Boundary layer flow and heat transfer in a thin-film second-grade nanoliquid embedded with graphene nanoparticles

Abstract

The two-dimensional magnetohydrodynamic unsteady movement and transmission of energy confined to finite domain layer of the second-grade nanofluid embedded with graphene nanoparticles on an expanding space are studied. Graphene nanoparticles have continuous electrical conductivity because the charge carrier movement in graphene bears extremely peak points compared to the available nanomaterials. The well-known system of equations for movement and energy of the second-grade nanofluid film accompanying the additional information have been transformed into the fourth-order coupled differential systems accompanying the auxiliary facts on behalf of simplifying substitutions. The simplified systems are evaluated via an efficient approach through homotopy analysis method which provides very clear relations for the motion and energy representatives. All the potential factors of the output are debated and portrayed pictorially. The results are useful in the analysis, design of coating, and cooling/heating processes.

Introduction

Recently, the world confronts major challenges such as consistent energy provision, capacitance, and utilization. The main aim is not only to propose innovative and consistent work ability means but also, probably even more significantly, to save energy sufficiently and provide it on need, always for moving utilization such as systems of transportation and portable electronic devices. Carbon is the most suitable and of small weight element employed for energy saving which is essentially made into diverse shapes to deliver big surface capacity and rate of doing work capability. There exists a rich concentration on the application of carbon nanotubes (CNTs) for the rate of doing work capability saving tool fabrication. Apart from keeping low weight, CNTs have several good applications. They keep a sufficient enhanced surface area up to 1315 m² g⁻¹ for single-walled CNTs, in contrast to other elements, for example, graphite which has a typical surface area of 10–20 m² g⁻¹. But it should be noted that CNTs show disadvantages such as the existence of hazardous remaining metallic impure things that are highly difficult to make clean and a vast production price.

Graphene at present appeared as another energy-saving substance keeping remarkable aspects, such as light weight, inertness due to chemical aspects, and minimum cost. Graphene, keeping two-dimensional sp² carbon atoms packed in a honeycomb lattice, having great attraction throughout the world application after its finding. Intrinsically, graphene has a step-by-step layer shape, extreme stiffness, and exceptionally very tight and popular. The basic physical aspects of graphene particularly mechanical stiffness, tensile bearing capability, heat conduction, and ratio of aspect are remarkably very great. These popular aspects of graphene enable it a high advantage of delivering entity for diverse utilizations such as graphene polymer composites having conduction aspect, pressurizing sensor, rate of doing work-reliant usages, and the biofield. The surface area of graphene is 2630 m² g⁻¹ that is significantly suitable for the rate of doing work capacity utilizations. Based on extensive applications of graphene, Zuhra et al. ¹ made a research in the time-reliant movement and transmission of energy characteristics having electronic capability water-reliant non-Newtonian thin liquid film (of different structures) nanoliquid dispersion of graphene tiny size particles surpassing an expanding material in the existence of magnet acting at the right angle and the provision of dissimilar energy production/accumulation. They proved that heat transfer in non-Newtonian dispersion provides a leading activity in technical fields and in nature because of its tangential force looseness, tangential force weakening, and strength aspects. Khan et al. ² analytically studied the hydromagnetic CuO–H₂O thin nanoliquid finite domain confined layer coated on an expanding material in the hydrothermal environment, manifesting that the thermal conductivity enhances by introducing tiny/small-sized particle materials in dispersion and coating rate rises with amplifying the film size. Pumera ³ discussed the development that is fulfilled with the involvement of chemical relation, electrochemical and electricity rate of doing work-saving environments through graphene. Eswaraiah et al. ⁴ performed the experimental work using the graphene. They have been prepared a small layer of graphene through the oxide of some element through a nanodispersion procedure on the mentioned process emitted from the sun; in addition, graphene-reliant engine oil nanodispersion is created and their aspects due to friction, antiwear, and maximum force per unit features are computed. Khan et al. ⁵ provided the discoveries on the joint flow equation of a two-dimensional steady laminar gravity employed non-Newtonian Casson and Williamson nanofluid films accompanying a type of heat conducting on solid non-horizontal wall accompany involvement of Cu tiny particles and small organisms using actively boundary conditions and the Buongiorno model. They have obtained interesting results regarding fluids flow, Cu nanoparticles, gyrotactic microorganisms, and heat transfer in analyzing the comparison of Casson and Williamson nanoliquid. Meharali et al. ⁶ prepared nitrogen-doped graphene (NDG) nanodispersion by adopting the special step procedure in a water dispersion of 0.025 wt% Triton X-100 as an impurity at some saturations (0.01, 0.02, 0.04, 0.06 wt%). They experimentally proved that due to electrical conduction of the water-dependent NDG tiny particles, dispersion exhibits a straight forward reliance on the saturation and developed up to 1814.96% for taking the amount of 0.06 wt% NDG tiny particles. Khan ⁷ presented the study about thermal conductivity using transmission due to heat in a non-Newtonian rat-type tiny particle dispersion movement retaining Cu tiny particles and small organisms. His work is related to energy sources such as biomass and biogas. Akhavan ⁸ investigated the preparation of graphene confined to finite domain layer on the Si₃N₄/Si material through minimization of exfoliation due to chemical reactions in graphene oxide tiny particle spaces in a minimizing situation, keeping several temperatures values, and following few distinct energy utilization techniques.

