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Abstract

The low thermal efficiency of the base liquids is the main issue among the researchers and to resolve this issue, scientists use the small-sized (1–100 nm) metal solid particles in the base liquids to increase the thermal efficiency of the base solvents. In the recent article, a theoretical study has been carried out for the thermal application functioning of graphene-oxide-water-based and graphene-oxide-ethylene-glycol-based nanofluids under the impact of the Marangoni convection. The nanofluid flow is also subjected to thermal radiation and magnetic field. The problem has been solved through optimal homotopy analysis method. The impacts of the embedded parameters over the velocity and temperature pitches have been analysed. Due to strong thermophysical properties of graphene-oxide-ethylene-glycol-based nanofluid, it is observed that the heat transfer rate of this sort of nanofluid is more efficient as compared to the graphene-oxide-water-based nanofluids. All the obtained outputs have been presented graphically and numerically.

Keywords

Graphene oxide graphene-oxide-water-based/graphene-oxide-ethylene-glycol-based nanofluid flow Marangoni convection magnetohydrodynamic thermal radiation OHAM-BVPh 2.0 package

Introduction

The huge interest in energy assets is the toughest issue for the recent researchers to meet the growing energy demand in advanced scientific processes. The scientists are trying to develop new sources which are simply accessed and economical for the thermal and cooling applications. The easily accessible resource of the renewable energy is the solar energy in the universe. The researchers discovered the solar collector to store the solar energy radiation effectively at large scale, and they utilized the convectional fluid flow as a heat conduction medium. However, the common liquids such as water, vegetable oil, mineral oil and ethylene glycol (EG) are the convective heat transfer fluids and play a vital role in numerous technological and industrial approach such as air-conditioning, heat generation, microelectronics and transportation of chemical production. Heat transfer fluids are primarily suffering from less thermal conductivity, and it is a serious restriction for a device achievement. To overcome this demerit, nanofluids are introduced by Choi¹ to improve the heat transfer rate and thermal conductivity of nanofluids through nanoparticles. Nanofluids are colloidal solutions incorporating nanoparticles of size (10–50 nm) in the base fluids. The important properties of nanofluid are thermal properties, accessibility, compatibility with the base fluid, chemical stability, toxicity and cost. Common nanomaterials are metal oxides, metal and carbon materials. Hummers and Offeman² developed a speedy and comparatively safe technique for the production of graphite oxide.

Graphene is structured as a single layer of carbon which makes it a striking material with exceptional physical and chemical properties. Graphene is discussed for the first time by Novoselov et al.³ The rich chemical and electrical properties of graphene oxide (GO) and good cycling stability, large capacitance values and exceptional surface area make it interesting for the researcher. These effective properties of GO play an important role in electronic and electrical equipment such as sensor, transistor and fuel cell. Poor solubility of water is its main disadvantage. GO is well soluble in water because it contains more oxygen and belongs to a functional group such as carboxyl and hydroxyl; therefore, we used GO instead of graphene. Vinayan et al.⁴ has experimentally examined the thermal conductivity and thermal properties of a special type of graphene called hydrogen exfoliated graphene (HEG) mixed with deionized (DI) water. Water and EG were used as the base solvents for the preparation of nanofluids in their study. Li et al.⁵ have organized the polyaniline (PANI)/GO mixtures of GO with a ratio of 6:1 to exhibit the uppermost capacitance of 422 F/g at a present density of 1 A/g. Paredes et al.⁶ have examined numerous organic diluents (DMF, THF, NMP and Eg) in which they organized graphite oxide diffusions with maximum stability. Shen et al.⁷ have elaborated the preparation of GO from the chemical reaction of graphite with benzoyl peroxide (BPO), comprehensive alteration of GO to GO-sheets.

The high thermal conductivity and characteristic lubricity of graphene make it a perfect applicant for the alteration of functional fluids. The solid particles having efficient thermal conductivity are assorted to the base liquid to enhance the overall thermal efficiency of the fluid as depicted in Maxwell.⁸ Balandin et al.⁹ have examined the thermal efficiency of single-layer graphene in different solvents. Wei et al.¹⁰ were the pioneers to express the use of GO in C₂H₆O₂ to enhance the thermal efficiency of the nanoliquid. The GO nanosheets were set and isolated in EG and water at 5% capacity concentration to enhance the thermal conductivity up to 60% compared to the base liquid EG.

