Rotating flow of nanofluid due to exponentially stretching surface: An optimal study

Abstract

In this article, the presented study is based on a modification in Gegenbauer wavelets method. The modeled problem is presented to analyze the phenomena of transfer of heat of rotating nanofluids in which the flow is produced by an exponentially stretching sheet. The purpose of this study is to examine the simultaneous effects of rotation of nanofluid and exponentially stretching on the shear stresses and heat transfer rate, cooling proficiency of water-based nanofluids containing Ag, Cu, Al₂O₃, TiO₂, and CuO nanoparticles, and modification in Gegenbauer wavelets method to obtain the numerical solution of the said problem. A comparative analysis is presented among the outcomes obtained by modified Gegenbauer wavelets method, Runge–Kutta method of order-4, and already existing methods. The comparison shows that this modification is extremely efficient, and proposed technique could be extended for other physical problems.

Introduction

Motivation in the rotating flows originates from different applications in engineering and geophysics, for example, rotating machinery, anti-cyclonic flow circulations, magma flow in earth’s mantle, centrifugal filtration process, geological stretching of tectonic plate beneath rotating ocean, food and chemical processing, rotor-stator system, viscometry, and dynamics of hurricanes and tornadoes. Wang ¹ was the pioneer, who explored flow adjacent to a stretching surface immersed in rotating fluid. Such flow problems contain a parameter $k$ that is the ratio between stretching rate and rotation rate of the sheet. He used a perturbation approach to determine the velocity distributions for the small values of parameter $k$ . Takhar and Nath ² made numerical analysis for Magneto hydro dynamics (MHD) rotational flow over an unsteady stretching sheet. Nazar et al. ³ studied an induced unsteady flow due to a stretching surface in a rotating fluid. They used Keller-box scheme to analyze the solution of said problem. The unsteadiness in the flow is due to the sudden stretchiness of the surface. Asymptotic solutions for higher values of $k$ resulted and found in a qualitative agreement with the numerical results. Kumari et al. ⁴ analyzed the rotating flow of power-law fluids over a stretching surface. They found that power-law index n is non-monotonic for variation in the velocity distributions. Javed et al. ⁵ provided the local similarity solutions for rotating viscous flow over an exponentially stretching flat surface. Zaimi et al. ⁶ numerically analyzed the boundary layer flow over a stretching surface in a rotating viscoelastic fluid and solved the proposed model by means of the Keller-box method. They concluded that both the velocity and skin friction coefficients in the direction of $x$ increased with an increase in the rotation and viscoelastic parameters. Turkyilmazoglu ⁷ studied the conventional Bödewadt flow when the disk is exposed to uniform stretched in the radial direction. He adopted spectral Chebyshev collocation technique to obtain the numerical results. Khan et al. ⁸ studied the MHD squeezing flow between two infinite plates. Transfer of heat in the rotating flow via the Cattaneo–Christov heat flux theory that defines the features of imperative thermal relaxation time was examined by Mustafa. ⁹ He provided an HAM-based solution and described that both analytical and numerical results are well-matched. Rotating flow over an exponentially shrinking sheet with suction is investigated by Rosali et al. ¹⁰ Stefani and Kirillov ¹¹ studied the disruption of rotating flows besides positive shear by azimuthal magnetic fields. Patel models of thermal conductivity and Hamilton Crosser for rotating flow induced by exponential variable velocity of the surface were presented by Ahmad and Mustafa. ¹² They simulated the problem for five various types of water-based nanofluids.

Recently, the developments in nanotechnology have opened new areas to study the phenomena of heat transfer in the nanofluid. The nanoparticles which are distributed in the base fluid normally selected from oxides and metals that increase the convection and conduction coefficients and hence increase the transfer of heat rate for coolants. Nanofluids have special application in the transfer of heat for example coldness of nuclear reactor, heating of solar water, hybrid-powered engines, domestic chillers/refrigerators, grinding, cooling the electronic systems, heat exchangers, storage of thermal energy, drag reduction, and many other. The prospective applications of nanofluids can be found in the following works.^13–17 Nanofluids containing the iron-based nanoparticles (also stated as ferro-fluids) are more beneficial due to their thermophysical characteristics and can be modified by the magnetic field. Moreover, ferro-fluids also perform in some biomedical situations like in contrast agent enhancement in magnetic resonance imaging, the removal of cancer tumors, bleeding reduction during surgeries, and heat conduction in tissues. First, Buongiorno ¹⁸ gave an idea about two-component nanofluid transport model accounting for the mutual effects of thermophoresis and Brownian diffusion. Later, single-phase model by conveying nanofluid characteristics as functions of the base fluids properties and its constituents are presented by Tiwari and Das. ¹⁹ The research community has constantly used these transport models for the past several years. Nield and Kuznetsov ²⁰ used Buongiorno’s model to address the Cheng–Minkowycz problem for natural convection in nanofluid filling a porous space. Using Buongiorno’s model, Mohyud-Din et al. ²¹ studied the heat and mass transfer of nanofluids arising in biosciences. Recently, Mushtaq et al. ²² numerically investigated the rotating nanofluids flow caused by an exponentially stretching sheet. Some recent and qualitative study in the domain of nanofluids has been cited by Sheikholeslami et al.^23,24 and Usman et al.^25,26

The theoretical investigation by using mathematical techniques has a significant role. Previously, many techniques developed or extended to find the better solution of above-discussed physical problems and successfully applied to many problems.^27–33 On the other hand, these methods involve some deficiencies to obtain the better solution for these problems. Literature survey witnesses that the wavelets theory and related methods are very rare to find the solution of physical problems especially when problem contains infinite boundary conditions. In the current work, we introduced a modification in Gegenbauer wavelets method. Previously, the Gegenbauer wavelets method is used by various authors.^32–35 Here, we analyze the boundary layer flow of water-based nanofluid driven by exponentially stretching plate via Gegenbauer wavelets coupled with the method of moments. The flow problem composed of five various kinds of nanoparticles including Ag, Cu, Al₂O₃, TiO₂, and CuO. Numerical solutions obtained via modified Gegenbauer wavelets method (MGWM). A comparison between results via MGWM, already published^5,22 and RK (order-4) is presented. Observation shows that these results are in good agreement and compatible. Physical changes also deliberated via graphical plots. Moreover, error and convergence analysis has been presented to show the efficiency of MGWM. This proposed method could be extended for other nonlinear problems.

Mathematical and geometrical analysis

Let us explore the effects of heat transfer process of nanofluid over an exponentially stretching surface along the x-axis, prevailing in the plane where $z=0$ and the fluid rotate with an angular velocity $\text{Ω}={\text{Ω}}_0\exp -x/L$ along the z-axis in which L is the characteristic length and ${\text{Ω}}_0$ is an average angular velocity. Let $T_{\infty }$ indicates the fluid temperature at the ambient and let $T_w$ is the constant temperature at the surface. The geometry of the problem is provided in Figure 1. The flow includes five various types of nanoparticles: i.e. Ag, Cu, Al₂O₃, TiO₂, and CuO. Assuming the Tiwari and Das ¹⁹ nanofluid transport model and after using traditional boundary layer approximations, the conservation equations can be stated as

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

(1)

$${\rho }_{nf}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}-2\text{Ω}v\right)={\mu }_{nf}\frac{{\partial }^{2}u}{\partial z^2}$$

(2)

$${\rho }_{nf}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}-2\text{Ω}u\right)={\mu }_{nf}\frac{{\partial }^{2}v}{\partial z^2}$$

(3)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}={\alpha }_{nf}\frac{{\partial }^{2}T}{\partial z^2}$$

(4)

Figure 1.