The movement and transmission of energy confined to finite domain non-Newtonian fluids past expanding space keep attraction and enough attention because of its vast utilizations in various departments of scientific and technological nature. Some applications are wire and fiber spraying, in thrusting out the conducting material and carbon substances, nourishments processing, nonstop provision of materials having high iron content, generation of plastic substances, exchanging of substances, making low temperature through the act of passing through pores, reactor, chemically making instrument, and so on. Nearly, all painting/spraying techniques need smoothness on the surface to fulfill the needs for existence, minimum roughness, transparency, and strength. In the view of these applications, Khan et al. ⁹ described the parametric study of diverse phenomenon accompanying energy and quantity of matter transmission flow of an external deriving expertness of an agent on the finite domain–confined layer second-grade dispersion of temperature-dependent features on expanding space. They added the homotopy procedure to solve the challenging issue and illustrated graphically the important contributions of all parameters, the dispersion changeable aspects such as heating conduction and viscousness in the platform of finite domain originated due to temperature. Lin et al. ¹⁰ analyzed the movement and transposition of energy of pseudo-plastic tiny particle dispersion in a finite domain on a time-reliant expanding space with the dispersion changeable aspects such as heating conduction and viscousness in the platform of finite domain originated due to temperature by investigating four several kinds of tiny particles such as Cu, CuO, Al₂O₃, and TiO₂ accompanying sodium carboxymethyl cellulose (CMC)-H₂O added like a classic dispersion. Khan et al. ¹¹ solved the problem using similarity solutions for the non-Newtonian second-order dispersion immersed in a permeable space on an expanding space providing that motion and heat transmission become lower, even though the high values to all of the parameters are assigned. They authenticated the computation by finding numerical comparisons, residual errors and showing the correlation with the published work. Abbas et al. ¹² employed a homotopy analysis method (HAM) to investigate the movement of confined finite domain rat-type dispersion fluid on a time-dependent expanding sheet in which the sequential computation attached to the complicated fourth-order differential issue is obtained and the potentialities of diverse representatives on movement and skin friction are illustrated. Zuhra et al. ¹³ investigated the magnetohydrodynamic (MHD) second-order nanodispersion of finite domain layer movement retaining copper tiny particles and small organisms accompanying the type of convection heating. They showed that skin friction quantity amplifies with a huge amount under the existence of amplifying positive quantities of the second-grade nanofluid representative. Khan et al. ¹⁴ considered Buongiorno model of nanofluid by investigating the tiny particle movement and dragging of particles control of MHD-mixed convection finite domain–confined layer second-grade nanodispersion movement accompanying intense external impact and transmission of energy past an expanding space by delivering the output that the system resumes axial and transverse velocities, thereby making three-dimensional motion on account of strong applied magnetic field. They obtained the interesting MHD results in which one result implies that the transverse velocity is due to the Hall effect and it vanishes as soon as the Hall effect is removed. Palwasha et al. ¹⁵ reported the creation of a tiny particle system environment, microorganisms in the activism of magnet prevailing, considered as a relevant class for usages in diverse fields including energy and medicine. Their study decided that magnetotactic microorganisms are the backbone of the interpreted system.

To get insight into the research, a study is to discuss the time-reliant special type of motion and transposition of energy in the second-grade confined to finite domain layer nanoliquid embedded with graphene tiny particles on an expanding space. The governing system is shifted into dimensionless form by the application of appropriate transformations and then treated with an efficient and well-known procedure, ¹⁶ which does not miss any parameter in different expressions of different profiles. The potentialities of each and every pertinent representative on motion and energy fields are portrayed in figures and elucidated.

Methods

Basic equations

A time-reliant, electrical capability dispersion finite domain–confined movement of an external force–controlled non-Newtonian second-grade nanofluid on an expanding space is treated. A flexible sheet begins working at a slit of small width determining its position on the place (0, 0) under the measuring dimensions (x, y). Note that x-axis is adopted accompanying the expanding space keeping the expanded displacement rate U_w(x, t) = cx(1 – αt)⁻¹ being c and α manifest constants in a style that c > 0 (expanding rate) also αt < 1. y-axis rests at the right angle to expanding space. Note that T₀ and T_ref are the initial and temperature at the reference position, and T_w(x, t) = T₀ + T_ref (cx²/2v_f) (1 – αt)^−1/2 manifests the wall energy of the dispersion, in which ${\nu }_f={\mu }_f/{\rho }_f$ manifests kinematic viscousness where f manifests the notation for base fluid. From outside, an agent (magnet) having field B(t) = B₀(1 – αt)^−1/2 is exerted at the right angle to the expanding space which is revealed in plot 1.

The potential of the induced magnet environment is very low under the considered system.

The Cauchy stress tensor in the second-grade dispersion as in the study by Abbas et al. ¹² is

$$\mathit{T}=-P\mathit{I}+\mu {\mathit{A}}_1+{\alpha }_1{\mathit{A}}_2+{\alpha }_2{\mathit{A}}_1^2$$

(1)

where P manifests the pressure, $\mathit{I}$ is the unit rate of tangential force, $\mu$ is the dynamic viscosity, ${\alpha }_1$ , ${\alpha }_2$ manifest the material constants, and ${\mathit{A}}_1$ , ${\mathit{A}}_2$ manifest the first Rivilin–Ericksen tangential forces formulated as

$${\mathit{A}}_1=\left(\mathrm{grad}\mathit{V}\right)+{\left(\mathrm{grad}\mathit{V}\right)}^T$$

(2)

$${\mathit{A}}_2=\frac{d{\mathit{A}}_1}{\mathrm{dt}}+{\mathit{A}}_1\left(\mathrm{grad}\mathit{V}\right)+{\left(\mathrm{grad}\mathit{V}\right)}^T{\mathit{A}}_1$$