Recently, Gul and Firdous¹¹ examined experimentally and numerically the full solubility of GO in water and they also examined the numerical study. Sheikholeslami and colleagues^12–15 have examined the nanofluid flow utilizing various mathematical models. The temperature-dependent convection situation is called Marangoni. This phenomenon was first discussed by Thomson¹⁶ in 1855, and later in 1865, Marangoni¹⁷ fruitfully discussed this phenomenon. This phenomenon is applicable in soluble, insoluble, and partially soluble materials. The effects of Marangoni have many uses in welding purposes, drying, soap film, silicon wafer crystal development, equilibrium and electron beam melting of metals. Arifin et al.¹⁸ investigated the radiation properties of the flow of nanofluids with Marangoni convection. Golia and Viviani¹⁹ have studied the Marangoni convection effect in the non-isobaric boundary layers. Chamkha et al.²⁰ have studied the idea of the mixed convection using the Marangoni convection in a boundary layer flow. Pop et al.²¹ have examined the forced convection idea utilizing the Marangoni convection. Christopher and Wang²² have observed the effect of the Prandtl number model over an inclined plate under the influence of Marangoni convection. Hamid et al.²³ completely discussed the Marangoni convection and radiation effects over a plane surface. The effect of Marangoni convection and thermos gradient situation in a penetrable medium is discussed by Al-Mudhaf and Chamkha.²⁴ Ellahi et al.²⁵ scrutinized the influence of EG base aqueous solution of nanofluids. The magnetic effect of the carbon nanotube (CNT) nanofluids on the Marangoni convection flow has been examined by Tiwari et al.²⁶ Magnetohydrodynamic nanofluid in a stretching sheet is discussed by Hamad.²⁷ Bataller²⁸ discussed the effect of viscoelastic fluid under the influence of heat source and thermal emission phenomenon over an extending surface.

Keeping in mind the above fruitful discussion, the current study examines the graphene-oxide-water-based (GO-W) and graphene-oxide-ethylene-glycol-based (GO-EG) nanofluids for the thermal applications under the Marangoni convection. The experimental results of Gul and Ferdous¹¹ have been utilized for the GO-W and GO-EG nanofluids under the influence of Marangoni convection.

The basic governing equations for the fluid flow pattern are converted into a set of nonlinear equations. The optimal homotopy analysis method (OHAM)^11,29–34 has been used for the solution of the problems. The BVPh 2.0 package has been utilized in the error analysis. Many researchers have published many articles using the Marangoni convection, but the present work explained fruitfully the effect of GO-W and GO-EG nanofluid flow and thermal radiation under the influence of magnetic field. The comparison of the two sorts of nanofluids has also been investigated. The BVPh 2.0 package up to the 15th-order approximation has been used and the sum of the total square residual error has been calculated numerically. The numerical study of GO-W and GO-EG nanofluids under the influence of Marangoni convection is new and there is no literature that exists.

Mathematical formulation

Consider the incompressible and the two-dimensional nanofluid flow under the influence of Marangoni convection. The nanofluids consist of GO where water and EG are used as the base fluids. The constant magnetic field $B_{0}$ is applied to the flow field through an angle $Ω = π / 3$ , with respect to x-axis. The interface of the curve and the axis is shown in the geometry of the problem (Figure 1). Keeping in view all the above assumptions, our governing equations become

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

(1)

u (\frac{\partial u}{\partial x}) + v (\frac{\partial u}{\partial y}) = υ_{nf} \frac{\partial^{2} u}{\partial y^{2}} - \frac{σ_{nf} B_{o}^{2}}{ρ_{nf}} (u) {Sin}^{2} (Ω)

(2)

u (\frac{\partial T}{\partial x}) + v (\frac{\partial T}{\partial y}) = \frac{k_{nf}}{{(ρ c_{p})}_{nf}} (\frac{\partial^{2} T}{\partial y^{2}}) - \frac{1}{{(ρ c_{p})}_{nf}} (\frac{\partial q^{rad}}{\partial y})

(3)

Figure 1.

Graphical representation of the problem.