Flow analysis and diagram of the problem.

Associated with BCs

$$\text{ for }z=0,u=U_w\left(x\right)=U\exp \frac{x}{L},v=w=0,T=T_w$$

(5)

$$\text{as}z\to \infty ,u,v\to 0,T\to T_{\infty }$$

(6)

Brinkman ³⁶ proposed the effective dynamic viscosity ${\mu }_{nf}$ for nanofluids. Moreover, the effective heat capacitance ${\left(\rho C_p\right)}_{nf}$ and effective density ${\rho }_{nf}$ are stated in equation (8) ³⁷

$$\begin{array}{l}{\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\varphi \right)}^{5/2}},\\ {\left(\rho C_p\right)}_{nf}=\left(1-\varphi \right){\left(\rho C_p\right)}_f+\varphi {\left(\rho C_p\right)}_s,\\ {\rho }_{nf}=\left(1-\varphi \right){\rho }_f+\varphi {\rho }_s\end{array}$$

Maxwell model expected the effective thermal conductivity $k_{nf}$ of nanofluid stated under Khanafer et al., ³⁷ in which $\varphi$ indicates the nanoparticle volume fraction and subscripts $s$ and $f$ show solid and fluid phase, respectively

$$\frac{k_{nf}}{k_f}=\frac{\left(k_s+2k_f\right)-2\phi \left(k_f-k_s\right)}{\left(k_s+2k_f\right)+\phi \left(k_f-k_s\right)}$$

(7)

The thermophysical properties of water and considered nanoparticles are stated in Table 1. The following non-dimensionless variables are used ⁵

$$\begin{array}{l}\eta =z\sqrt{U/2{\nu }_{f}L}\exp \frac{x}{2L},u=U\exp \frac{x}{L}{f}^{\prime }\left(\eta \right),\\ w=-\sqrt{\frac{{\nu }_{f}U}{2L}}\exp \frac{x}{2L}\left(f+\eta f^{\prime }\left(\eta \right)\right),\\ \theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},v=U\exp \frac{x}{L}g\left(\eta \right)\end{array}$$

Table 1.

Thermophysical properties of base fluid and nanoparticles.^5,22

Thermophysical properties	$k \left(\frac{\text{W}}{\text{mK}}\right)$	$\rho \left(\frac{\text{kg}}{{\text{m}}^3}\right)$	$c_{p }\left(\frac{\text{J}}{\text{kg\hspace{0.17em}K}}\right)$
Water	0.613	997.1	4179
Ag	429	10,500	235
Cu	400	8954	383
CuO	76.5	6320	531.5
Al2O3	40	3970	765
TiO2	8.9538	4250	686.2

Equation (1) is identically satisfied while equations (2) to (6) can be expressed as given below

$$\begin{array}{l}\frac{1}{{\left(1-\varphi \right)}^{\frac{5}{2}}\left(1-\varphi +\varphi \frac{{\rho }_s}{{\rho }_f}\right)}\frac{{\text{d}}^3}{\text{d}{\eta }^3}f\left(\eta \right)-2{\left(\frac{\text{d}}{\text{d}\eta }f\left(\eta \right)\right)}^2\\ +f\left(\eta \right)\frac{{\text{d}}^2}{\text{d}{\eta }^2}f\left(\eta \right)+4\lambda g\left(\eta \right)=0\end{array}$$

(8)

$$\begin{array}{l}\frac{1}{{\left(1-\varphi \right)}^{\frac{5}{2}}\left(1-\varphi +\varphi \frac{{\rho }_s}{{\rho }_f}\right)}\frac{{\text{d}}^2}{\text{d}{\eta }^2}g\left(\eta \right)+f\left(\eta \right)\frac{\text{d}}{\text{d}\eta }g\left(\eta \right)\\ -2\frac{\text{d}}{\text{d}\eta }f\left(\eta \right)g\left(\eta \right)-4\lambda \frac{\text{d}}{\text{d}\eta }f(\eta )=0\end{array}$$

(9)

$$\frac{\frac{k_{nf}}{k_f}}{\left(1-\varphi +\varphi \frac{{\left(\rho C_p\right)}_s}{{\left(\rho C_p\right)}_f}\right)}\frac{{\text{d}}^2}{\text{d}{\eta }^2}\theta \left(\eta \right)+\text{Pr}f\left(\eta \right)\frac{\text{d}}{\text{d}\eta }\theta \left(\eta \right)=0$$

(10)

$$\text{ for }\eta =0,f\left(\eta \right)=g\left(\eta \right)=0,\frac{\text{d}}{\text{d}\eta }f\left(\eta \right)=\theta \left(\eta \right)=1$$

(11)

$$\text{as}\eta \to \infty ,\frac{\text{d}}{\text{d}\eta }f\left(\eta \right)=g\left(\eta \right)=\theta \left(\eta \right)=0$$

(12)

In the above expression, $\text{Pr}=\frac{{\nu }_f}{{\alpha }_f},\lambda =\frac{{\text{Ω}}_{0}L}{U}$ indicates Water’s Prandtl number $(\text{Pr}=6.2)$ and the rotation parameter, respectively. The local Nusselt number $N{u}_x$ and skin friction coefficient are defined as follows

$$C_{fx}=\frac{{\tau }_{wx}}{{\rho }_f{U}_w^2},C_{fy}=\frac{{\tau }_{wy}}{{\rho }_f{U}_w^2},N{u}_x=\frac{x{q}_w}{k_f}\frac{1}{T_w-T_{\infty }}$$

The wall heat flux $q_w$ and the shear stresses for the wall $\left({\tau }_{wx},{\tau }_{wy}\right) \text{are}$ given below

$$\begin{array}{l}{q}_w=-k_{nf}{\left(\frac{\partial T}{\partial z}\right)}_{z=0}+q_r,{\tau }_{wx}={\mu }_{nf}{\left(\frac{\partial u}{\partial z}\right)}_{z=0},\\ {\tau }_{wy}={\mu }_{nf}{\left(\frac{\partial v}{\partial z}\right)}_{z=0}\end{array}$$

By means of the similarity transformation in above physical quantities, we get the following expression, wherein $R{e}_x=U_{w}L/{\nu }_f$ shows local Reynolds number

$$\begin{array}{l}\sqrt{2R{e}_x}{C}_{fx}=\frac{1}{{\left(1-\varphi \right)}^{\frac{5}{2}}}\frac{{\text{d}}^2}{\text{d}{\eta }^2}f\left(0\right),\\ \sqrt{2R{e}_x}{C}_{fy}=\frac{1}{{\left(1-\varphi \right)}^{\frac{5}{2}}}\frac{\text{d}}{\text{d}\eta }g\left(0\right),\frac{N{u}_x}{\sqrt{2R{e}_x}}=-\frac{k_{nf}}{k_f}\frac{\text{d}}{\text{d}\eta }\theta \left(0\right)\end{array}$$