(3)

in which $d/\mathrm{dt}$ manifests the material time differentiability and $\mathit{V}$ is the velocity vector. It is carefully noted that the Clausius–Duhem inequality is justified, and Helmholtz-free heat is very low in the balanced form to the dispersion locally at static position for the following conditions as in the study by Abbas et al. ¹²

$$0\le \mu ,0\le {\alpha }_1,{\alpha }_1+{\alpha }_2=0$$

(4)

Invoking the fact that for ${\alpha }_1={\alpha }_2=0$ , the governing system of the second-order dispersion alters to that of viscous dispersion. Taking the previous consideration, the nano-thermal system in time-reliant shape as in the study by Abbas et al. ¹² is derived as

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

(5)

$$\begin{array}{c}{\rho }_{\mathrm{nf}}\left[\begin{array}{c}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\hfill \end{array}\right]={\mu }_{\mathrm{nf}}\frac{{\partial }^{2}u}{\partial y^2}\hfill \\ +{\mu }_{\mathrm{nf}}{\alpha }_1\left[\begin{array}{c}\frac{{\partial }^{3}u}{\partial t\partial y^2}+u\frac{{\partial }^{3}u}{\partial x\partial y^2}+\frac{\partial u}{\partial x}\frac{{\partial }^{2}u}{\partial y^2}+\frac{\partial u}{\partial y}\frac{{\partial }^{2}v}{\partial y^2}+v\frac{{\partial }^{3}u}{\partial y^3}\hfill \end{array}\right]\hfill \\ -{\sigma }_{\mathrm{nf}}{B}^2\left(t\right)u\hfill \end{array}$$

(6)

$$\begin{array}{c}{\left(\rho c_P\right)}_{\mathrm{nf}}\left(\begin{array}{c}\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\hfill \end{array}\right)\hfill \\ =k_{\mathrm{nf}}\frac{{\partial }^{2}T}{\partial y^2}+\frac{k_f{u}_w}{x{\nu }_f}\left[\begin{array}{c}{A}_3(T_w-T_0)f\prime \left(\zeta \right)+A_4(T-T_0)\hfill \end{array}\right]\hfill \end{array}$$

(7)

Note that in the coordinate system, x-and y-directions have u and v velocities components, respectively. ${\rho }_{\mathrm{nf}}$ manifests the density of the nanofluid, B(t) manifests the magnet from outside, T manifests the dispersion energy state, $k_{\mathrm{nf}}$ manifests the conduction due to the heat of nanodispersion, $k_f$ manifests the conduction due to the heat of base dispersion, and $(c_P{)}_{\mathrm{nf}}$ manifests the energy capacity of the nanodispersion. $A_3$ and $A_4$ manifest the space and temperature-reliant heat source/sink representatives, respectively.

The auxiliary information are

$$u=U_w,v=V=0,T=T_w,\mathrm{at}y=0$$

(8)

$$\frac{\partial u}{\partial y}=\frac{\partial T}{\partial y}=0\mathrm{at}y=H,v=\frac{\mathrm{dH}}{\mathrm{dt}}\mathrm{at}y=H\left(t\right)$$

(9)

in which H(t) manifests the size of the dispersion.

Considering the experimental utilization, the graphene tiny particle saturation is very low. By taking advantage of Taylor’s series, the nanodispersion items are simplified as follows

$$\begin{array}{c}\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}=\frac{1}{(1-2.5\varphi )},\frac{{\rho }_{\mathrm{nf}}}{{\rho }_f}=1-\varphi +\varphi r,\hfill \\ \frac{{\left(\rho c_P\right)}_{\mathrm{nf}}}{{\left(\rho c_P\right)}_f}=1-\varphi +\varphi d\hfill \end{array}$$

(10)

$$\frac{k_{\mathrm{nf}}}{k_f}=1+\frac{3(k-1)\varphi }{k+2},\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}=\frac{3(\sigma -1)\varphi }{(\sigma +2)-(\sigma -1)\varphi }+1$$

(11)

here $r={\rho }_{\mathrm{sd}}/{\rho }_f$ , $d=(\rho c_P{)}_{\mathrm{sd}}/(\rho c_P{)}_f$ , $k=k_{\mathrm{sd}}/k_f$ , $\sigma ={\sigma }_{\mathrm{sd}}/{\sigma }_f$ , $\varphi$ manifests the volume fraction of the graphene tiny particles, ${\rho }_f$ and ${\rho }_{\mathrm{sd}}$ manifest the viscosity of the base dispersion and solid tiny particles, $k_f$ and $k_{\mathrm{sd}}$ manifest the heat conductions of the classical dispersion and solid tiny particles, ${\sigma }_f$ and ${\sigma }_{\mathrm{sd}}$ manifest the electrical conductivities of base dispersion and solid tiny particles, and $(c_P{)}_f$ and $(c_P{)}_{\mathrm{sd}}$ manifest the particular heat capacities of the base dispersion and solid tiny particles at fixed value of pressure, respectively. Note that the sign “sd” manifests the solid tiny particles. The heat relevant physical aspects of H₂O and graphene tiny particles are provided in Table 1.

Table 1.

Heat-reliant aspects of H₂O as well as graphene tiny particles.