Exposed boundary conditions are

\begin{matrix} v = 0, T = T_{o}, \frac{μ_{nf}}{μ_{f}} (\frac{\partial u}{\partial y}) = (\frac{\partial T}{\partial x}), at y \to 0 \\ u = U (x), T = T_{\infty}, at y \to \infty \end{matrix}

(4)

The Marangoni boundary conditions are shown in equation (4), the surface tension is $σ^{\otimes} = σ_{o} [1 - γ (T - T_{\infty})]$ and $γ = (- 1 / σ_{o}) (\partial σ^{\otimes} / \partial T) > 0$ .^20,21 Here, $σ_{o}$ denotes the constant surface tension at the origin and $γ$ represents the surface tension temperature coefficient. T, $T_{o}$ and $T_{\infty}$ be the temperature distribution, temperature at the origin and ambient temperature of the nanofluid, respectively. Different parameters of the existing model are thermal conductivity ( $k_{nf}$ ), electrical conductivity ( $σ_{nf}$ ), specific heat capacity ( $(ρ c_{p})_{nf}$ ) and dynamic viscosity ( $μ_{nf}$ ) of the nanoliquid²

\begin{matrix} \frac{{(ρ c_{p})}_{nf}}{{(ρ c_{p})}_{f}} = [1 - ϕ + \frac{{(ρ c_{p})}_{s}}{{(ρ c_{p})}_{f}} ϕ], \\ \frac{μ_{nf}}{μ_{f}} = \frac{1}{{(1 - ϕ)}^{2.5}}, \frac{ρ_{nf}}{ρ_{f}} = [(1 - ϕ) + (\frac{ρ_{s}}{ρ_{f}}) ϕ] \\ α_{nf} = \frac{k_{nf}}{{(ρ c_{p})}_{nf}}, σ = \frac{σ_{s}}{σ_{f}}, \frac{σ_{nf}}{σ_{f}} = (1 + \frac{3 (σ - 1) ϕ}{(σ + 2) - (σ - 1) ϕ}) \end{matrix}

(5)

Here, $ϕ$ is the volume fraction of the solid particles and $(ρ c_{p})_{f}$ , $ρ_{f}$ and $σ_{f}$ are the specific heat capacity, density and electrical conductivity of the base liquids, respectively. The subscript $s$ is for solid nanoparticles.

Electrical conductivity model

Mathematically, we can write the relative heat flux $q^{rad}$ through Rosseland approximation as $q^{rad} = - \frac{4 ε^{x}}{3 K_{o}^{x}} \frac{\partial T^{4}}{\partial y}$ . Using Taylor series, $T^{4}$ can be expanded in the form $T^{4} = 4 T_{\infty}^{3} T - 3 T_{\infty}^{4}$ and $\frac{\partial q^{rad}}{\partial y} = - \frac{16 ε^{x}}{3 K_{o}^{x}} T_{\infty}^{3} \frac{\partial^{2} T}{\partial y^{2}}$ . Therefore, using the above-mentioned assumption, equation (3) becomes

u (\frac{\partial T}{\partial x}) + v (\frac{\partial T}{\partial y}) = α_{nf} (\frac{\partial^{2} T}{\partial y^{2}}) + \frac{1}{{(ρ c_{p})}_{nf}} (\frac{16 ε^{x}}{3 K_{o}^{x}} T_{\infty}^{3} \frac{\partial^{2} T}{\partial y^{2}})

(6)

The transformations are suggested as defined in Tiwari et al.²⁶

\begin{matrix} u (x, y) = u_{o} x^{(\frac{2 r - 1}{3})} \frac{df (η)}{d η}, v (x, y) = \frac{1}{3} u_{o} p_{o} x^{(\frac{r - 2}{3})} \\ [(2 - r) η \frac{df (η)}{d η} - (1 + r) f (η)] \\ T (x, y) = T_{\infty} - h_{o} x^{r} (Θ (η)), η = \frac{y}{p_{o}} [x^{(\frac{r - 2}{3})}] \end{matrix}

(7)

where $p_{o}, u_{o}, h_{o}$ and r represent the constant terms. Moreover, if we put $h_{o} = 1$ , then $u_{o}$ and $p_{o}$ take the resulting exclusive form²⁶ as

u_{o} = {(\frac{3}{1 + r})}^{\frac{1}{3}} r^{\frac{2}{3}}, p_{o} = {(\frac{3}{1 + r})}^{\frac{1}{3}} r^{\frac{- 1}{3}}

(8)

Similarly, the constants $p_{o}, u_{o} and h_{o}$ satisfy the following conditions