Gegenbauer polynomials and their properties

In this section, Gegenbauer polynomials, its connection with other polynomials, and properties are discussed. Gegenbauer polynomials $G_m^{\nu }\left(\eta \right)$ are of order $m$ which satisfy the following singular Sturm–Liouville equation in [–1]

$$\begin{array}{l}\frac{\text{d}}{\text{d}\eta }\left({\left(1-{\eta }^2\right)}^{\nu +\frac{1}{2}}\frac{\text{d}}{\text{d}\eta }{G}_m^{\nu }\left(\eta \right)\right)\\ \hspace{1em}+m\left(m+2\nu \right){\left(1-{\eta }^2\right)}^{\nu -\frac{1}{2}}{G}_m^{\nu }\left(\eta \right)=0\end{array}$$

and defined on the interval [–1] and can be calculated using subsequent relations

$$\begin{array}{l}{G}_0^{\nu }\left(\eta \right)=1,G_1^{\nu }\left(\eta \right)=2\nu \eta ,\\ G_{m+1}^{\nu }\left(\eta \right)=\frac{1}{m+1}[2\eta \left(\nu +m\right)G_m^{\nu }\left(\eta \right)\\ -\left(2\nu +m-1\right)G_{m-1}^{\nu }\left(\eta \right)],m\ge 1\end{array}$$

Rodrigues formula for the Gegenbauer polynomials is given as^32–35

$$\begin{array}{l}{G}_m^{\nu }\left(\eta \right)={\left(-\frac{1}{2}\right)}^m\frac{{\left(2\nu \right)}_m}{m!{\left(\nu +\frac{1}{2}\right)}_m}{\left(1-{\eta }^2\right)}^{-\nu +\frac{1}{2}}\\ \times \frac{{\text{d}}^m}{\text{d}{\eta }^m}{\left(1-{\eta }^2\right)}^{m-\nu -\frac{1}{2}},\nu >-\frac{1}{2},\nu \ne 0\end{array}$$

Here, ${\left(\theta \right)}_m$ is the Pochhammer symbol. Moreover, the Gegenbauer polynomials orthogonal^32–35 with respect to the $L^2$ inner product on the interval $\left[-1,1\right]$

$$\begin{array}{l}{\displaystyle {\int }_{-1}^1{G}_n^{\nu }\left(\eta \right)G_m^{\nu }\left(\eta \right)\vartheta \text{d}\eta }\\ =\{\begin{array}{c} 0. \mathrm{for} m\ne n\\ \frac{\pi 2^{1-2\nu }\text{Γ}\left(m+2\nu \right)}{\text{Γ}\left(m+1\right)\left(m+\nu \right){\left[\text{Γ}\left(\nu \right)\right]}^2},\end{array}\text{ for }m=n\end{array}$$

with weight function ${\vartheta }^{\nu }={\left(1-{\eta }^2\right)}^{\nu -\frac{1}{2}}$ . Chebyshev polynomials of the first kind $T_m\left(\eta \right)$ and the second kind $U_m^{\text{*}}\left(\eta \right)$ , Jacobi polynomials $P_m\left(\eta \right),$ and the Legendre polynomials, $L_m\left(\eta \right)$ all are special cases of Gegenbauer polynomials and have the following relations^32–35

$$G_0^{\nu }=1=L_0=T_0=U_0^{\text{*}}$$

and

$$\begin{array}{l}{L}_m=G_m^{\frac{1}{2}}\left(\eta \right),T_m=\frac{m}{2}\underset{\nu \to 0}{\lim }\frac{G_m^{\nu }\left(\eta \right)}{\nu },U_m^{\text{*}}=G_m^1\left(\eta \right),\\ G_m^{\nu }\left(\eta \right)=\frac{\text{Γ}\left(\nu +\frac{1}{2}\right)\text{Γ}\left(m+2\nu \right)}{\text{Γ}\left(\nu +m+\frac{1}{2}\right)\text{Γ}\left(2\nu \right)}{P}_m^{\left(\nu -\frac{1}{2},\nu -\frac{1}{2}\right)}\left(\eta \right)\end{array}$$

An explicit formula to calculate the Gegenbauer polynomials is given below^32–35

$$G_m^{\nu }\left(\eta \right)={\displaystyle \sum }_{i=0}^m\frac{{\left(-1\right)}^{m-i}{\left(2\nu \right)}_{m+i}}{i!\left(m-i\right)!{\left(\nu +\frac{1}{2}\right)}_i}{\left(\frac{\eta +1}{2}\right)}^i$$

The generating function^32–35 is defined as

$${\displaystyle \sum }_{m=0}^{\infty }{G}_m^{\nu }\left(\eta \right)t^m={\left(1-2\eta t+t^2\right)}^{-\nu },\left|t\right|<1$$

First, four Gegenbaure polynomials are given as

$$\begin{array}{l}{G}_0^{\nu }\left(\eta \right)=1,\hspace{1em}{G}_1^{\nu }\left(\eta \right)=2\nu \eta ,\\ G_2^{\nu }\left(\eta \right)=\frac{{\left(\nu \right)}_2}{2!}{\left(2\eta \right)}^2-\nu ,\hspace{1em}{G}_3^{\nu }\left(\eta \right)=\frac{{\left(\nu \right)}_3}{3!}{\left(2\eta \right)}^3-2{\left(\nu \right)}_2\eta \end{array}$$

The curves of first 10 Gegenbauer polynomials for the fixed value of $\nu =0.5$ are presented in Figure 2, and Figure 3 indicates the curves of 10th-order Gegenbauer polynomials for different values of $\nu$ .

Figure 2.

Curves of first tenth-order Gegenbauer polynomials with $\nu =\frac{1}{2}.$

Figure 3.

Curves of the 10th-order Gegenbauer polynomial for different values of $\nu$ .

The Gegenbauer polynomials have the following properties^32–35

$\frac{\text{d}}{\text{d}\eta }{G}_m^{\nu }\left(\eta \right)=2\nu G_{m-1}^{\nu +1}\left(\eta \right),m\ge 1$

$\frac{{\text{d}}^k}{\text{d}{\eta }^k}{G}_m^{\nu }\left(\eta \right)=2^k{\nu }^k{G}_{m-k}^{\nu +k}\left(\eta \right)$

Wavelets and Gegenbauer wavelets

There are four influences which are involved in Gegenbauer wavelets over the interval [0,1] i.e. ${\psi }_{p,q}\left(t\right)=\psi \left(k,p,q,t\right)$ , where $k\in Z^{+}$ , $p=1,\dots ,2^{k-1}$ , $q$ is the order of Gegenbauer polynomials, and $t$ is the normalized time. It can be defined as

$${\psi }_{p,q}\left(t\right)=\{\begin{array}{cc}{2}^{\frac{k}{2}} {\tilde{G}}_q^{\nu }\left(2^{k}t-\left(2p-1\right)\right),& \frac{p-1}{2^{k-1}}\le t<\frac{p}{2^{k-1}}\\ 0& \mathrm{otherwise}\end{array}$$

where

$${\tilde{G}}_j^{\nu }=\frac{\text{Γ}\left(m+1\right)\left(m+\nu \right){\left[\text{Γ}\left(\nu \right)\right]}^2}{\pi 2^{1-2\nu }\text{Γ}\left(m+2\nu \right)}{G}_j^{\nu }$$

(13)

where $q=0,1,2,\dots ,M-1$ and $p=0,1,\dots ,2^{k-1}$ . In the above expression, $G_q^{\nu }\left(\eta \right)$ are Gegenbauer polynomials of order $q$ , and the coefficient in equation (13) is for orthonormality. It is important to remember while dealing with Gegenbauer wavelets that the weight function ${\vartheta }^{\nu }\left(t\right)={\left(1-t^2\right)}^{\nu -\frac{1}{2}}$ has to be dilated and translated as given below

$${\vartheta }_p^{\nu }\left(t\right)={\vartheta }^{\nu }\left(2^{k}t-\left(2p-1\right)\right)$$