Thermo-physical properties	Pure water	Graphene
$\rho$ (kg/m3)	997	2250
$c_P$ (J/(kg K))	4076	2100
k (W/(m K))	0.605	2500
$\sigma$ (Ω m−1)	0.005	1 × 107

The following substitutions simplify the functions f, $\theta$ , and similarity notation $\zeta$ as

$$\begin{array}{c}\psi (x,y,t)=x{\left(\begin{array}{c}\frac{c{\nu }_f}{1-\alpha t}\hfill \end{array}\right)}^{\frac{1}{2}}f\left(\zeta \right),\hfill \\ T(x,y,t)=T_0-T_{\mathrm{ref}}\left[\begin{array}{c}\frac{c{x}^2}{2{\nu }_f}\hfill \end{array}\right]{(1-\alpha t)}^{-1.5}\theta \left(\zeta \right),\hfill \\ \zeta ={\left[\begin{array}{c}\frac{c}{{\nu }_f(1-\alpha t)}\hfill \end{array}\right]}^{\frac{1}{2}}y=\frac{\beta }{H\left(t\right)}y\hfill \end{array}$$

(12)

note that ψ(x, y, t) manifests the stream function which provides $u=\partial \psi /\partial y$ and $v=-\partial \psi /\partial x$ . The quantity $\beta$ manifests the non-dimensional dispersion layer. Equation (12) correctly satisfies equation (5). Through picking up quantity from equation (12), equations (6)–(9) create the following equations (13)–(16)

$$\begin{array}{cc}\multicolumn{1}{c}{{\varphi }_{1}f″\prime }& +{\varphi }_2{\beta }^2\left[\begin{array}{c}\mathrm{ff}″-{(f\prime )}^2-S(f\prime +\frac{\zeta }{2}f″)\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{}& +{\varphi }_1\gamma \left[\begin{array}{c}2\mathrm{ff}″+S(2f″\prime +\frac{\zeta }{2}{f}^{″″})-{(f″)}^2-f{f}^{″″}\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{}& -{\varphi }_3\mathrm{Mf}\prime =0\hfill \end{array}$$

(13)

$$\begin{array}{cc}\multicolumn{1}{c}{{\varphi }_4\theta ″}& -{\varphi }_5\mathrm{Pr}{\beta }^2(2f\prime \theta -f\theta \prime +\frac{3}{2}S\theta +\frac{1}{2}S\zeta \theta \prime )\hfill \\ \multicolumn{1}{c}{}& +A_{3}f\prime +A_4\theta =0\hfill \end{array}$$

(14)

$$f=\theta =0,f\prime =1\mathrm{at}\zeta =0$$

(15)

$$f=\frac{\beta S}{2},\theta \prime =f″=0\mathrm{at}\zeta =\beta$$

(16)

here the prime (′) manifests the differentiation concept in terms of $\zeta$ , ${\varphi }_i$ (i = 1–5) manifest the concentration quantities provided as ${\varphi }_1={\mu }_{\mathrm{nf}}/{\mu }_f$ , ${\varphi }_2={\rho }_{\mathrm{nf}}/{\rho }_f$ , ${\varphi }_3={\sigma }_{\mathrm{nf}}/{\sigma }_f$ , ${\varphi }_4=k_{\mathrm{nf}}/k_f$ , and ${\varphi }_5=(\rho c_P{)}_{\mathrm{nf}}/(\rho c_P{)}_f$ . $\gamma =c{\alpha }_1/\mu (1-\alpha t)$ is the non-dimensional second-grade fluid parameter, $S=\alpha /c$ manifests the unsteadiness representative, $M={\sigma }_f{B}_0^2/c{\rho }_f$ manifests the magnetic field representative, and $\mathrm{Pr}=(\mu c_P{)}_f/k_f$ manifests the Prandtl number.

Transformed equations solution through HAM

The scheme (HAM) is utilized to evaluate the transformed equations. The initial guesses of evaluation and auxiliary linear operators are furnished as

$$f_0\left(\zeta \right)=\zeta ,{\theta }_0\left(\zeta \right)=1$$

(17)

$${\mathit{L}}_f=f″\prime ,{\mathit{L}}_{\theta }=\theta ″$$

(18)

The following aspects are satisfied with the linear operators

$${\mathit{L}}_f\left[\begin{array}{c}{C}_1+C_2\zeta +C_3{\zeta }^2\hfill \end{array}\right]=0,{\mathit{L}}_{\theta }\left[\begin{array}{c}{C}_4+C_5\zeta \hfill \end{array}\right]=0$$

(19)

where $C_i(i=1-5)$ are the arbitrary constants.

Deformation equations of zeroth order

Inserting the nonlinear operator $\aleph$ as

$$\begin{array}{cc}\multicolumn{1}{c}{{\aleph }_f\left[f(\zeta ,p)\right]=}& {\varphi }_1\frac{{\partial }^{3}f(\zeta ,p)}{\partial {\zeta }^3}+{\varphi }_2{\beta }^2\left[\begin{array}{c}f(\zeta ,p)\frac{{\partial }^{2}f(\zeta ,p)}{\partial {\zeta }^2}-{\left(\begin{array}{c}\frac{\partial f(\zeta ,p)}{\partial \zeta }\hfill \end{array}\right)}^2-S\left(\begin{array}{c}\frac{\partial f(\zeta ,p)}{\partial \zeta }+\frac{\zeta }{2}\frac{{\partial }^{2}f(\zeta ,p)}{\partial {\zeta }^2}\hfill \end{array}\right)\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{}& +{\varphi }_1\gamma \left[\begin{array}{c}2f(\zeta ,p)\frac{{\partial }^{2}f(\zeta ,p)}{\partial {\zeta }^2}+S\left(\begin{array}{c}2\frac{{\partial }^{3}f(\zeta ,p)}{\partial {\zeta }^3}+\frac{\zeta }{2}\frac{{\partial }^{4}f(\zeta ,p)}{\partial {\zeta }^4}\hfill \end{array}\right)-{\left(\begin{array}{c}\frac{{\partial }^{2}f(\zeta ,p)}{\partial {\zeta }^2}\hfill \end{array}\right)}^2-f(\zeta ,p)\frac{{\partial }^{4}f(\zeta ,p)}{\partial {\zeta }^4}\hfill \end{array}\right]-{\varphi }_{3}M\frac{\partial f(\zeta ,p)}{\partial \zeta }\hfill \end{array}$$

