]. There are studies that emphasize the thin film liquid flow of nanofluids. Lin et al. [15] numerically scrutinized the thin film Pseudo-plastic nano liquid flow with the impact of internal heat generation by utilizing R–K scheme and Newton’s method. Later, Lin et al. [16] extended this study to the impacts of viscous dissipation and temperature reliant thermal conductivity. The nano-liquid thin film flow comprising graphene nanoparticles under the influence of aligned magnetic effect is discussed by Sandeep [17]. Zhang et al. [18] studied the Oldroyd-B nanofluid thin film flow analytically with two types of nanoparticles, i.e., silver and copper and found that nanofluid containing silver nanoparticles has a better thermal conductivity in comparison to the copper nanoparticles. Zhang et al. [19] also deliberated the power law nano liquid thin film flow with the slip using the differential transform method. The problem of nanofluid thin films flowing past an elastic stretched sheet is solved using the least square method (LSM) by Fakour et al. [20]. Ishaq et al. [21] deliberated the analytical solution of Powell-Eyring nano liquid thin film flow with thermal radiation past a permeable stretched surface. The flow of Darcy-Forchheimer nanofluid thin film comprising SWCNTs past an unsteady stretched surface is studied by Nasir et al. [22].

There are numerous applications of heat transfer in industrial and engineering processes. These include cooling towers, fuel cells, microelectronics, and nuclear reactors. The fundamental essence in all these processes is that the value of thermal conductivity is presumed to be a constant. However, this value varies with temperature and other factors. Pal [23] and Vajravelu et al. [24] observed that the thermal conductivity varies linearly as the temperature is altered from 0° to 400° F. Initially, the Fourier law of heat conduction has been used in the modeling of heat transfer applications but the system encounters an initial disturbance due to the “parabolic energy equation” which is referred to as “paradox in heat conduction”. This shortcoming in the Fourier’s model was addressed by Cattaneo [25] who introduced the thermal relaxation time in the Fourier law of heat conduction. Cattaneo’s act helped to represent the temperature profile via the hyperbolic energy equation and heat transport propagation using thermal waves with a controlled speed. This heat transport mechanism is employed in diverse practical scenarios, ranging from nano-liquid flow models to skin burn injury models [26]. Moreover, several materials possess a large thermal relaxation time, such as biological tissues having a relaxation time of 91–100 s and sand of 21 s. To uphold the material invariant formulation, Christov altered the Maxwell-Cattaneo model by swapping the time derivative with Oldroyd’s upper convected derivative. This improved version is nowadays being termed as the Cattaneo-Christov (C-C) heat flux model. Later, Han et al. [27] introduced an analytical solution for the viscoelastic material including the velocity slip boundary along with the C-C heat flux. Mustafa [28] analyzed the rotating flow of the Maxwell fluid with an upper convected derivative and C-C heat flux over a linearly stretched surface using both the analytical and numerical methods. A similar case was examined by Khan et al. [29] considering an exponentially stretched surface. The squeezed flow of the C-C heat flux with CNTs between two parallel disks is studied by Lu et al. [30]. Ramzan et al. [31] studied the flow of the Williamson fluid flow numerically with C-C heat flux associated with the convective boundary condition and homogeneous-heterogeneous reactions. The flow of the magnetohydrodynamics (MHD) second-grade fluid over a stretched cylinder with C-C heat flux is discussed by Alamri et al. [32]. Ramzan et al. [33,34] deliberated the Maxwell and third-grade fluid flows with homogeneous-heterogeneous reactions and C-C heat flux. The flow of aqueous based nanotubes with homogeneous-heterogeneous reactions past a Darcy-Forchheimer three-dimensional flow is studied by Alshomrani and Ullah [35]. Saleem et al. [36] discussed the squeezing three-dimensional nanofluid flow comprising of nanotubes in a Darcy-Forchheimer medium with thermal radiation and heat generation/absorption. There are numerous explorations that discuss on the flow of nanofluid amalgamated with carbon nanotubes in various scenarios but there are fewer that address the thin film flow. Some more explorations focusing on carbon nanotube or nanofluid flow may be found in References [37,38,39,40] and many therein.