A function $f\left(\eta \right)$ from $L^2\left(\mathbb{R}\right)$ -space express in [0,1) may be expanded ³⁵ as the Gegenbauer wavelets

$$f\left(\eta \right)={\displaystyle \sum }_{i=1}^{\infty }{\displaystyle \sum }_{j=0}^{\infty }{\tau }_{i,j}{\psi }_{i,j}\left(\eta \right)$$

where

$${\tau }_{ij}={\displaystyle {\int }_0^{1}f\left(\eta \right){\psi }_{ij}\left(\eta \right){\varpi }_i\text{d}\eta =〈\hspace{0.17em}f\left(\eta \right),{\psi }_{i,j}{\left(\eta \right)〉}_{L_{\varpi }^2\left[0,1\right)}}$$

where the expression $〈·,{·〉}_{L_{\varpi }^2\left[0,1\right)}$ represents the inner product space in $L_{\varpi }^2\left[0,1\right)$ . Since the above expression is an infinite series, and for approximate solution, we should truncate it; therefore, it can be written as

$$f\left(\eta \right)={\displaystyle \sum }_{i=1}^{2^{k-1}}{\displaystyle \sum }_{j=0}^{M-1}{\tau }_{ij}{\psi }_{ij}\left(\eta \right)={\mathbf{C}}^T\mathbf{\Psi }\left(\eta \right)$$

where the matrices $\mathbf{C}$ and $\mathbf{\Psi }\left(\eta \right)$ are of order $2^{k-1}M\times 1$ and given by Iqbal et al. ³⁵

Convergence of Gegenbauer wavelets

Theorem: The series solution ${\displaystyle \sum }_{i=1}^{2^{k-1}}{\displaystyle \sum }_{j=0}^{M-1}{\tau }_{ij}{\psi }_{ij}$ converges to the solution $f\left(\eta \right)$ as $2^{k-1},M\to \infty$ .

Proof: We can write the inner product of $f(\eta )$ and ${\psi }_{i,j}\left(\eta \right)$ w.r.t the weight function as in the form

$${\tau }_{ij}={\displaystyle {\int }_0^{1}f\left(\eta \right){\psi }_{i,j}\left(\eta \right){\varpi }_i\text{d}\eta =f(\eta ),{\psi }_{i,j}\left(\eta \right)}$$

Now, consider that ${\lambda }_1=2^{k_1-1},{\lambda }_2=2^{k_2-1},{\theta }_1=M,\text{ and }{\theta }_2=N,$ where $k_1$ and $k_2$ represent the level of resolutions, and $M$ and $N$ shows the order of Gegenbauer polynomials. Let the sequence of partial sums of ${\tau }_{i,j}{\psi }_{i,j}\left(\eta \right)$ is given by ${\delta }_{{\lambda }_1,{\lambda }_2}$ , we prove that ${\delta }_{{\lambda }_1,{\lambda }_2}$ is a Cauchy sequence in Hilbert space $L^2[0,1)$ and then afterward, we show that ${\delta }_{{\lambda }_1,{\lambda }_2}$ converges to $f(\eta )$ when ${\lambda }_1,{\lambda }_2\to \infty .$

To show ${\delta }_{{\lambda }_1,{\lambda }_2}$ is a Cauchy sequence, we first assume that ${\delta }_{{\lambda }_1{\hspace{0.17em}}^{\prime },{\lambda }_2{\hspace{0.17em}}^{\prime }}$ be any arbitrary sums of ${\tau }_{i,j}{\psi }_{i,j}\left(\eta \right)$ with ${\lambda }_1>{\lambda }_1{\hspace{0.17em}}^{\text{′}},{\lambda }_2>{\lambda }_2{\hspace{0.17em}}^{\text{′}}$

$$\begin{array}{l}\Vert {{\delta }_{{\lambda }_1,{\lambda }_2}-{\delta }_{{\lambda }_1{\hspace{0.17em}}^{\text{′}},{\lambda }_2{\hspace{0.17em}}^{\text{′}}}\Vert }^2=\Vert {\displaystyle \sum }_{i={\lambda }_1{\hspace{0.17em}}^{\text{′}}+1}^{{\lambda }_1}\hspace{0.17em}{\displaystyle \sum }_{j={\lambda }_2{\hspace{0.17em}}^{\text{′}}}^{{\lambda }_2}{\tau }_{i,j}{\psi }_{i,j}{\left(\eta \right)}^2\Vert ,\\ =〈{\displaystyle \sum }_{i={\lambda }_1{\hspace{0.17em}}^{\text{′}}+1}^{{\lambda }_1}{\displaystyle \sum }_{j={\lambda }_2{\hspace{0.17em}}^{\text{′}}}^{{\lambda }_2-1}{\tau }_{i,j}{\psi }_{i,j}\left(\eta \right),{\displaystyle \sum }_{m={\lambda }_1{\hspace{0.17em}}^{\text{′}}+1}^{{\lambda }_1}{\displaystyle \sum }_{n={\lambda }_2{\hspace{0.17em}}^{\text{′}}}^{{\lambda }_2-1}{\tau }_{m,n}{\psi }_{m,n}\left(\eta \right)〉,\\ ={\displaystyle \sum }_{i={\lambda }_1{\hspace{0.17em}}^{\text{′}}+1}^{{\lambda }_1}{\displaystyle \sum }_{j={\lambda }_2{\hspace{0.17em}}^{\text{′}}}^{{\lambda }_2-1}{\displaystyle \sum }_{m={\lambda }_1{\hspace{0.17em}}^{\text{′}}+1}^{{\lambda }_1}{\displaystyle \sum }_{n={\lambda }_2{\hspace{0.17em}}^{\text{′}}}^{{\lambda }_2-1}{\tau }_{i,j}{\tau }_{m,n}〈{\psi }_{i,j}\left(\eta \right),{\psi }_{m,n}\left(\eta \right)〉,\\ ={\displaystyle \sum }_{i={\lambda }_1{\hspace{0.17em}}^{\text{′}}+1}^{{\lambda }_1}{\displaystyle \sum }_{j={\lambda }_2{\hspace{0.17em}}^{\text{′}}}^{{\lambda }_2-1}{\left|{\tau }_{i,j}\right|}^2\end{array}$$

Since from the Bessel’s inequality, we found that ${\displaystyle \sum }_{i=1}^{\infty }{\displaystyle \sum }_{j=1}^{\infty }{\left|{\tau }_{i,j}\right|}^2$ is convergent and hence

$$\Vert {\delta }_{{\lambda }_1,{\lambda }_2}-{\delta }_{{\lambda }_1{\hspace{0.17em}}^{\text{′}},{\lambda }_2{\hspace{0.17em}}^{\text{′}}}{\Vert }^2\to 0\text{as}{\lambda }_1,{\lambda }_2,{\lambda }_1{\hspace{0.17em}}^{\text{′}},{\lambda }_2{\hspace{0.17em}}^{\text{′}}\to \infty$$

This implies that ${\delta }_{{\lambda }_1,{\lambda }_2}$ is a Cauchy sequence, and it converges to a function say $g(\eta )\in L^2[0,1)$ . Now, we need to show that $g\left(\eta \right)=f\left(\eta \right)$

$$\begin{array}{l}〈g\left(\eta \right)-f\left(\eta \right),{\psi }_{i,j}\left(\eta \right)〉=〈g\left(\eta \right),{\psi }_{i,j}\left(\eta \right)〉-〈\hspace{0.17em}f\left(\eta \right),{\psi }_{i,j}\left(\eta \right)〉,\\ =\underset{{\lambda }_1,{\lambda }_2\to \infty }{\lim }〈{\delta }_{{\lambda }_1,{\lambda }_2},{\psi }_{i,j}\left(\eta \right)〉-{\tau }_{i,j},\\ ={\tau }_{i,j}-{\tau }_{i,j}=0\end{array}$$

Hence ${\displaystyle \sum }_{i=1}^{{\lambda }_1}{\displaystyle \sum }_{j=0}^{{\lambda }_2-1}{\tau }_{i,j}{\psi }_{i,j}\left(\eta \right)$ converges to $f(\eta )$ as ${\lambda }_1,{\lambda }_2\to \infty$ . This completes the proof.