(20)

$$\begin{array}{cc}\multicolumn{1}{c}{{\aleph }_{\theta }[f(\zeta ,p),\theta (\zeta ,p)]=}& {\varphi }_4\frac{{\partial }^2\theta (\zeta ,p)}{\partial {\zeta }^2}+{\varphi }_5\mathrm{Pr}{\left(\beta \right)}^2\left(\begin{array}{c}2\frac{\partial f(\zeta ,p)}{\partial \zeta }\theta (\zeta ,p)-f(\zeta ,p)\frac{\partial \theta (\zeta ,p)}{\partial \zeta }+\frac{3}{2}S\theta (\zeta ,p)+\frac{1}{2}S\zeta \frac{\partial \theta (\zeta ,p)}{\partial \zeta }\hfill \end{array}\right)\hfill \\ \multicolumn{1}{c}{}& +A_3\frac{\partial f(\zeta ,p)}{\partial \zeta }+A_4\theta (\zeta ,p)\hfill \end{array}$$

(21)

where p shows an embedding representative in a manner p ∈ [0, 1].

Considering equations of the zeroth-order deformation, the following is obtained

$$(1-p){\mathit{L}}_f[f(\zeta ,p)-f_0\left(\zeta \right)]=p\hslash {\aleph }_f\left[f(\zeta ,p)\right]$$

(22)

$$(1-p){\mathit{L}}_{\theta }[\theta (\zeta ,p)-{\theta }_0\left(\zeta \right)]=p\hslash {\aleph }_{\theta }[f(\zeta ,p),\theta (\zeta ,p)]$$

(23)

where $\hslash$ denotes the helping non-zero representative.

Equation (22) holds the auxiliary information

$$f(0,p)=0,f\prime (0,p)=1,f″(\beta ,p)=0,f(\beta ,p)=\frac{\beta S}{2}$$

(24)

Equation (23) holds the auxiliary information

$$\theta (0,p)=1,\theta \prime (\beta ,p)=0$$

(25)

For p = 0 and p = 1, the following is obtained

$$p=0\Rightarrow f(\zeta ,0)=f_0\left(\zeta \right)\mathrm{and}p=1\Rightarrow f(\zeta ,1)=f\left(\zeta \right)$$

(26)

$$p=0\Rightarrow \theta (\zeta ,0)={\theta }_0\left(\zeta \right)\mathrm{and}p=1\Rightarrow \theta (\zeta ,1)=\theta \left(\zeta \right)$$

(27)

$f(\zeta ,p)$ shifts $f_0(\zeta )$ to $f(\zeta )$ for p keeping the values through 0 to 1. Also, $\theta (\zeta ,p)$ shifts ${\theta }_0(\zeta )$ to $\theta (\zeta )$ with the quantities of p changing from 0 to 1. Following the Taylor series expansion for equations (22) and (23), it takes the form

$$f(\zeta ,p)=f_0\left(\zeta \right)+\sum _{m=1}^{\infty }{f}_m\left(\zeta \right)p^m,f_m\left(\zeta \right)=\frac{1}{m!}\frac{{\partial }^{m}f(\zeta ,p)}{\partial p^m}{|}_{p=0}$$

(28)

$$\theta (\zeta ,p)={\theta }_0\left(\zeta \right)+\sum _{m=1}^{\infty }{\theta }_m\left(\zeta \right)p^m,{\theta }_m\left(\zeta \right)=\frac{1}{m!}\frac{{\partial }^m\theta (\zeta ,p)}{\partial p^m}{|}_{p=0}$$

(29)

The coincidence for the series in equations (28) and (29) is strongly dependent on $\hslash$ . It is necessary to employ $\hslash$ with a quantity when the series in equations (28) and (29) converges at p = 1; hence, equations (28) and (29) take the form

$$f\left(\zeta \right)=f_0\left(\zeta \right)+\sum _{m=1}^{\infty }{f}_m\left(\zeta \right)$$

(30)

$$\theta \left(\zeta \right)={\theta }_0\left(\zeta \right)+\sum _{m=1}^{\infty }{\theta }_m\left(\zeta \right)$$

(31)

Problems of deformation keeping mth order

Lines of deformations of mth order

Employing differentiation up to m times regarding p of equations (22) and (24), performing division through m!, and putting p = 0 develop the furnished lines

$${\mathit{L}}_f[f_m\left(\zeta \right)-{\chi }_m{f}_{m-1}\left(\zeta \right)]=\hslash {\Re }_m^f\left(\zeta \right)$$

(32)

$$f_m\left(0\right)=f_m\left(\beta \right)=f{\prime }_m\left(0\right)={f″}_m\left(\beta \right)=0$$

(33)

$$\begin{array}{cc}\multicolumn{1}{c}{{\Re }_m^f\left(\zeta \right)=}& {\varphi }_{1}f″{\prime }_{m-1}+{\varphi }_2{\beta }^2\left[\begin{array}{c}\sum _{k=o}^{m-1}{f}_{m-1-k}{f″}_k-\sum _{k=o}^{m-1}f{\prime }_{m-1-k}f{\prime }_k-S\left(\begin{array}{c}f{\prime }_{m-1}+\frac{\zeta }{2}{f″}_{m-1}\hfill \end{array}\right)\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{}& +{\varphi }_1\gamma \left[\begin{array}{c}2\sum _{k=o}^{m-1}{f}_{m-1-k}{f″}_k+S\left(\begin{array}{c}2f″{\prime }_{m-1}+\frac{\zeta }{2}{f}_{m-1}^{″″}\hfill \end{array}\right)-\sum _{k=o}^{m-1}{f″}_{m-1-k}{f″}_k-\sum _{k=o}^{m-1}{f}_{m-1-k}{f}_k^{″″}\hfill \end{array}\right]-{\varphi }_{3}Mf{\prime }_{m-1}\hfill \end{array}$$

