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Abstract

In this study, the transport of Al₂O₃ nanoparticles in ethylene glycol conventional fluid over a linearly stretching cylinder is investigated. The current research employs a convective surface boundary condition for heat transfer exploration. Flux model proposed by Rosseland is employed to examine effect of thermal radiations. The governing flow problem comprises highly nonlinear ordinary differential equations. Similarity transformations are used to reduce the equations in similar forms, which are then solved by Runge–Kutta–Fehlberg fourth-fifth order numerical scheme with shooting algorithm in MATLAB software. In order to authenticate the accuracy of our results, we have contrasted results with those obtained by Ishak et al., Wang, and Pandey and Kumar and found that they are in better concord, as revealed in Table 2. The impact of numerous emerging parameters on velocity distribution and heat transfer distribution are argued in all aspects and depicted through graphs.

Keywords

Boundary layer flow nanofluid convective heating condition stretching cylinder

Introduction

The terminology of “nanofluid” invented by Choi and Eastman¹ describes engineered colloids incorporating a conventional liquid (e.g. air, water, oil and ethylene glycol (EG)) and nanoparticles. Later, Buongiorno² offered a new model to examine heat transfer properties of conventional fluid. In this model, impact of Brownian motion and thermophoretic are considered. Nanofluids typically employ metals, metal oxides, carbides, or carbon nanotubes nanoparticles, and the conventional fluid includes EG, kerosene oil, and water. To evaluate heat transfer enhancements, nanofluid usually contains nanoparticles (e.g. gold, copper, titanium and aluminum) volume fraction up to 5%. In the past few years, hydromagnetic transport of nanoparticles in base fluid has become an attractive area due to their extensive range of applications. Specifically, the magnetohydrodynamic (MHD) nanofluids have applications like optical modulators, transformer oil, and solar water heater. Nanofluids also play a vital role in biomedicine applications like sink float separation, cancer therapy, and magnetic cell separation. Some recent efforts related to this article can be revealed by the investigators^3–8 and a number of references therein.

Sheikholeslami et al.⁹ investigated MHD impacts on Cu-H₂O nanofluid in an enclosure. They conclude that due to magnetic field, Nusselt number becomes smaller versus inclination angle. In another scientific exploration, Sheikholeslami and Ellahi¹⁰ examined the impact of electrohydrodynamics on Fe₃O₄/EG nanofluid in a sinusoidal enclosure. Turkyilmazoglu¹¹ considered the flow problem of nanofluid past a rotating disk. In his research, water-based nanofluid comprising volume fraction of (Ag, Cu, CuO, Al₂O₃, TiO₂) nanoparticles are considered in description. Ellahi et al.¹² used modified Darcy’s law flow modeling to study non-Newtonian nanofluid with coaxial cylinders. Mabood et al.¹³ performed numerical analysis of MHD boundary layer flow due to a nonlinear stretching sheet. They found that due to magnetic number, Nusselt number falls in magnitude, while the skin friction coefficient rises. Rashidi et al.¹⁴ in his research focused on MHD flow over porous disk immersed in an incompressible nanofluid incorporating three varieties of nanoparticles, Cu, Cuo, Al₂O₃, and water as base fluid. Hussain et al.¹⁵ explored the problem of MHD flow of third-grade nanofluid with prescribed heat flux condition (PHF) and prescribed concentration flux conditions (PCF). In another computational work, Ellahi et al.¹⁶ investigated the convective nanofluid flow over wedge including the influence of porous media. They employed Nimonic 80a metal particles of various shapes and sizes. Zeeshan et al.¹⁷ studied the convective MHD nanofluid flow along with inverted cone. Makinde and Aziz¹⁸ examined boundary layer nanofluid past a stretching sheet by employing convective surface condition for heat transfer explorations. Hayat et al.¹⁹ analyzed the flow problem of MHD nanofluid in porous media by utilizing a convective surface heating condition. Rashidi et al.²⁰ theoretically assessed boundary layer nanofluid flow by applying differential transform method (DTM).

The literature survey reveals that there are only few articles related to hydromagnetic transport of nanoparticles over stretching cylinder with convective boundary conditions. Some authors investigated flow problems due to stretching surfaces.^21–24 Mustafaa et al.²⁵ and Makinde and Aziz¹⁸ studied the boundary layer flow of nanofluid over a stretching sheet with convective boundary conditions. Yao et al.²⁶ analyzed the heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions. Ramzan et al.²⁷ discussed the influence of homogeneous–heterogeneous reactions on MHD three-dimensional (3D) Maxwell fluid flow with Cattaneo–Christov heat flux and convective boundary condition. Recently, few authors made their contribution about this topic.^19,28,29

The aim of the present problem is to examine the influence of thermal radiation and Ohmic heating on hydromagnetic transport of $A l_{2} O_{3}$ nanoparticles in EG base fluid over a convectively heated linearly stretching cylinder. As per author’s information, this type of study is still unexplored. The main objective of present research is to use a convective condition to explore heat transfer effects. The influence of non-uniform electric field is also taken into account. The effects of sundry parameters like magnetic parameter, electric parameter, Eckert number, radiation number, and Biot number have been analyzed on flow and thermal characteristics (Tables 1 and 2).