{(p_{o})}^{2} u_{o} = \frac{3}{1 + r}, \frac{p_{o} h_{o}}{u_{o}} = \frac{1}{r}

(9)

Keeping in mind all the above assumptions, equation (1) is satisfied successfully, while equations (2)–(5) are transformed into the following forms

\begin{matrix} \frac{1}{φ_{1}} \frac{d f^{3} (η)}{d η^{3}} + f (η) \frac{d f^{2} (η)}{d η^{2}} - \frac{2 r - 1}{r + 1} {(\frac{df (η)}{d η})}^{2} \\ - M^{2} (\frac{df (η)}{d η}) {Sin}^{2} (Ω) = 0 \end{matrix}

(10)

\frac{φ_{3}}{φ_{2}} \frac{d Θ^{2} (η)}{d η^{2}} + \Pr [f (η) \frac{d Θ (η)}{d η}] - \frac{3 r}{1 + r} \Pr [\frac{df (η)}{d η} Θ (η)] = 0

(11)

\begin{matrix} f = 0, {(1 - ϕ)}^{- 2.5} \frac{d f^{2}}{d η^{2}} = - 1, Θ = 1, at η = 0 \\ \frac{df}{d η} = 1, Θ = 0, at η \to \infty \end{matrix}

(12)

where $M, Rd and \Pr$ indicate the transformation of magnetic parameter, radiation parameter and Prandtl number defined individually as

\begin{matrix} M^{2} = \frac{σ_{f} B_{o}^{2}}{μ_{f} p_{f}}, Rd = \frac{4 ε^{x} T_{\infty}^{3}}{K_{o}^{x} k_{f}}, \Pr = \frac{ν_{f}}{α_{f}} = \frac{{(μ c_{p})}_{f}}{k_{f}}, \\ φ_{1} = {(1 - ϕ)}^{2.5} {(1 - ϕ) + \frac{ρ_{s}}{ρ_{f}} ϕ} \\ φ_{2} = {(1 - ϕ) + \frac{{(ρ c_{p})}_{s}}{{(ρ c_{p})}_{f}} ϕ}, \\ φ_{3} = (\frac{k_{nf}}{k_{f}} + \frac{4}{3} Rd) \end{matrix}

(13)

The local Nusselt number ( $N u_{x}$ ) is

N u_{x} = \frac{r {- {(\frac{\partial T}{\partial y})}_{y = 0} + {(q^{rad})}_{y = 0}}}{[T (x, 0) - T (x, \infty)]}

(14)

The dimensional form of the Nusselt number is

N u_{x} {Re}^{- 0.5} = - (\frac{k_{nf}}{k_{f}} + \frac{4}{3} Rd) \frac{d Θ (0)}{d η}, Re = u_{0} x

(15)

OHAM solution

Equations (10) and (11) with the physical conditions in equation (12) have been tackled through the BVPh 2.0 package of OHAM.^29–31 The residual errors have been minimized using the 15th-order approximation for the stable convergence of the attained results. The auxiliary constraint $h$ has been calculated to regulate the convergence of the proposed problem. The mentioned linear operators and initial guess which have an important role in the OHAM solution are selected as

f_{0} (η) = η + {(1 - ϕ)}^{2.5} (1 - \exp (- η)), Θ_{0} (η) = \exp (- η)

(16)

The linear terms are represented by $L_{f}$ and $L_{Θ}$ defined as

L_{f} = \frac{d^{3} f}{d η^{3}} - \frac{df}{d η}, L_{Θ} = \frac{d^{2} Θ}{d η^{2}} - Θ

(17)

The general results of $L_{f}$ and $L_{Θ}$ are

\begin{matrix} L_{f} [℘_{1} + ℘_{2} \exp (- η) + ℘_{3} \exp (η)] = 0 \\ L_{Θ} [℘_{4} + ℘_{5} \exp (- η) + ℘_{6} \exp (η)] = 0 \end{matrix}

(18)

Here, $℘_{i} (i = 1 - 6)$ are any arbitrary constants.