(34)

Employing differentiation up to m times with respect to p of equations (23) and (25), performing division through m! and putting p = 0 develop the furnished lines

$${\mathit{L}}_{\theta }[{\theta }_m\left(\zeta \right)-{\chi }_m{\theta }_{m-1}\left(\zeta \right)]=\hslash {\Re }_m^{\theta }\left(\zeta \right)$$

(35)

$${\theta }_m\left(0\right)=\theta {\prime }_m\left(\beta \right)=0$$

(36)

$$\begin{array}{c}{\Re }_m^{\theta }\left(\zeta \right)={\varphi }_4{\theta ″}_{m-1}-{\varphi }_5\mathrm{Pr}{\beta }^2\hfill \\ \left[\begin{array}{c}2\sum _{k=o}^{m-1}f{\prime }_{m-1-k}\theta {\prime }_k-\sum _{k=o}^{m-1}{f}_{m-1-k}\theta {\prime }_k+\frac{3}{2}S{\theta }_{m-1}+\frac{1}{2}S\zeta \theta {\prime }_{m-1}\hfill \end{array}\right]\hfill \\ +A_{3}f{\prime }_{m-1}+A_4{\theta }_{m-1}\hfill \end{array}$$

(37)

$${\chi }_m=\{\begin{array}{cc}0,\hfill & m\le 1\hfill \\ \multicolumn{1}{c}{1,}& m>1\hfill \end{array}$$

(38)

If $f_m^{*}(\zeta )$ , ${\theta }_m^{*}(\zeta )$ are the particular evaluations, then the general evaluations of equations (32) and (35) are

$$f_m\left(\zeta \right)=f_m^{*}\left(\zeta \right)+C_1+C_2\zeta +C_3{\zeta }^2$$

(39)

$${\theta }_m\left(\zeta \right)={\theta }_m^{*}\left(\zeta \right)+C_4+C_5\zeta$$

(40)

Results and discussion

The simplified equations (13)–(16) due to similarity transformations are treated through symbolic evaluating software MATHEMATICA following HAM code. The activities of different profiles such as velocity $f(\zeta )$ and temperature $\theta (\zeta )$ are reliant on the potentialities of all the dominating representatives. Figures 4–10 and 11–22 are specified for motion and energy, respectively. The issues are outlined in Figure 1. Liao ¹⁶ composed $\hslash$ curves to resume the coincidence of the sequential evaluations of the issue; consequently, the allowed $\hslash$ curves for $f(\zeta )$ and $\theta (\zeta )$ exist to receive the intervals $-6.0\le \hslash \le -0.0$ and $-3.5\le \hslash \le -0.0$ in Figures 2 and 3, respectively.

Figure 1.

Geometry of the physical model.

Figure 2.

$\hslash$ region of $f(\zeta )$ .

Figure 3.

$\hslash$ region of $\theta (\zeta )$ .

Velocity profile

Movement $f(\zeta )$ of the liquid finite–confined domain second-grade nanofluid reduces due to the existence of graphene nanoparticles, as shown in Figure 4. It is established that the values of finite-confined domain representative $\beta$ increase in the presence of graphene nanoparticles; the layer and mass of the base liquid are enhanced which decrease the velocity. Figure 5 reveals the effect of the second-grade nanoliquid film parameter $\gamma$ at the simplified movement profile $f(\zeta )$ . The motion of the second-grade liquid film increases consisting of the graphene nanoparticles. A schematic representation of the potentiality of unsteadiness representative S on the simplified movement profile $f(\zeta )$ is portrayed in Figure 6. It shows that movement gets speed with the rise of S. The simplified movement representative $f(\zeta )$ becomes stable with metallic graphene nanoparticles with the passage of time.

Figure 4.

Movement curves for $\hslash =-3.00$ , ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₃ = 0.80, S = 0.70, M = 1.00, γ = 0.60 while some assignments to $\beta$ .

Figure 5.

Movement curves for $\hslash =-3.00$ , ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₃ = 0.80, S = 0.70, M = 1.00, β = 0.50 while some assignments to $\gamma$ .

Figure 6.

Movement curves for $\hslash =-2.00$ , ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₃ = 0.80, γ = 0.50, M = 1.00, β = 0.50 while some assignments to S.

Magnet environment regulates the graphene tiny particle flow. During the present study, the graphene nanoparticles flow is enhanced on the magnet environment amplification from 0.70 to 1.00, as shown in Figure 7. At low magnetic field strength, there is a formation of small dipolar chains. For the peak magnitude of magnet environment, the continuity extent is high, forming in evenly distanced one tiny particle continuity throughout the tiny particle dispersion volume. When the magnet environment is intensified, there observed the zipping of the two-pole continuities in microscopic pictures.

Figure 7.

Movement curves for $\hslash =-2.00$ , ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₃ = 0.80, γ = 0.50, S = 0.90, β = 0.50 while some assignments to M.