Table 1.

Properties of ethylene glycol and Al₂O₃.

Physical properties	Ethylene glycol	Al₂O₃
$ρ (kg m^{- 3})$	1110	3980
$c_{p} (J k g^{- 1} K^{- 1})$	2400	840
$k (W m^{- 1} K^{- 1})$	0.26	33

Table 2.

Comparison of numerous values of Nusselt number $- θ' (1)$ , for copper–water nanofluid and $Re = 10, N_{r} = 0, A_{1} = 1, E_{1} = 0, M = 0, Ec = 0, Bi \to \infty$ .

Pr	Wang³⁰	Ishak et al.²⁴	Pandey and Kumar³¹	Present
0.7	1.568	1.5683	1.58679	1.5962
2	3.035	3.0360	3.03534	3.0259
7	6.160	6.1592	6.15590	6.1198
10	10.77	7.4668	7.46230	7.4088

Convective transport equations

In the current communication, an incompressible, steady, boundary layer flow of conducting viscous EG base fluid with Alumina Al₂O₃ nanoparticles over a linear stretching cylinder is considered. The x-axis and r-axis are considered along the cylinder axis and along the radial directions, respectively. The magnetic field $\vec{B} = (0, B_{0}, 0)$ and electric field $\vec{E} = (0, 0, - E_{0})$ are applied to interpret flow region. To generate flow, the elastic cylinder is stretched linearly such that its velocity is linear function in the axial direction as shown in Figure 1.

Figure 1.

Schematic of flow.

From Maxwell’s equations, $\nabla . \vec{B} = 0$ and $\nabla \times \vec{E} = 0$ .

When magnetic field is so weak then magnetic and electric field may obey Ohm’s law $\vec{J} = σ_{0} (\vec{E} + \vec{V} \times \vec{B})$ , where $\vec{V}$ is the velocity field vector. The induced magnetic field is neglected due to its small value. The influence of magnetic field and electric field is taken in the flow problem. The bottom boundary of cylinder is heated convectively by a liquid of temperature $T_{f}$ , which gives a coefficient h. Under above-mentioned physical situations, the governing equations can be described as

\frac{\partial (ru)}{\partial x} + \frac{\partial (rv)}{\partial r} = 0

(1)

ρ_{nf} (u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial r}) = \frac{μ_{nf}}{r} \frac{\partial}{\partial r} (r \frac{\partial u}{\partial r}) + σ_{0} (E_{0} B_{0} - B_{0}^{2} u)

(2)

\begin{array}{l} {(ρ c_{p})}_{n f} (u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial r}) \\ = \frac{k_{n f}}{r} \frac{\partial}{\partial r} (r \frac{\partial T}{\partial r}) + μ_{n f} {(\frac{\partial u}{\partial r})}^{2} + σ_{0} {(u B_{0} - E_{0})}^{2} - \frac{\partial Q_{r}}{\partial r} \end{array}

(3)

The applicable boundary conditions for above reflected problem are taken as

u = 2 cx, v = 0, - k_{f} \frac{\partial T}{\partial r} = h (T_{f} - T), at r = a

(4)

u \to 0, T \to T_{\infty} as r \to \infty

(5)

The radiation flux model proposed by Rosseland can be written as

Q_{r} = - \frac{4 σ^{*}}{3 k^{*}} \frac{\partial T^{4}}{\partial r}

(6)

By the Taylor series expansion of $T^{4}$ about $T_{\infty}$ and omitting higher terms, we got

T^{4} ≅ 4 T_{\infty}^{3} T - 3 T_{\infty}^{4}

(7)

\frac{\partial Q_{r}}{\partial r} = - \frac{16 σ^{*} T_{\infty}^{3}}{3 k^{*}} \frac{\partial^{2} T}{\partial r^{2}}

(8)

The models for nanofluid density $ρ_{nf}$ and heat capacitance $(ρ c_{p})_{nf}$ are given by

\begin{array}{l} ρ_{n f} = (1 - ϕ) ρ_{f} + ϕ ρ_{p}, {(ρ c_{p})}_{n f} \\ = (1 - ϕ) {(ρ c_{p})}_{f} + ϕ {(ρ c_{p})}_{p} \end{array}

(9)

The models for effective dynamic viscosity and thermal conductivity of nanofluid are proposed by Brinkman and Maxwell Grannet, respectively, which are given by