The literature review reveals that the flow of a thin film with the Newtonian/non-Newtonian fluids is scarce in the literature and this subject gets even narrower if we talk about the thin film flows of nanofluids. Very few explorations are available that discuss the thin film flows of nanofluid-comprising nanotubes. Keeping in mind the importance of hydrodynamic flows, the idea of nanoliquid thin films in comparatively new and fewer explorations are available in the literature (see

Table 1. The studies on nanoliquid film flow.

Authors	Nanofluid Models	Film Thickness	Nanotubes SWCNTs/MWCNTs	C-C Heat Flux
Lin et al. [15]	Tiwari and Das	√	×	×
Sandeep [17]	Tiwari and Das	√	×	×
Nasir et al. [22]	Tiwari and Das	√	SWCNTs	×
Narayana and Sibanda [41]	Tiwari and Das	√	×	×
Xu et al. [42]	Tiwari and Das	√	×	×
Qasim et al. [43]	Buongiorno’s	√	×	×
Present	Tiwari and Das	√	√	√

(√) means effect is present and (×) means effect is absent.

). This presented model is solved numerically and will present an estimated solution. The other limitation of the flow is that it is discussed in 2D and can be extended to 3D with some more novel effects like homogeneous-heterogeneous reactions, etc. The model presented here is an amalgamation of C-C heat flux and entropy generation in the thin film flows of the nanofluids comprising of both types of nanotubes (SWCNTs/MWCNTs) and has not yet been discussed in the literature. The numerical solution of the problem is achieved. A comparison with an already established result in the limiting case is also given and an excellent agreement between both is found. This corroborates our presented results. The graphical illustrations and numerically calculated values of the physical parameters are also added to the problem.

2. Mathematical Modeling

Let us assume a thin film flow of a nanoliquid flow comprising CNTs past a time dependent linearly stretched surface. The elastic sheet emerges from a slender slit at the Cartesian coordinate system’s origin (Figure 1). The surface moves along the x-axis (y = 0) with a velocity $u_w(x,t)=\frac{bx}{(1-\mathsf{\alpha }t)},$ with b and a being the constants in the y-direction and temperature T_w(x, y). The stream function ξ is considered such that u = ξ_y, and v = −ξ_x.

The thin film is of width h(x, y). The flow is laminar and incompressible. A magnetic field $B(x,t)=B_o{(1-at)}^{-\frac{1}{2}},$ is employed normal to the extended surface. The governing unsteady conservation equations [17] under the aforementioned assumptions are appended as follows:

$$\frac{{\partial }^2\mathsf{\xi }}{\partial x\partial y}-\frac{{\partial }^2\mathsf{\xi }}{\partial y\partial x}=0$$

(1)

$$\frac{{\partial }^2\mathsf{\xi }}{\partial t\partial y}-\frac{\partial \mathsf{\xi }}{\partial y}\frac{{\partial }^2\mathsf{\xi }}{\partial x\partial y}-\frac{\partial \mathsf{\xi }}{\partial x}\frac{{\partial }^2\mathsf{\xi }}{\partial y^2}=v_{nf}\frac{{\partial }^3\mathsf{\xi }}{\partial y^3}+\frac{{\mathsf{\sigma }}_{nf}}{{\mathsf{\rho }}_{nf}}{B}^{{}^2}(t)\frac{\partial \mathsf{\xi }}{\partial y}{\cos }^2\mathsf{\epsilon },$$

(2)

$${(\mathsf{\rho }{C}_p)}_{nf}\left(\frac{\partial T}{\partial t}+\frac{\partial \mathsf{\xi }}{\partial y}\frac{\partial T}{\partial x}-\frac{\partial \mathsf{\xi }}{\partial x}\frac{\partial T}{\partial y}\right)+{\mathsf{\lambda }}_2{\mathsf{\Omega }}_2=\left(k_{nf}+\frac{16T_{\infty }{}^3{\mathsf{\sigma }}^{\ast }}{3k^{\ast }}\right)\frac{{\partial }^{2}T}{\partial y^2}+q{}^{‴}$$

(3)