Solution procedure

This section is devoted to discussing the methodology and implementation of the MGWM to find an optimal solution of the said problem (equations (8) to (12)). The proposed modification has two major advantages over the Gegenbauer wavelets method: first, it has less computational work since it condenses the unknown present in trial solutions, and second, the trial solution suggestion in modified version must satisfy the boundary conditions. The proposed method has the following steps to solve equations (8) to (12).

Step 1: Consider the nonlinear system (8–10)

$$\begin{array}{l}\frac{1}{{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_s}{{\rho }_f}\varphi \right)}{f}^{\text{″′}}\left(\eta \right)-2f^{\text{\hspace{0.17em}′}}{\hspace{0.17em}}^2\left(\eta \right)\\ +f\left(\eta \right)f^{\text{\hspace{0.17em}″}}\left(\eta \right)+4\lambda g\left(\eta \right)=0\end{array}$$

(14)

$$\begin{array}{l}\frac{1}{{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_s}{{\rho }_f}\varphi \right)}{g}^{\text{″}}\left(\eta \right)-2g\left(\eta \right)f^{\text{\hspace{0.17em}′}}\left(\eta \right)\\ +f\left(\eta \right)g^{\text{′}}\left(\eta \right)-4\lambda f^{\text{′}}\left(\eta \right)=0\end{array}$$

(15)

$$\frac{1}{\left(1-\varphi +\frac{{\left(\rho c_p\right)}_s}{{\left(\rho c_p\right)}_f}\varphi \right)}{\theta }^{\text{″}}\left(\eta \right)+\text{Pr}f\left(\eta \right){\theta }^{\text{′}}\left(\eta \right)=0$$

(16)

Step 2: Consider the following trial solutions to investigate the nonlinear system (14–16) via Gegenbauer wavelets method ³⁵

$$\tilde{f}\left(\eta \right)={\displaystyle \sum }_{i=1}^{2^{k-1}}{\displaystyle \sum }_{j=0}^{M-1}{\tau }_{ij}^1{\psi }_{ij}\left(\eta \right)={\mathbf{C}}_1^T\mathbf{\Psi }\left(\eta \right)$$

(17)

$$\tilde{g}\left(\eta \right)={\displaystyle \sum }_{i=1}^{2^{k-1}}{\displaystyle \sum }_{j=0}^{M-1}{\tau }_{ij}^2{\psi }_{ij}\left(\eta \right)={\mathbf{C}}_2^T\mathbf{\Psi }\left(\eta \right)$$

(18)

$$\tilde{\theta }\left(\eta \right)={\displaystyle \sum }_{i=1}^{2^{k-1}}{\displaystyle \sum }_{j=0}^{M-1}{\tau }_{ij}^3{\psi }_{ij}\left(\eta \right)={\mathbf{C}}_3^T\mathbf{\Psi }\left(\eta \right)$$

(19)

where

$${\mathbf{C}}_i={\left[{\tau }_{1,0}^i,{\tau }_{1,1}^i,{\tau }_{1,2}^i,{\tau }_{1,3}^i,\dots \right]}^T,i=1,2,3$$

and

$$\begin{array}{l}\mathbf{\Psi }\left(\eta \right)=[\sqrt{2}P\left(0\right),2\sqrt{2}\left(2\eta -1\right)\alpha P\left(1\right),\\ \times \sqrt{2}\left(8\alpha {\eta }^2+8{\eta }^2-8\alpha \eta -8\eta +2\alpha +1\right)P\left(2\right),\\ \times \frac{1}{3\sqrt{2}}\left(8\alpha {\eta }^2-8\alpha \eta +16{\eta }^2-16\eta +2\alpha +1\right),\\ {\times \left(2x-1\right)\alpha \left(\alpha +1\right)P\left(3\right),\dots ]}^T\end{array}$$

The trial solutions (17–19) stated in above can be rewritten as

$$\tilde{f}\left(\eta \right)={\mathbf{\Lambda }}_1^T\mathbf{\chi }\left(\eta \right),\tilde{g}\left(\eta \right)={\mathbf{\Lambda }}_2^T\mathbf{\chi }\left(\eta \right),\tilde{\theta }\left(\eta \right)={\mathbf{\Lambda }}_3^T\mathbf{\chi }\left(\eta \right)$$

(20)

The matrices ${\mathbf{\Lambda }}_i,i=1,2,3$ in above expression (20) are defined as

$${\mathbf{\Lambda }}_{\mathit{i}}={\left[\begin{array}{c}\sqrt{2}\left({\tau }_{0,1}^{i}P\left(0\right)-2{\tau }_{1,1}^i\alpha P\left(1\right)+2{\alpha }^2{\tau }_{1,2}^{i}P\left(2\right)+\cdots \right),\\ \sqrt{2}(4\alpha P\left(1\right){\tau }_{1,1}^i-8{\alpha }^{2}P\left(2\right){\tau }_{1,2}^i\\ -8\alpha P\left(2\right){\tau }_{1,2}^i+\cdots ),\\ \sqrt{2}(8\alpha P\left(2\right){\tau }_{1,2}^i+8{\alpha }^{2}P\left(2\right){\tau }_{1,2}^i+\cdots ),\\ \sqrt{2}(32{\alpha }^{2}P\left(3\right){\tau }_{1,3}^i+\cdots ),\dots \end{array}\right]}^T$$

for $i=1,2,3,$ and $\mathbf{\chi }\left(\eta \right)={\left[1,\eta ,{\eta }^2,{\eta }^3,\dots \right]}^T$

where

$$P\left(m\right)=\sqrt{\frac{\text{Γ}\left(m+1\right)\left(m+\alpha \right){\left(\text{Γ}\left(\alpha \right)\right)}^2}{2^{1-\alpha }\text{Γ}\left(m+\alpha \right)\pi }}$$

To reduce the unknown constants, we introduced a new set of constants as $\left\{{\tilde{\tau }}_0^i,{\tilde{\tau }}_1^i,{\tilde{\tau }}_2^i,\dots \right\},$ i = 1,2,3, which must satisfy the following expressions

$$\begin{array}{l}\sqrt{2}\left({\tau }_{0,1}^{i}P\left(0\right)-2{\tau }_{1,1}^i\alpha P\left(1\right)+\cdots \right)\\ ={\tilde{\tau }}_0,\sqrt{2}\left(4\alpha P\left(1\right){\tau }_{1,1}^i-8{\alpha }^{2}P\left(2\right){\tau }_{1,2}^i+\cdots \right)={\tilde{\tau }}_1\end{array}$$