\frac{μ_{nf}}{μ_{f}} = \frac{1}{{(1 - ϕ)}^{2.5}}, \frac{k_{nf}}{k_{f}} = \frac{k_{p} + 2 k_{f} + 2 ϕ (k_{p} - k_{f})}{k_{p} + 2 k_{f} - ϕ (k_{p} - k_{f})}

(10)

To transform the boundary value problem in self-similar form, the following similarity transformations are utilized

η = {(\frac{r}{a})}^{2}, u = 2 cxf' (η), v = - \frac{ca}{\sqrt{η}} f (η), θ = \frac{T - T_{\infty}}{T_{w} - T_{\infty}}

(11)

Using above-mentioned transformations, the problem in non-dimensional form can be presented as

A_{1} (η f ″' + f ″) + A_{2} Re (ff ″ - {f'}^{2}) + M (E_{1} - f') = 0

(12)

\begin{array}{l} A_{3} (θ^{'} + η θ^{″}) + N_{r} η θ^{″} + A_{4} R e P r θ^{'} f \\ + A_{1} η E c P r {f^{″}}^{2} + M P r E c {(f^{'} - E_{1})}^{2} = 0 \end{array}

(13)

The corresponding boundary conditions are

f (1) = 0, f' (1) = 1, θ' (1) = - Bi [1 - θ (1)]

(14)

f' (\infty) \to 0, θ (\infty) \to 0

(15)

The constant terms in equations (12) and (13) are given as

\begin{matrix} A_{1} = \frac{1}{{(1 - ϕ)}^{2.5}}, A_{2} = (1 - ϕ) + ϕ \frac{ρ_{p}}{ρ_{f}}, \\ A_{3} = \frac{k_{p} + 2 k_{f} + 2 ϕ (k_{p} - k_{f})}{k_{p} + 2 k_{f} - ϕ (k_{p} - k_{f})}, A_{4} = (1 - ϕ) + ϕ \frac{{(ρ c_{p})}_{p}}{{(ρ c_{p})}_{f}} \end{matrix}

Here, $Re = c a^{2} / 2 ν$ the Reynolds number, $M = σ_{0} B_{0}^{2} a^{2} / 4 μ_{f}$ the magnetic number, $\Pr = μ_{f} c_{p} / k_{f}$ the Prandtl number, $Ec = (4 c^{2} x^{2}) / c_{p} (T_{w} - T_{\infty})$ the Eckert number, $E_{1} = \frac{E_{0}}{2 {cxB}_{0}}$ the electric parameter, $N_{r} = (16 σ^{*} T_{\infty}^{3}) / (3 k^{*} k_{f})$ the radiation number, and $Bi = ha / 2 k$ the Biot number are the dimensionless parameters occurring in governing equations, and prime denote the derivative w.r.t $η$ . The expressions for coefficient of skin friction $C_{f}$ and Nusselt number $Nu$ can be specified as

C_{f} = \frac{τ_{w}}{\frac{1}{2} ρ_{f} u_{w}^{2}}, Nu = \frac{r Q_{w}}{k_{f} (T_{w} - T_{\infty})}

(16)

where τ_{w} = μ_{f} {(\frac{\partial u}{\partial r})}_{r = a}, Q_{w} = - {[(k_{f} + \frac{16 σ^{*} T_{\infty}^{3}}{3 k^{*}}) \frac{\partial T}{\partial r}]}_{r = a}

(17)

Now in the dimensionless form, the expression for $C_{f}$ and $Nu$ are given as

C_{f} (\frac{Re x}{a}) = f ″ (1), Nu = - 2 (1 + N_{r}) θ' (1)

(18)

Numerical solution

In this study, the resulting flow phenomenon is described by a system of nonlinear coupled ordinary differential equations (ODEs) (equations (12) and (13)) along with the boundary conditions (equations (14) and (15)). The system of equations (12) and (13) are first reduced into system of first-order ODEs, which are then solved by Runge–Kutta–Fehlberg numerical integration scheme with shooting algorithm. The selection of $η_{\infty}$ varies from 2 to 10, subject to the involving parameters so that no numerical oscillations would occur. Assuming

f = F_{1}, f' = F_{2}, f ″ = F_{3}, θ = F_{4}, θ' = F_{5}

The system of five simultaneous first-order equations can be written as follows

[\begin{matrix} {F_{1}}^{'} \\ {F_{2}}^{'} \\ {F_{3}}^{'} \\ {F_{4}}^{'} \\ {F_{5}}^{'} \end{matrix}] = [\begin{matrix} F_{2} \\ F_{3} \\ \frac{- 1}{A_{1} η} {A_{1} F_{3} + A_{2} Re (F_{1} F_{3} - {F_{2}}^{2}) + M (E_{1} - F_{2})} \\ F_{5} \\ \frac{- 1}{(A_{3} + N_{r}) η} {A_{3} F_{5} + A_{4} RePr F_{1} F_{5} + A_{1} η {PrEcF}_{3}^{2} + MPrEc {(F_{2} - E_{1})}^{2}} \end{matrix}]