With the following corresponding boundary conditions

$$\begin{array}{c}-{\mathsf{\xi }}_x=0,{\mathsf{\xi }}_y=u_w,T=T_s,\mathrm{at}y=0,\\ {\mathsf{\xi }}_{yy}=0,-{\mathsf{\xi }}_x=h_t,T=0,\mathrm{as}y=h(t).\end{array}$$

(4)

The Cattaneo-Christov term is defined as

$$\begin{array}{r}{\mathsf{\Omega }}_2=\frac{{\partial }^{2}T}{\partial t^{{}^2}}+\frac{\partial u}{\partial t}\frac{\partial T}{\partial x}+2\frac{\partial \mathsf{\xi }}{\partial y}\frac{{\partial }^{2}T}{\partial t\partial x}-2\frac{\partial \mathsf{\xi }}{\partial x}\frac{{\partial }^{2}T}{\partial t\partial y}+\frac{\partial v}{\partial t}\frac{\partial T}{\partial y}+\frac{\partial \mathsf{\xi }}{\partial y}\frac{{\partial }^{{}^2}\mathsf{\xi }}{\partial x\partial y}\frac{\partial T}{\partial x}+\frac{\partial \mathsf{\xi }}{\partial x}\frac{{\partial }^2\mathsf{\xi }}{\partial y\partial x}\frac{\partial T}{\partial y}\\ +{\left(\frac{\partial \mathsf{\xi }}{\partial y}\right)}^2\frac{{\partial }^{{}^2}T}{\partial x^{{}^2}}+{\left(\frac{\partial x}{\partial x}\right)}^2\frac{{\partial }^{2}T}{\partial y^{{}^2}}-2\frac{\partial \mathsf{\xi }}{\partial y}\frac{\partial \mathsf{\xi }}{\partial x}\frac{{\partial }^{{}^2}T}{\partial x\partial y}-\frac{\partial \mathsf{\xi }}{\partial y}\frac{{\partial }^2\mathsf{\xi }}{\partial x^{{}^2}}\frac{\partial T}{\partial y}-\frac{\partial \mathsf{\xi }}{\partial x}\frac{{\partial }^2\mathsf{\xi }}{\partial y^2}\frac{\partial T}{\partial x}\hfill \end{array}$$

(5)

The heat source/sink “$q{}^{‴}$” is represented by

$$q{}^{‴}=\frac{k_f{u}_w(T_s-T_0)}{x{\mathsf{\nu }}_f}\left(A^{\ast }{f}^{\prime }+B^{\ast }\frac{(T-T_0)}{(T_s-T_0)}\right)$$

(6)

The thermophysical attributes (specific heat C_p, density ρ and thermal conductivity k) of the base fluid (H₂O) and carbon nanotubes (SWCNTs /MWCNTs) are appended in

Table 2. The thermophysical physiognomies of the fluid and CNTs [30].

Physical Characteristics	Conventional Fluid	Nano Particles
–	H2O	SWCNTs	MWCNTs
Cp (J/kg K)	4179	425	796
ρ (kg/m3)	997	2600	1600
k (W/mK)	0.613	6600	3000

.

The hypothetical relations are characterized as follows:

$${\mathsf{\mu }}_{nf}=\frac{{\mathsf{\mu }}_f}{{(1-\mathsf{\varphi })}^{2.5}},v_{nf}=\frac{{\mathsf{\mu }}_{nf}}{{\mathsf{\rho }}_{nf}},$$

(7)

$${\mathsf{\rho }}_{nf}=(1-\mathsf{\varphi }){\mathsf{\rho }}_f+\mathsf{\varphi }{\mathsf{\rho }}_{{}_{CNT}},{\mathsf{\alpha }}_{nf}=\frac{k_{nf}}{{\mathsf{\rho }}_{nf}{(c_p)}_{nf}}$$

(8)

$$\frac{{\mathsf{\sigma }}_{nf}}{{\mathsf{\sigma }}_f}=1+\frac{3\mathsf{\sigma }\mathsf{\varphi }-3\mathsf{\varphi }}{\mathsf{\sigma }+2-\mathsf{\sigma }\mathsf{\varphi }+\mathsf{\varphi }},\mathsf{\sigma }=\frac{{\mathsf{\sigma }}_{CNT}}{{\mathsf{\sigma }}_f},$$