(21)

$$\begin{array}{l}\sqrt{2}\left(8\alpha P\left(2\right){\tau }_{1,2}^i+8{\alpha }^{2}P\left(2\right){\tau }_{1,2}^i+\cdots \right)\\ ={\tilde{\tau }}_2,\sqrt{2}\left(32{\alpha }^{2}P\left(3\right){\tau }_{1,3}^i+\cdots \right)={\tilde{\tau }}_3\end{array}$$

(22)

$$⋮$$

Using the expressions (21) and (22) into equation (20), we get

$$\tilde{f}\left(\eta \right)={\displaystyle \sum }_{n=0}^M{\tilde{\tau }}_n^1{\eta }^n,\tilde{g}\left(\eta \right)={\displaystyle \sum }_{n=0}^M{\tilde{\tau }}_n^2{\eta }^n,\tilde{\theta }\left(\eta \right)={\displaystyle \sum }_{n=0}^M{\tilde{\tau }}_n^3{\eta }^n$$

(23)

Since the trial solution suggested by the proposed method must fulfill the boundary conditions, the trial solution (23) takes the following form after applying the boundary conditions

$$\tilde{f}\left(\eta \right)=\eta -\frac{1}{2{\eta }_{\infty }}{\eta }^2+{\displaystyle \sum }_{n=1}^{M-3}{\tilde{\tau }}_n^1\left[{\eta }^n-\frac{n+2}{2}{\eta }_{\infty }^n\right]{\eta }^2$$

(24)

$$\tilde{g}\left(\eta \right)={\displaystyle \sum }_{n=1}^{M-2}{\tilde{\tau }}_n^2\left[{\eta }^n-{\eta }_{\infty }^n\right]\eta$$

(25)

$$\tilde{\theta }\left(\eta \right)=1-\frac{1}{{\eta }_{\infty }}\eta +{\displaystyle \sum }_{n=1}^{M-2}{\tilde{\tau }}_n^3\left[{\eta }^n-{\eta }_{\infty }^n\right]\eta$$

(26)

where ${\eta }_{\infty }\left(=\infty \right)$ .

Step 3: Putting the trial solutions (24–26) into equations presnt in step 1, we found the following system of algebraic equations

$$\begin{array}{l}{\mathbf{R}}_f=\frac{1}{{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_s}{{\rho }_f}\varphi \right)}{\tilde{f}}^{\text{\hspace{0.17em}″′}}\left(\eta \right)-2{\tilde{f}}^{\text{\hspace{0.17em}′}}{\hspace{0.17em}}^2\left(\eta \right)\\ +\tilde{f}\left(\eta \right){\tilde{f}}^{\text{\hspace{0.17em}″}}\left(\eta \right)+4\lambda \tilde{g}\left(\eta \right)\end{array}$$

(27)

$$\begin{array}{l}{\mathbf{R}}_g=\frac{1}{{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_s}{{\rho }_f}\varphi \right)}{\tilde{g}}^{\text{″}}\left(\eta \right)-2\tilde{g}\left(\eta \right){\tilde{f}}^{\text{\hspace{0.17em}′}}\left(\eta \right)\\ +\tilde{f}\left(\eta \right){\tilde{g}}^{\text{′}}\left(\eta \right)-4\lambda {\tilde{f}}^{\text{\hspace{0.17em}′}}\left(\eta \right)\end{array}$$

(28)

$${\mathbf{R}}_{\theta }=\frac{1}{\left(1-\varphi +\frac{{\left(\rho c_p\right)}_s}{{\left(\rho c_p\right)}_f}\varphi \right)}{\tilde{\theta }}^{\text{″}}\left(\eta \right)+\text{Pr}\tilde{f}\left(\eta \right){\tilde{\theta }}^{\text{′}}\left(\eta \right)$$

(29)

Step 4: The method of moment’s concept is used to investigate the set of unknown constants $\left\{{\tilde{\tau }}_0^i,{\tilde{\tau }}_1^i,{\tilde{\tau }}_2^i,\dots \right\},i=1,2,3$ and deliberated as

$$\begin{array}{c}{\mathbf{E}}_f^n={\displaystyle {\int }_0^{{\eta }_{\infty }}{\mathbf{R}}_f{\eta }^n\text{d}\eta ,n=0,1,2,\dots ,2^{k-1}M-4,}\\ {\mathbf{E}}_g^n={\displaystyle {\int }_0^{{\eta }_{\infty }}{\mathbf{R}}_g{\eta }^n\text{d}\eta ,n=0,1,2,\dots ,2^{k-1}M-3,}\\ {\mathbf{E}}_{\theta }^n={\displaystyle {\int }_0^{{\eta }_{\infty }}{\mathbf{R}}_{\theta }{\eta }^n\text{d}\eta ,n=0,1,2,\dots ,2^{k-1}M-3}\end{array}$$

A system of nonlinear algebraic equations is generated with the help of above system.

Step 5: The solution nonlinear system found in step 4 leads to the values of $\tilde{\tau }$ ’s. For the investigation of $\tau$ ’s inserting the values of $\tilde{\tau }$ ’s in equations (21) and (22) and solve it by backward substitution. Finally, approximate solution is obtained by using the values of $\tau$ ’s in equations (8) to (12).

The solution of discussed problem by means of the proposed methodology is given below when $\varphi =0.05,\lambda =1,\mathrm{Pr}=6.2,{\eta }_{\infty }=5$ , and the order of approximation is k = 1 and M = 9

$$\begin{array}{l}\tilde{f}\left(\eta \right)=\frac{2}{\sqrt{\pi }}[0.88622693\eta -0.886058717{\eta }^2\hfill \\ +0.379141079{\eta }^3-0.0514739938{\eta }^4\\ -0.013935928{\eta }^5+0.0062027{\eta }^6\\ -0.00086145351{\eta }^7+0.0000427942266{\eta }^8],\\ \tilde{g}\left(\eta \right)=\frac{2}{\sqrt{\pi }}[-1.198248206348\eta +2.1712611{\eta }^2\\ -1.62466396{\eta }^3+0.6476932{\eta }^4\\ \begin{array}{l}-0.147178514298{\eta }^5+0.01874337{\eta }^6\\ \hspace{1em}\hspace{1em}-0.0011970136{\eta }^7+0.000027056196{\eta }^8],\end{array}\\ \begin{array}{l}\tilde{\theta }\left(\eta \right)=\frac{2}{\sqrt{\pi }}[0.8862269-1.15394074917\eta \\ \hspace{1em}\hspace{1em}+\hspace{0.17em}0.0781719678{\eta }^2+0.8559565{\eta }^3\end{array}\\ \hspace{1em}\hspace{1em}-\hspace{0.17em}0.7631968{\eta }^4+0.31113642{\eta }^5\\ \hspace{1em}\hspace{1em}-\hspace{0.17em}0.0678436{\eta }^6+0.00765188{\eta }^7-0.000351016{\eta }^8]\end{array}$$

Results and discussion

Modified methodology based on Gegenbauer wavelets coupled with method of moments is proposed to explore the numerical solution of the rotating flow of nanofluid due to exponentially stretching surface. In this section, Figures 4 to 9 have been plotted to investigate the variation in the velocities “ $f^{\text{′}}\left(\eta \right)$ ,” “ $g\left(\eta \right),$ ” and temperature “ $\theta \left(\eta \right)$ ” profiles for various values of the physical parameters involving the nanoparticles volume fraction “ $\varphi$ ” and rotation “ $\lambda$ ” parameters along with a comprehensive discussion and graphical representation. Figures 10 to 15 demonstrate the behavior of physical quantities, coefficient of skin friction, and Nusselt number under the influence of nanoparticle volume fraction and rotation parameters. Figure 16 plots to explain the study of square residual error for various values of order of approximants. Comparison of the achieved results via proposed method with numerical method RK-4 is depicted in Figure 17.