The addition of graphene nanoparticles brings change in flow, as shown in Figure 8. It is evident that with the graphene tiny particle saturation ${\varphi }_1$ increment, the movement $f(\zeta )$ of the non-Newtonian second-grade nanofluid elevates. It can be visualized very clearly that at the time saturation, ${\varphi }_1$ upsurges from 0.5 to 0.8, and the size of moment of particular motion part increases. In graphene nanoliquid, graphene nanoparticles perform abnormal heated movement on behalf of the Brownian movement and the strength of the graphene tiny particle movement is reliant to the body shape and thickness. The tiny particles speed up very high while the movement speed of big-sized particles is very low. Figure 9 shows the effect of graphene nanoparticles concentration ${\varphi }_2$ due to the density of graphene nanoliquid. With the inclusion of graphene nanoparticles, ${\varphi }_2$ increases which makes hurdle to the fluid flow. The dispersion behavior of water-based graphene nanoparticles in contrast to concentration ${\varphi }_3$ is shown in Figure 10. It portrays that at the time the scalar quantities of ${\varphi }_3$ magnify, the velocity $f(\zeta )$ also enhances. The volume or mass of the sedimentation expresses the consistency of the graphene nanoparticle nanosuspension. Graphene nanoparticle nanofluids are measured to be consistent at the time; the graphene nanoparticle saturation or particle dimensions of the surfactant particles are not changeable with the passage of time.

Figure 8.

Movement curves for $\hslash =-2.00$ , M = 1.00, ϕ₂ = 0.60, ϕ₃ = 0.80, γ = 0.50, S = 0.90, β = 0.50 while some assignments to ${\varphi }_1$ .

Figure 9.

Movement curves for $\hslash =-2.00$ , M = 1.00, ϕ₁ = 0.50, ϕ₃ = 0.80, γ = 0.50, S = 0.90, β = 0.50 while some assignments to ${\varphi }_2$ .

Figure 10.

Movement curves for $\hslash =-2.00$ , M = 1.00, ϕ₁ = 0.50, ϕ₂ = 0.60, γ = 0.50, S = 0.90, β = 0.50 while some assignments to ${\varphi }_3$ .

Temperature profile

Due to crystal form of matter, energy is taken through the expansion of lattice movements, that is, phonons, which are irregularly generated, expanded in irregular sides, and dispersed from one another. The potentiality of a finite-confined domain representative $\beta$ in Figure 11 shows that temperature decreases with increasing the thin film parameter because heat absorbs in the thick layer of the second-grade fluid containing graphene nanoparticles.

Figure 11.

Energy curves for $\hslash =-3.00$ , ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₃ = 0.80, ϕ₄ = 0.90, ϕ₅ = 0.60, S = 0.70, M = 1.00, γ = 0.60, Pr = 10.00, A₃ = 0.10, A₄ = 0.20 while some assignments to $\beta$ .

Figure 12 visualizes that heat energy $\theta (\zeta )$ achieves minimum position with various states of the second-grade thin liquid film parameter $\gamma$ . Graphene tiny particles keep big heat-related conduction because heat diffuses in graphene nanoparticles with a large diameter. Also, graphene nanoparticles have small interparticle distance in nanofluid so heat reaches sharply in nearby particles, resulting in a large thermal conductivity. All these things provide a diffusion of heat in the second-grade graphene nanofluid. It happens in Figure 13 that the energy $\theta (\zeta )$ decreases due to the time-reliant representative S. As time passes, heat exhausts in the fluid and environment. The effect of magnet field representative M on energy $\theta (\zeta )$ profile is portrayed in Figure 14. Magnet environment of intensity B₀ is exerted perpendicularly versus the flow which resists and controls the flow. In the present study, the fluid is the second grade and the existence of graphene nanoparticles makes high the temperature $\theta (\zeta )$ in the existence of magnet environment.

Figure 12.

Energy curves for $\hslash =-1.10$ , ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₃ = 0.80, ϕ₄ = 0.90, ϕ₅ = 0.60, S = 0.70, M = 1.00, β = 0.50, Pr = 10.00, A₃ = 0.10, A₄ = 0.20 while some assignments to $\gamma$ .

Figure 13.

Energy curves for $\hslash =-1.10$ , ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₃ = 0.80, ϕ₄ = 0.90, ϕ₅ = 0.60, γ = 0.60, M = 1.00, β = 0.50, Pr = 10.00, A₃ = 0.10, A₄ = 0.20 while some assignments to S.

Figure 14.

Energy curves for $\hslash =-1.10$ , ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₃ = 0.80, ϕ₄ = 0.90, ϕ₅ = 0.60, γ = 0.60, S = 0.70, β = 0.50, Pr = 10.00, A₃ = 0.10, A₄ = 0.20 while some assignments to M.

In Figure 15, the energy $\theta (\zeta )$ decreases at the rising scalar magnitude of ${\varphi }_1$ . The effect of graphene nanofluid constant ${\varphi }_2$ on temperature $\theta (\zeta )$ is shown in Figure 16 where it is clear that energy $\theta (\zeta )$ shows increasing behavior for the values of 0.60–3.60 of ${\varphi }_2$ . On behalf of the zigzag movement, the graphene nanoparticles are sufficient near and so elevate the heating, flowing within the graphene nanoparticles. The graphene nanoparticles are not necessary to exist in physical combination, but at particular space, enabling quick transposition of movement among graphene nanoparticles. On behalf of large surface capacity-to-volume ratio, the graphene tiny particles exist in exothermal reaction. Furthermore, the inclusion of chemical reacting activity also elevates the heat passage. The clustering of graphene nanoparticles is responsible for the minimum thermal resistivity and enhances the heat conduction prominently. With the passage of time, big clusters of graphene tiny particles precipitate in the base dispersion water which make the nonchemical space of graphene nanoparticles too little than the dominant space of the clusters. Conduction is very rapid in these clusters and so the thermal conductivity increases effectively. Figure 17 depicts the effect of graphene nanofluid constant ${\varphi }_3$ against energy $\theta (\zeta )$ . It is evident that temperature depreciates due to the electrical conductivity of nanofluid. Similarly, there is a decreasing behavior in Figure 18 for temperature in contrast with ${\varphi }_4$ . It is seen that when ${\varphi }_4$ increases from 0.90 to 1.20, the temperature decreases due to the involvement of thermal conductivities of nanofluid and base fluid. Figure 19 shows that energy $\theta (\zeta )$ mitigates versus the increasing positive values of nanofluid constant ${\varphi }_5$ . It is because of the particular heat capacitances of the classic dispersion and solid graphene tiny particles on the fixed force per unit area.