The OHAM-BVPh 2.0 package^29–31 has been applied and equations (10) and (11) are validated as

ε_{m}^{f} = \frac{1}{k + 1} \sum_{j = 1}^{k} {[ℵ_{f} {(\sum_{i = 1}^{m} f (η))}_{ξ = j δ ξ}]}^{2}

(19)

ε_{m}^{Θ} = \frac{1}{k + 1} \sum_{j = 1}^{k} {[ℵ_{Θ} {(\sum_{i = 1}^{m} f (η), \sum_{i = 1}^{m} Θ (η))}_{ξ = j δ ξ}]}^{2}

(20)

Liao^29,30 defined the total squared residual error as

ε_{m}^{t} = ε_{m}^{f} + ε_{m}^{Θ}

(21)

The auxiliary parameters have been used, and the total residual error $ε_{m}^{t}$ has been obtained for the temperature and velocity profiles, respectively. The approximate values obtained from optimal constraints $h_{f} = - 0.9897, h_{Θ} = - 1.3682$ have been archived in the situation of GO-W, while $h_{f} = - 0.9801, h_{Θ} = - 1.4326$ have been obtained in the case of GO-EG nanofluids.

Result and discussion

The OHAM-BVPh 2.0 package as mentioned in Liao^29,30 has been utilized to attain the results of the proposed problem for the 15th-order approximation. The desired accuracy of the modelled problem with the possible range of the embedding parameters for the GO-W and GO-EG nanofluids has also been obtained for the temperature and velocity pitch.

The residual error has been depicted in Figures 2 and 3 for velocity and temperature fields. The approximate values obtained from optimal convergence control parameters $h_{f} = - 0.9897, h_{Θ} = - 1.3682$ have been archived in the situation of the GO-W, while $h_{f} = - 0.9801, h_{Θ} = - 1.4326$ have been obtained in the case of GO-EG nanofluids.

Figure 2.

Residual error of $f (η)$ versus the approximation order m.

Figure 3.

Residual error of $Θ (η)$ versus the approximation order m.

The impact of magnetic parameter $M$ versus $f' (η)$ for the GO-W and GO-EG nanofluids is depicted in Figure 4. The increasing values of $M$ lead to the decline of the velocity pitch due to effect of Lorentz force which always opposes the motion phenomenon. This decelerating force controls the GO-W and GO-EG nanofluid velocities which are practically used in several engineering and industrial applications, for example, heat transportation and cooling radiator. Furthermore, GO-EG shows dominant results than GO-W.

Figure 4.

Influence of M on $f' (η)$ .

The increasing values of $ϕ$ decline the velocity pitch using the GO-W and GO-EG nanofluids as shown in Figure 5. Physically, the intermolecular forces among the nanoplatelets become weak with the larger amount of $ϕ$ , which enhances the velocity pitch. Furthermore, GO-EG displays dominant results than GO-W for the identical values of the solid volume fraction. The nanoparticle $ϕ$ is basically used as the heat transport agent parameter, and its larger amount boosts up the temperature profile as displayed in Figure 6. Actually, the cohesive forces among the liquid molecules release with the increasing amount of $ϕ$ which enhances the thermal boundary layer. This effect is comparatively efficient in the GO-EG nanofluids due to their enriched thermophysical properties. Figure 7 describes the effect of radiation parameter ( $Rd$ ) versus temperature field. The radiation term is also the heat enhancement agent, and its increasing value increases the temperature profile. We noted that in case of GO-EG and GO-W, the temperature distribution is dominant and almost very closed. The detailed study of the GO-W and GO-EG nanofluids for the $Ω$ versus magnetic parameter ( $M$ ) and $η$ has been shown in the contour Figures 8 –11. The thermophysical properties of EG nanoparticles are stronger than the base fluid water, and therefore, the strong thermal efficiency of the GO-EG nanofluid has been observed.

Figure 5.

Influence of nanoparticle $ϕ$ on $f' (η)$ .

Figure 6.

Influence of nanoparticle $ϕ$ on $Θ (η)$ .

Figure 7.

Influence of Rd on $Θ (η)$ .

Figure 8.

Influence of $Ω$ on $M$ in GO-W nanofluid when $\Pr = 10.1, r = K = 0.2, ϕ = 0.1 .$

Figure 9.

Influence of $Ω$ on $M$ in GO-EG nanofluid when $\Pr = 10.1, r = K = 0.2, ϕ = 0.1 .$

Figure 10.

Influence of $Ω$ on $η$ in GO-EG nanofluid when $\Pr = 10.1, r = M = K = 0.2, ϕ = 0.1 .$

Figure 11.