(9)

$$\frac{k_{nf}}{k_f}=\frac{(1-\mathsf{\varphi })+2\mathsf{\varphi }\frac{k_{CNT}}{k_{CNT}-k_f}\ln (\frac{k_{CNT}+k_f}{2k_f})}{(1-\mathsf{\varphi })+2\mathsf{\varphi }\frac{k_f}{k_{CNT}-k_f}\ln (\frac{k_{CNT}+k_f}{2k_f})}$$

(10)

Using the similarity transformations

$$\begin{array}{c}\mathsf{\eta }=\frac{1}{\mathsf{\beta }}{\left(\frac{b}{{\mathsf{\nu }}_f(1-at)}\right)}^{\frac{1}{2}}y,\mathsf{\Psi }=\mathsf{\beta }{\left(\frac{b{\mathsf{\nu }}_f}{(1-at)}\right)}^{\frac{1}{2}}xf(\mathsf{\eta }),\mathsf{\theta }=\frac{T-T_0}{T_s-T_{{}_0}}\\ T=T_{{}_0}-T_r\left(\frac{b{x}^2}{2{\mathsf{\nu }}_f}\right){(1-at)}^{-1.5}\mathsf{\theta }(\mathsf{\eta }),\end{array}$$

(11)

The requirement of Equation (1) is fulfilled undoubtedly and Equations (2) and (3) yield

$$f{}^{‴}+{(1-\mathsf{\varphi })}^{2.5}(1-\mathsf{\varphi }+\mathsf{\varphi }\frac{{\mathsf{\rho }}_{CNT}}{{\mathsf{\rho }}_f})\mathsf{\lambda }\left\{ff{}^{″}-f{}^{\prime 2}-S\left(f^{\prime }+\frac{1}{2}\mathsf{\eta }f{}^{″}\right)\right\}-{(1-\mathsf{\varphi })}^{{}^{2.5}}\frac{{\mathsf{\sigma }}_{nf}}{{\mathsf{\sigma }}_f}M{f}^{\prime }{\cos }^2\mathsf{\epsilon }=0$$

(12)

$$\begin{array}{l}\frac{\left(\frac{k_{nf}}{k_f}+\frac{4}{3}R\right)}{P_r\left[1-\mathsf{\varphi }+\mathsf{\varphi }\frac{{(\mathsf{\rho }{C}_p)}_{CNT}}{{(\mathsf{\rho }{C}_p)}_f}\right]}\mathsf{\theta }{}^{″}-\mathsf{\lambda }\left[2f^{\prime }\mathsf{\theta }-f\mathsf{\theta }{}^{\prime }+\frac{S}{2}\left(3\mathsf{\theta }+\mathsf{\eta }\mathsf{\theta }{}^{\prime }\right)\right]+\frac{1}{P_r\left[1-\mathsf{\varphi }+\mathsf{\varphi }\frac{{(\mathsf{\rho }{C}_p)}_{CNT}}{{(\mathsf{\rho }{C}_p)}_f}\right]}(A^{\ast }{f}^{{}^{\prime }}+B^{\ast }\mathsf{\theta })\\ +\mathsf{\gamma }\left\{\begin{array}{l}-\frac{15}{2}{S}^2\mathsf{\theta }-\frac{7}{2}{S}^2\mathsf{\eta }\mathsf{\theta }\prime -\frac{1}{4}{S}^2{\mathsf{\eta }}^2\mathsf{\theta }{}^{″}-8Sf{}^{\prime }\mathsf{\theta }-\mathsf{\eta }Sf{}^{″}\mathsf{\theta }-\frac{3}{2}\mathsf{\eta }Sf{}^{\prime }\mathsf{\theta }{}^{\prime }\\ +\frac{9}{2}Sf\mathsf{\theta }{}^{\prime }+S\mathsf{\eta }f\mathsf{\theta }{}^{″}-4f{\prime }^2\mathsf{\theta }+3ff{}^{\prime }\mathsf{\theta }{}^{\prime }-f{\prime }^2\mathsf{\theta }{}^{″}+2ff{}^{″}\mathsf{\theta }\end{array}\right\}=0,\end{array}$$