A comparative analysis of the obtained results via MGWM with the existing techniques^5,22 is also presented. The thermophysical properties of the base fluid and different nanoparticles are given in Table 1.

Figures 4 and 5 are plotted to show the behavior of the dimensionless x-component of the velocity profile “ $f^{\text{′}}\left(\eta \right)$ ” for different values of the non-dimensional parameters, nanoparticle volume fraction, and rotation parameters by considering silver and copper nanoparticles, respectively. It is observed that as increasing the volume fraction of silver and copper nanoparticles, the velocity profile and the thickness of boundary layer are gradually decreased. Same behavior obtained for copper oxide nanoparticle while the opposite behavior is noted when volume fraction of alumina and titania nanoparticles is increased.

Figure 4.

Behavior of $f^{\text{′}}\left(\eta \right)$ for numerous values of $\lambda$ and $\varphi$ for the case of Ag-water.

Figure 5.

Behavior of $f^{\text{′}}\left(\eta \right)$ for numerous values of $\lambda$ and $\varphi$ for the case of Cu-water.

In all the cases, variation in the rotation parameter causes a decrease in the velocity profile and the thickness of the boundary layer. The oscillatory behavior obtained nearby the wall due to the rotation effect while velocity reduces as $\eta$ increases.

The effects of different values of nanoparticle volume fraction and rotation parameters on the y-component of the non-dimensional velocity profile are illustrated in Figures 6 and 7. As enhancing the volume fraction of silver and copper nanoparticles, the y-component of velocity profile and the thickness of boundary layer is decreased gradually. Copper oxide nanoparticles of the y-component demonstrate the same behavior although the opposite behavior is observed while increasing the volume fraction of alumina and titania nanoparticles.

Figure 6.

Behavior of $g\left(\eta \right)$ for numerous values of $\lambda$ and $\varphi$ for the case of Ag-water.

Figure 7.

Behavior of $g\left(\eta \right)$ for numerous values of $\lambda$ and $\varphi$ for the case of Cu-water.

An increase in rotation parameter reduced the magnitude of velocity field. Moreover, parabolic behavior of the velocity field is obtained when $\lambda$ is sufficiently small and oscillatory behavior is observed when $\lambda$ is sufficiently large near the wall.

Figures 8 and 9 are plotted to show the behavior of the nanoparticle volume fraction and rotation parameters on the non-dimensional temperatures profile for silver and copper nanoparticles, respectively. In all the cases, the nanoparticle volume fraction and the rotation parameters give the similar behavior. As both parameters enlarge, the temperature profile increases. The thickness of the boundary layer is also increased with an increase in the parameters. Physically, the enhancement of the rotation effects causes the viscous forces within the fluid to increase and resists the motion of the fluid, leading to increase the temperature of the fluid.

Figure 8.

Behavior of $\theta \left(\eta \right)$ for numerous values of $\lambda$ and $\varphi$ for the case of Ag-water.

Figure 9.

Behavior of $\theta \left(\eta \right)$ for numerous values of $\lambda$ and $\varphi$ for the case of Cu-water.

The behavior of Nusselt number and coefficient of the skin friction with increasing values of nanoparticle volume fraction and rotation parameters are depicted in Figures 10 to 15 for all nanoparticles. Figure 10 demonstrates the behavior of skin friction coefficient $\sqrt{2{\text{Re}}_x}{\text{C}}_{fx}$ for the numerous values of nanoparticle volume fraction. An increase in the nanoparticle volume fraction causes a decrease in the skin friction coefficient $\sqrt{2{\text{Re}}_x}{\text{C}}_{fx}$ . The least value of the skin friction coefficient $\sqrt{2{\text{Re}}_x}{\text{C}}_{fx}$ is obtained in the case of silver-water nanofluid, and the highest value of skin friction coefficient $\sqrt{2{\text{Re}}_x}{\text{C}}_{fx}$ is obtained in the case of alumina-water nanofluid. The effect of rotation parameter on the coefficient of the skin friction $\sqrt{2{\text{Re}}_x}{\text{C}}_{fx}$ is illustrated in Figure 11. It is observed that the absolute values of coefficient of the skin friction $\sqrt{2{\text{Re}}_x}{\text{C}}_{fx}$ enlarge as the rotating effects are increased.

Figure 10.

Behavior of skin friction coefficient $C_{fx}\text{ for }\text{numerous values of }\varphi \text{ and }\lambda =1$ .

Figure 11.

Behavior of skin friction coefficient $C_{fx}$ for numerous values of $\lambda$ and $\varphi =0.1$ .

Again least and highest values of the coefficient of the skin friction $\sqrt{2{\text{Re}}_x}{\text{C}}_{fx}$ are achieved for the case of silver-water and alumina-water nanofluid, respectively. According to an industrial source, the alumina-water nanoparticles are more feasible than the others because they preserve the uniform stretching least force. Figures 12 and 13 are plotted to see the variation coefficient of the skin friction $\sqrt{2{\text{Re}}_x}{\text{C}}_{fy}$ for different values of nanoparticle volume fraction and rotation parameters.

Figure 12.

Behavior of skin friction coefficient $C_{fy}$ for numerous values of $\varphi$ and $\lambda =1$ .

Figure 13.

Behavior of skin friction coefficient $C_{fy}$ for numerous values of $\lambda$ and $\varphi =0.1$ .

It is observed that almost similar variation in coefficient of the skin friction $\sqrt{2{\text{Re}}_x}{\text{C}}_{fy}$ is found as in Figures 10 and 11. Figures 14 and 15 deliberate the effect of the nanoparticle volume fraction and rotation parameters on the dimensionless Nusselt number for silver, copper, copper oxide, alumina, and titania nanoparticles. Almost linear behavior of the Nusselt number obtained as varying the nanoparticle volume fraction. The Nusselt number increases while increasing the nanoparticle volume fraction. Reverse variation of Nusselt number under the influence of rotation parameter is obtained in Figure 15, that is Nusselt number decreases while increasing the rotating effects. In both Figures 14 and 15, the least and highest values of the Nusselt number are obtained for the cases of silver-water nanofluid and alumina-water nanofluid, respectively.

Figure 14.

Behavior of Nusselt number for numerous values of $\varphi$ and $\lambda =0.4$ .

Figure 15.

Behavior of Nusselt number for numerous values of $\lambda$ and $\varphi =0.1$ .

Figure 16 deliberates the graphical behavior of the square residual error of $f\left(\eta \right),g\left(\eta \right),$ and $\theta \left(\eta \right)$ for the numerous values of M. As varying the order of approximation M, it is noted that the residual error of velocities and temperature goes to zero, which witnesses that the solution convergent. Comparison of the solution found by means of MGWM with the numerical method RK-4 is demonstrated in Figure 17. The obtained solution is well agreed with the numerical solution, which shows the efficiency of the proposed method.

Figure 16.