Figure 15.

Energy curves for $\hslash =-1.10$ , M = 1.00, ϕ₂ = 0.60, ϕ₃ = 0.80, ϕ₄ = 0.90, ϕ₅ = 0.60, γ = 0.60, S = 0.70, β = 0.50, Pr = 10.00, A₃ = 0.10, A₄ = 0.20 while some assignments to ${\varphi }_1$ .

Figure 16.

Energy curves for $\hslash =-1.10$ , M = 1.00, ϕ₁ = 0.50, ϕ₃ = 0.80, ϕ₄ = 0.90, ϕ₅ = 0.60, γ = 0.60, S = 0.70, β = 0.50, Pr = 10.00, A₃ = 0.10, A₄ = 0.20 while some assignments to ${\varphi }_2$ .

Figure 17.

Energy curves for $\hslash =-1.10$ , M = 1.00, ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₄ = 0.90, ϕ₅ = 0.60, γ = 0.60, S = 0.70, β = 0.50, Pr = 10.00, A₃ = 0.10, A₄ = 0.20 while some assignments to ${\varphi }_3$ .

Figure 18.

Energy curves for $\hslash =-1.10$ , M = 1.00, ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₃ = 0.80, ϕ₅ = 0.60, γ = 0.60, S = 0.70, β = 0.50, Pr = 10.00, A₃ = 0.10, A₄ = 0.20 while some assignments to ${\varphi }_4$ .

Figure 19.

Energy curves for $\hslash =-1.10$ , M = 1.00, ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₃ = 0.80, ϕ₄ = 0.90, γ = 0.60, S = 0.70, β = 0.50, Pr = 10.00, A₃ = 0.10, A₄ = 0.20 while some assignments to ${\varphi }_5$ .

Figure 20 focuses on the space-reliant energy yield representative A₃. Perceiving that the energy $\theta (\zeta )$ resumes devaluation in the nonnegative states of A₃. The reality happens that the developed positions of A₃ keep the attitude similar to energy sink and consequently drops down the heat relevant motion part. The potentiality of heat-reliant energy consumption representative A₄ is shown in Figure 21. Due to curves, it is portrayed that the energy $\theta (\zeta )$ keeps peek position against the peak magnitude of A₄. Graphene tiny particles keep maximum heat conduction because heat diffuses in graphene nanoparticles with a large diameter. Also, graphene nanoparticles have small interparticle distance in nanofluid so heat reaches sharply nearby particle, resulting in a large thermal conductivity. All these things provide a diffusion of heat in the second-grade graphene nanofluid. The potential of Prandtl number Pr with respect to energy $\theta (\zeta )$ is evident from Figure 22. Energy decreases with the increasing status of Prandtl number Pr from which it is concluded that Prandtl number Pr has significant potentialities in establishing a pleasant environment.

Figure 20.

Energy curves for $\hslash =-1.10$ , M = 1.00, ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₃ = 0.70, ϕ₄ = 0.80, ϕ₅ = 0.90, γ = 0.60, S = 0.70, β = 0.50, Pr = 10.00, ϕ₆ = 0.60, A₄ = 0.20 while some assignments to A₃.

Figure 21.

Energy curves for $\hslash =-1.10$ , M = 1.00, ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₃ = 0.70, ϕ₄ = 0.80, ϕ₅ = 0.90, γ = 0.60, S = 0.70, β = 0.50, Pr = 10.00, ϕ₆ = 0.60, A₃ = 0.10 while some assignments to A₄.

Figure 22.

Energy curves for $\hslash =-1.10$ , M = 1.00, ϕ₁ = 0.50, ϕ₂ = 0.60, ϕ₃ = 0.70, ϕ₄ = 0.80, ϕ₅ = 0.90, γ = 0.60, S = 0.70, β = 0.50, A₄ = 0.20, ϕ₆ = 0.60, A₃ = 0.10 while some assignments to Pr.

Conclusion

The aforementioned lines attribute to series evaluation of time-reliant movement and energy transmission aspects of electrical conduction nature H₂O residing the non-Newtonian second-grade nanofluid dispersion film mixed with graphene tiny particles on an expanding space in the occurrence of magnet exerted at the right angle and dissimilar energy yield/consumption. The evaluation of the issue performed through an efficient procedure (HAM) for the movement and energy profiles. The potentialities of each and every representative for movement and energy furnish quite real output to the movements of the second-grade nanofluid.

Supplemental Material

Scientific_Justification – Supplemental material for Boundary layer flow and heat transfer in a thin-film second-grade nanoliquid embedded with graphene nanoparticles

Supplemental material, Scientific_Justification for Boundary layer flow and heat transfer in a thin-film second-grade nanoliquid embedded with graphene nanoparticles by Noor Saeed Khan and Samina Zuhra in Advances in Mechanical Engineering