(13)

Additionally, the boundary conditions of Equation (4) become

$$f(0)=0,f^{\prime }(0)=1,\mathsf{\theta }(0)=1,f(1)=\frac{S}{2},f{}^{″}(1)=0,\mathsf{\theta }{}^{\prime }(1)=0$$

(14)

The values of various non-dimensional parameters are defined as follows:

$$P_r=\frac{{\mathsf{\upsilon }}_f}{{\mathsf{\alpha }}_f},S=\frac{\mathsf{\alpha }}{b},R=\frac{4{\mathsf{\sigma }}^{\ast }{T}_0{}^3}{k^{\ast }{k}_f},M=\frac{{\mathsf{\sigma }}_f{B}_{{}_0}{}^2}{b{\mathsf{\rho }}_f},\mathsf{\gamma }=\frac{{\mathsf{\lambda }}_{2}b}{1-\mathsf{\alpha }t},\mathsf{\lambda }={\mathsf{\beta }}^{{}^2}$$

(15)

Physical quantities like the Skin friction coefficient and the local Nusselt number are given as

$$\begin{array}{l}N{u}_x=\frac{x{q}_w(x)}{k_f(T_s-T_{{}_0})},C_f=\frac{{\mathsf{\tau }}_w}{{\mathsf{\rho }}_f{u}_w{}^2},\\ q_w(x)=-k_{nf}{(\frac{\partial T}{\partial y})}_{y=0},{\mathsf{\tau }}_w={\mathsf{\mu }}_{nf}{(\frac{\partial u}{\partial y})}_{y=0},\end{array}$$

(16)

Additionally, in dimensionless form, as follows:

$$\begin{array}{l}{C}_{f}R{e}_x{}^{1/2}=\frac{1}{\mathsf{\beta }{(1-\mathsf{\varphi })}^{{}^{2.5}}}f{}^{″}(0),\\ N{u}_{x}R{e}_x{}^{-1/2}=-\frac{1}{\mathsf{\beta }}\left(\frac{k_{nf}}{k_f}+\frac{4}{3}R\right){\mathsf{\theta }}^{{}^{\prime }}(0)\end{array}$$

(17)

3. Entropy Generation

The entropy generation under the aforementioned assumptions is given as below:

$$E_{gen}{}^{‴}=\frac{k_f}{T_{{}_0}{}^{{}^2}}\left[\frac{k_{nf}}{k_f}+\frac{16T_{\infty }{}^3{\mathsf{\sigma }}^{\ast }}{3k^{\ast }{k}_f}\right]{\left(\frac{\partial T}{\partial y}\right)}^2+\frac{{\mathsf{\mu }}_{nf}}{T_0}{\left(\frac{\partial u}{\partial y}\right)}^2+\frac{{\mathsf{\sigma }}_{nf}}{T_0}{B}^{{}^2}(t)u^2{\cos }^2\mathsf{\epsilon }$$

(18)

where all terms defined in Equation (15) portray the usual meaning. The entropy generation N_G is defined as

$$N_G=\left(\frac{k_{nf}}{k_f}+\frac{4}{3}R\right)R{e}_x\mathsf{\theta }{{}^{\prime }}^{{}^2}+\frac{1}{{(1-\mathsf{\varphi })}^{2.5}}\frac{BrR{e}_x}{\mathsf{\alpha }}f{}^{″}{}^2+\frac{BrM}{\mathsf{\alpha }}\frac{{\mathsf{\sigma }}_{nf}}{{\mathsf{\sigma }}_f}{\cos }^2\mathsf{\epsilon }f{\prime }^2{}^2$$

(19)

where $S_0{}^{‴}$ and $S_{gen}{}^{‴}$ are the characteristic entropy generation rate and the entropy generation rate. The parameters defined in the above equation are given as

$$\mathsf{\alpha }=\frac{\Delta T}{T_{w0}},Br=\frac{{\mathsf{\mu }}_f{u}_w{}^2}{k_f\Delta T},R{e}_x=\frac{u_{w}x}{{\mathsf{\nu }}_f}$$

(20)