(a)–(c): Analysis of square residual error for different values of M. (a) M = 10; (b) M = 20; (c) M = 30.

Figure 17.

(a), (b): Comparison of the obtained result with RK-4.

Tables 2 to 4 have been generated for the sake of comparison. These tables show the comparison of the obtained results for $-f^{\text{″}}\left(0\right),g^{\text{′}}\left(0\right)$ and Nusselt number with already published results.^4,22 Tables 2 to 4 evidence that the proposed method is highly effective, well-matched, and accurate to investigate the optimal solution of the above-discussed problem. Table 5 depicts the estimation of square residual error for different approximations. In this table, ${ϵ}_f,{ϵ}_g,$ and ${ϵ}_{\theta }$ are calculated from the following expressions

$$\begin{array}{l}{ϵ}_f={\displaystyle {\int }_0^{{\eta }_{\infty }}{\left[\begin{array}{l}\frac{1}{{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_s}{{\rho }_f}\varphi \right)}\frac{{\text{d}}^3}{\text{d}{\eta }^3}\tilde{f}\left(\eta \right)-2{\left(\frac{\text{d}}{\text{d}\eta }\tilde{f}\left(\eta \right)\right)}^2\\ +\tilde{f}\left(\eta \right)\frac{{\text{d}}^2}{\text{d}{\eta }^2}\tilde{f}\left(\eta \right)+4\lambda \tilde{g}\left(\eta \right)\end{array}\right]}^2}\text{d}\eta ,\\ {ϵ}_g={\displaystyle {\int }_0^{{\eta }_{\infty }}{\left[\begin{array}{l}\frac{1}{{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_s}{{\rho }_f}\varphi \right)}\frac{{\text{d}}^2}{\text{d}{\eta }^2}\tilde{g}\left(\eta \right)-2\tilde{g}\frac{\text{d}}{\text{d}\eta }\tilde{f}\left(\eta \right)\\ +\tilde{f}\left(\eta \right)\frac{\text{d}}{\text{d}\eta }\tilde{g}\left(\eta \right)-4\lambda \frac{\text{d}}{\text{d}\eta }\tilde{f}\left(\eta \right)\end{array}\right]}^2\text{d}\eta ,}\\ {ϵ}_{\theta }={\displaystyle {\int }_0^{n\infty }{\left[\frac{1}{\left(1-\varphi +\frac{{\left(\rho c_p\right)}_s}{{\left(\rho c_p\right)}_f}\varphi \right)}\frac{{\text{d}}^2}{\text{d}{\eta }^2}\tilde{\theta }\left(\eta \right)+\text{Pr}\tilde{f}\left(\eta \right)\frac{\text{d}}{\text{d}\eta }\tilde{\theta }\left(\eta \right)\right]}^2\text{d}\eta }\hfill \end{array}$$

Table 2.

Comparison of the obtained results via MGWM with already existing results. ² ,⁵²

	$-f^{″\prime }\left(0\right)$
$\lambda$	Javed et al. 5	Mushtaq et al. 22	Present	Numerical
0.2	1.3474169	1.3474203	1.3474202	1.3474203
0.5	1.5194131	1.5194196	1.5194197	1.5194195
2	2.2827966	2.2828128	2.2828129	2.2828126
5	3.3444338	3.3444609	3.3444606	3.3444607
10	4.6017220	4.6017646	4.6017652	4.6017649
50	10.058172	10.058261	10.058265	10.058264

Table 3.

Comparison of the obtained results via MGWM with already existing results. ² ,⁵²

	$-g^{\prime }\left(0\right)$
$\lambda$	Javed et al. 5	Mushtaq et al. 22	Present	Numerical
0.2	0.3701522	0.3701525	0.3701525	0.3701525
0.5	0.7625141	0.7625143	0.7625143	0.7625143
2	1.8485044	1.8485032	1.8485035	1.8485033
5	3.0609192	3.0609164	3.0609165	3.0609162
10	4.3990640	4.3990576	4.3990580	4.3990578
50	9.9668099	9.9667988	9.9667983	9.9667986

Table 4.

Comparison of the obtained results for the Nusselt number with already existing results. ²²

$\varphi$	$\lambda$	Mushtaq et al. 22	Present
0.1	0	2.04831	2.04831
	2	1.40376	1.40376
	10	0.69068	0.69068
	100	0.22017	0.22017
0	1	1.51678	1.51678
0.05		1.60879	1.60879
0.1		1.70561	1.70561
0.2		1.92719	1.92719

Table 5.

Square residual error estimation for the various order of approximations.

M	${ϵ}_f$	${ϵ}_g$	${ϵ}_{\theta }$	Time (s)
5	2.8907444 × 10–03	2.2943543 × 10–03	1.7002889 × 10–02	0.73
8	5.3294394 × 10–08	3.1854304 × 10–08	5.5359494 × 10–06	1.99
12	2.4643280 × 10–14	3.9660942 × 10–16	1.5953697 × 10–11	2.73
17	6.9262944 × 10–23	1.0422840 × 10–25	5.6251525 × 10–20	10.25
22	2.9222723 × 10–32	5.5511678 × 10–34	1.3098590 × 10–26	34.13
30	1.2563410 × 10–49	3.5621401 × 10–51	5.1478012 × 10–40	105.25

One can clearly observe that the square residual for velocities and temperature tends to zero while increasing the order of approximation. It is endorsing that the suggested method is reliable and accurate to find the solution of nonlinear physical problems. Moreover, the proposed method contains less computational work as compared to the traditional Gegenbauer wavelets method. ³⁵ The proposed modification reduces the unknown constant present in the trial solutions. These constants depend upon the total order of the system of differential equations like in our case total order of the system (8–10) is (3 + 2 + 2=) 7 which means in proposed method, seven fewer constants as compared to the original Gegenbauer wavelets method ³⁵ for arbitrary selection of M and k.

Conclusion

A model study is conducted to analyze the transfer of heat in rotating flow of nanofluids over an exponentially stretching surface. We examined the simultaneous effect of nanofluid rotation and exponential stretching on the shear stresses and heat transfer rate at the sheet that have noticeable role in polymer extrusion and metal-working processes. We also analyzed the cooling proficiency of water-based nanofluids containing Ag, Cu, Al₂O₃, TiO₂, and CuO nanoparticles for the said problem. The governing equations have been reduced to set of nonlinear ODE via appropriate transformations. The solution of the said model obtained by means of modified version of Gegenbauer wavelets method. Hence, major findings of our study are stated below:

Nanoparticle volume fraction $\varphi$ efficiently provisions the rate of heat transfer from the sheet.

The minimum and maximum values of wall shear stress occur, respectively, for Ag-water and Al₂O₃-water nanofluid.

Hydrodynamic boundary layer becomes thinner, and wall shear stress becomes larger when the rotation rate is increased.

Rotational influence severely decreases the rate of cooling from the sheet.

Maximum/minimum heat transfer rate is observed for Al₂O₃ Ag-water/water nanofluids.

The rate of cooling for metal-oxide kind nanoparticles is efficient than the metallic nanoparticles.

Comparative study presented among outcomes obtained by MGWM, RK (order-4), and already existing results. The comparison between methods, graphical plots, convergence, and error analysis endorses the credibility of this modification. The proposed technique can be extended for other physical problems.

The results disclose the appropriateness of nanoparticles inside water that enhance the cooling process of stretching surface, which would be advantageous in coating-related applications and polymer extrusion processes.