Influence of $Ω$ on $η$ in GO-W nanofluid when $\Pr = 10.1, r = M = K = 0.2, ϕ = 0.1 .$

The contour figures show the best seen of the physical parameters for the both sorts of nanofluids. Since these parameters are interlinked in the momentum boundary layer, the contour figures have been obtained from the velocity profile.

The thermophysical properties of the base solvents and solid particles are displayed in Table 1. The residual errors obtained for the temperature and velocity fields up to the 15th-order OHAM approximation for the two sorts of nanofluids are analysed in Table 2. The increasing order of the approximation reduces the residual error to authenticate the convergence of the problem. The higher values of the nanoparticle volume fraction ( $ϕ$ ) and radiation term ( $Rd$ ) enhance the temperature profile, and the cooling effect falls to reduce the local Nusselt number as shown in Table 3. The increasing amount of the Prandtl number declines the temperature field and as a result, the Nusselt number boosts up.

Table 1.

Thermophysical properties of GO-W/GO-EG nanofluids from the experiment.

Model	$ρ (kg / m^{3})$	$C_{p} (k g^{- 1} / k^{- 1})$	$k (W m^{- 1} k^{- 1})$
Water (W)	997.1	4179	0.613
Graphene oxide (GO)	1800	717	5000
Ethylene glycol (EG)	1.115	0.58	0.1490

Table 2.

Residual error for both sorts of nanofluids at the 15th-order OHAM approximation.

$m$	$ε_{m}^{f} (GO - W)$	$ε_{m}^{f} (GO - EG)$	$ε_{m}^{Θ} (GO - W)$	$ε_{m}^{Θ} (GO - EG)$
4	$5.71355 \times 10^{- 3}$	$5.76992 \times 10^{- 3}$	$2.33921 \times 10^{- 6}$	$2.35216 \times 10^{- 6}$
8	$4.83030 \times 10^{- 5}$	$4.92638 \times 10^{- 4}$	$1.35925 \times 10^{- 8}$	$1.38230 \times 10^{- 8}$
12	$4.08104 \times 10^{- 7}$	$4.13644 \times 10^{- 7}$	$1.24609 \times 10^{- 10}$	$1.24650 \times 10^{- 10}$
15	$4.67819 \times 10^{- 8}$	$4.61533 \times 10^{- 8}$	$9.32525 \times 10^{- 11}$	$1.50644 \times 10^{- 11}$

Table 3.

Numerical values for local Nusselt number for different physical parameters.

$ϕ$	$Rd$	$\Pr$	$- Θ' (0)$ , GO-W	$- Θ' (0)$ , GO-EG
0.1	0.3	0.6	0.818480	0.929591
0.2			0.807704	0.918815
0.3			0.798932	0.899843
0.1	0.3		0.818480	0.929591
	0.4		0.798709	0.889812
	0.5		0.780320	0.8712542
	0.3	0.6	0.818480	0.929591
		0.7	0.837086	0.948197
		0.8	0.855704	0.966815
		0.6	0.818480	0.929591

Conclusion

Graphene is a fascinating material of carbon and comprises exceptional mechanical and organic properties. The present research is related by means of the theoretical approach to the heat transfer applications of the GO-W/GO-EG nanofluids. The numerical and theoretical analyses have been carried out under the effect of Marangoni convection. The results have been calculated from the minimum residual error analysis using the OHAM-BVPh 2.0 package. The effects of the imperative parameters, such as magnetic parameter (M), radiation parameter (Rd) and Prandtl number (Pr), are pointed out as follows:

The increasing value of the Marangoni convection parameter creates hurdles in the fluid motion and declines the velocity field;

The greater values of $ϕ$ up to 5% of the base solvent decline the velocity field and boost up the temperature profile;

The impact of the local Nusselt number versus embedded parameters has been observed;

It has been concluded that the GO-EG nanofluids have higher thermal efficiency as compared to the GO-W nanofluids.

Footnotes

Appendix 1

Handling Editor: Ahmed Abdel Gawad

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iDs

Taza Gul

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Impact of the Marangoni and thermal radiation convection on the graphene-oxide-water-based and ethylene-glycol-based nanofluids

Abstract

Keywords

Introduction

Mathematical formulation

Electrical conductivity model

OHAM solution

Result and discussion

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

ORCID iDs

References