4. Results and Discussion

This section is devoted to witnessing the impression of numerous parameters on the involved profiles whilst keeping in view their physical significance. The MATLAB built-in function bvp4c is utilized to address the differential Equations (9), (10), and (16) with the associated boundary conditions of Equation (11). To solve these, first we have converted the 2nd and 3rd order differential equations to the 1st order by introducing new parameters. The tolerance for the existing problem is fixed as 10⁻⁶. The initial guess we yield must satisfy the boundary conditions asymptotically and the solution as well. The results show the influence of solid volume fraction (φ), dimensionless film thickness (λ), magnetic parameter (M), unsteadiness parameter (S), radiation parameter (R), thermal relaxation parameter (γ), and non-uniform heat source/sink parameter on the velocity, temperature and entropy generation profiles. Further, the numerical values for the Skin friction and Nusselt number are given in

Table 3. The comparison table of –θ’(0) with Sandeep [17] for varied estimates of S when R = M = γ = 0, P_r = 1.0.

S	φ = 0
	Sandeep [17]	Present Result
1.0	2.6772221621	2.677222
1.2	1.9995914260	1.999591
1.4	1.4477543611	1.447754
1.6	0.9566978443	0.956697
1.8	0.4845366320	0.484536

and

Table 4. The numerical value of the Skin friction with P_r = 6.2.

φ	S	λ	M	−CfRex½
–	–	–	–	SWCNTs	MWCNTs
0.1	0.1	0.1	1.0	0.53775	0.52995
0.2	–	–	–	0.57083	0.55402
0.3	–	–	–	0.60395	0.57701
–	0.1	–	–	0.53775	0.52995
–	0.2	–	–	0.54610	0.53761
–	0.3	–	–	0.55442	0.54524
–	–	0.1	–	0.53775	0.52995
–	–	0.2	–	0.62530	0.61054
–	–	0.3	–	0.70792	0.68693
–	–	–	0.0	0.12194	0.11183
–	–	–	0.5	0.34094	0.33214
–	–	–	1.0	0.53775	0.52995

for different parameters. The numerical values of the parameters are fixed as φ = 0.1, A* = B* = λ = γ = 0.5 = S, R = 1.0 = M, and P_r = 6.2. Figure 2 and Figure 3 display the impact of solid volume fraction (φ) on axial velocity and temperature distribution. For incremented values of the solid volume fraction (φ), the velocity and temperature profiles enhance in case of both SWCNTs and MWCNTs. Actually, the convective flow and the solid volume fraction are directly proportionate with each other and this is the main reason behind the enhancement of axial velocity and the temperature of the fluid.

Figure 4 and Figure 5 depict the behavior of axial velocity and the temperature field for the growth estimates of the film thickness parameter λ. It is found that both velocity and temperature profiles diminish for increasing values of the film thickness parameter λ. In fact, the more the film thickness, the lesser the fluid motion. This is because of the fact that higher values of film thickness dominate the viscous forces, eventually diminishing the fluid velocity. Similar behavior is observed for the temperature field.

The effect of the magnetic parameter M on the velocity and temperature fields can be visualized in Figure 6 and Figure 7. Figure 6 displays the impact of the magnetic parameter M on axial velocity. The is the axial velocity of the declining function of the magnetic parameter M. Physically, by enhancing the magnetic parameter M, the Lorentz force is strengthened in the flow, which has a tendency to resist the fluid’s motion and slow it down. This force also creates heat energy in the flow. Consequently, the temperature distribution increases both the SWCNTs and MWCNTs, which is displayed in Figure 7.

Figure 8 and Figure 9 show the effect of the unsteadiness parameter S on the velocity and temperature distributions. It is found that with the increase of the unsteadiness parameter S, the axial velocity diminishes. Physically, the bouncy effect acts on the flow and diminishes it due to the increase in the unsteadiness parameter S. Therefore, the thermal and momentum boundary layer thicknesses decrease.

Figure 10 determines the consequence of the thermal relaxation parameter γ on the temperature of the fluid. It is concluded that the temperature diminishes for increased values of the thermal relaxation parameter γ. The temperature tends to be sharper near the boundary as the value of γ is higher than the points on the growth in the wall slope of the temperature profile.

Figure 11 demonstrates the impact of the radiation parameter R on the temperature profile. It is comprehended that the temperature field is an increasing function of the radiation parameter R. It is also concluded that the thermal boundary layer thickness for both carbon nanotubes is increased. In fact, larger estimates of the radiation parameter reduce the mean absorption coefficient and enhance the radiative heat flux’s divergence. Due to this, the temperature of the fluid is upsurged.

The influence of non-uniform heat source/sink parameters A* and B* on the temperature distribution is shown in Figure 12 and Figure 13. It can be understood that the temperature profile augments the boosted estimates of non-uniform heat source/sink parameters.

The effect of Brinkman number (B_r), magnetic parameter (M) and Reynolds number (Re_x) on the averaged entropy generation number is demonstrated in Figure 14, Figure 15 and Figure 16. It is concluded that the entropy generation number increases for mounting estimations of Brinkman number (B_r), magnetic parameter (M) and Reynolds number (Re_x) for both SWCNT and MWCNT.

Table 3 is erected to envision the precision of the presented model by comparing it with Sandeep [17] who discusses the flow of nanofluids past a thin film under the influence of the magnetic field. To make a comparison, we have neglected the impacts of the volume fraction, electrical conductivity, and thermal relaxation parameters. Excellent alignment is achieved between both results.

Table 4 shows the estimates of the Skin friction coefficient for different parameters. It is seen that the Skin friction coefficient increases for growing values of the magnetic parameter, solid volume fraction, unsteadiness parameter, and film thickness.

Table 5. The numerical value of the Nusselt number with γ = 0.1, P_r = 6.2.

Λ	R	φ	A*	B*	S	NuxRex−½
–	–	–	–	–	–	SWCNTs	MWCNTs
0.1	0.1	0.1	0.1	0.1	0.1	2.11090	2.06990
0.2	–	–	–	–	–	3.27730	3.16440
0.3	–	–	–	–	–	4.46510	4.24750
–	0.1	–	–	–	–	2.36350	2.06990
–	0.3	–	–	–	–	2.36770	2.07060
–	0.5	–	–	–	–	2.37230	2.07130
–	–	0.1	–	–	–	2.11090	2.06990
–	–	0.2	–	–	–	2.59370	2.54840
–	–	0.3	–	–	–	2.84440	2.84000
–	–	–	0.1	–	–	2.11800	2.06990
–	–	–	0.3	–	–	1.93540	1.89220
–	–	–	0.5	–	–	1.75270	1.71440
–	–	–	–	0.1	–	2.11800	2.06990
–	–	–	–	0.3	–	1.88910	1.85770
–	–	–	–	0.5	–	1.66430	1.25340
–	–	–	–	–	0.1	2.11090	2.06990
–	–	–	–	–	0.2	2.81360	2.72250
–	–	–	–	–	0.3	3.66060	3.48490

demonstrates the numerical values of Nusselt number for numerous parameters. It is determined that the Nusselt number increases with augmented values of the dimensionless film thickness, radiation parameter, solid volume fraction, and unsteadiness parameter, while it diminishes for growing values of non-uniform heat source/sink.

5. Conclusions

The thin film flow of nanofluid comprising of CNTs of both types (SWCNTs/MWCNTs) is studied whilst keeping in view the important applications of CNTs in many engineering applications. The flow is supported by the additional effects like C-C heat flux and entropy generation. The model is solved numerically with the support of the MATLAB software function bvp4c. The highlights of the existing study are

• Velocity and temperature distributions are mounting functions of the solid volume fraction for both types of CNTs in case of the thin film flow.
• For growing estimates of the thin film thickness parameter, the axial velocity diminishes.
• The velocity and temperature distributions show an opposite trend for the strong magnetic field in a thin film flow model.
• Larger estimates of heat source/sink parameter lead to an increase in the temperature of the fluid.
• The temperature of the fluid is decreased for higher values of the thermal relaxation parameter.
• With an increase in the estimates of film thickness, the magnetic parameter and the Skin friction coefficient show mounting behavior.
• The Nusselt number shows declining behavior for growing values of non-uniform heat source/sink.
• Entropy generation in the case of thin film flow is higher for larger estimates of the Brinkman number and the magnetic parameter.