The boundary conditions are written as follows

\begin{matrix} F_{1} = 0, F_{2} = 1, F_{5} = - Bi (1 - F_{4}) at η = 1 \\ F_{2} \to 0, F_{4} \to 0 as η \to \infty \end{matrix}

Since $F_{3} (0) and F_{4} (0)$ are not given, we have to start with the initial values of $F_{3} (0) = g_{10} and F_{4} (0) = g_{20}$ . Let $γ_{1} and γ_{2}$ are the accurate $F_{3} (0) and F_{4} (0)$ , accordingly. The resultant system of five ODEs is tackled using fifth-order Runge–Kutta–Fehlberg method and represent $F_{3} and F_{4} at η = η_{\infty}$ by $F_{3} (g_{10}, g_{20}, η_{\infty})$ and $F_{4} (g_{10}, g_{20}, η_{\infty})$ , respectively. Since $F_{3} and F_{4}$ at $η = η_{\infty}$ are clearly functions of $γ_{1} and γ_{2}$ , they are expanded in Taylor series about $γ_{1} - g_{10} and γ_{2} - g_{20}$ , respectively. After solving the system of Taylor series expansions for $δ γ_{1} = γ_{1} - g_{10} and δ γ_{2} = γ_{2} - g_{20}$ , we obtained the new estimates $g_{11} = g_{10} + δ g_{10} and g_{21} = g_{20} + δ g_{20}$ . Next the complete procedure is reiterated with $F_{1} (0), F_{2} (0), g_{11}, g_{21}, F_{5} (0)$ as initial conditions. With the latest estimates of $γ_{1}, γ_{2}$ , the whole procedure is reiterated up to prescribed boundary conditions are satisfied. Finally, $g_{1 n} = g_{1 (n - 1)} + δ g_{1 (n - 1)} and g_{2 n} = g_{2 (n - 1)} + δ g_{2 (n - 1)}$ for $n = 1, 2, 3, \dots$ are achieved, which appeared to be the most preferred approximate initial values of $F_{3} (0) and F_{4} (0)$ . By this technique, when all needed conditions are obtained, the resulting system of five first-order equations has been solved by Runge–Kutta–Fehlberg fourth-fifth (RKF45) numerical technique and the simulation error is chosen $10^{- 5}$ in order to induce asymptotic convergence. A comparison of present results with those already published is revealed in Table 2 for validation purpose.

Discussion of results

In this article, the impact of thermal radiation and Ohmic heating on hydromagnetic transport of alumina nanoparticles in EG over linearly stretching cylinder is analyzed. The convective boundary condition is also employed for heat transfer effects. Differential equations (12) and (13) with boundary conditions (equations (14) and (15)) are solved by RKF45 with shooting algorithm in MATLAB software. Role of various involving parameters like magnetic field number $(M)$ , Eckert number $(Ec)$ , electric field parameter $(E_{1})$ , radiation number $(N_{r})$ , and Biot number $(Bi)$ on flow and heat transfer features has been debated in detail and presented through graphs. Throughout the calculations, the values of parameters are taken as Pr = 135, Re = 7.0. The thermophysical properties of the base fluids (Ethylene glycol) and the nanoparticles (alumina) are listed in Table 1.

Figures 2 and 3 represent variations of momentum and thermal boundary layer versus $η$ for numerous magnetic number $(M)$ values. Figure 2 displays that the momentum boundary layer thickness drops by rising magnetic number $(M)$ . Figure 3 shows that the temperature profiles are enhanced as magnetic field grows stronger because of the fact that magnetic field produces a Lorentz force which opposes the flow and so the temperature increases. Figures 4 and 5 describe the impact of electric parameter $(E_{1})$ on flow and temperature distributions. From Figure 4, it is examined that influence of electric parameter $(E_{1})$ on the velocity is more expressively away from the stretching surface and prominent throughout the boundary layer. This figure expose that the effect of $(E_{1})$ is to move the flow patterns away from the stretched boundary. Transferring of streamlines away from the stretching sheet is more prominent. Figure 5 represents the impact of electric field parameter $(E_{1})$ on temperature profiles. We note that the temperature distribution rises with increasing values of the electric parameter $(E_{1})$ . Figure 6 exhibits the impact of radiation number $(N_{r})$ on temperature distributions. We concluded that the temperature is directly proportional to thermal radiation number $(N_{r})$ . A special case $N_{r} = 0$ relates to the flow investigation without thermal radiation. Figure 7 represents the variation in thermal boundary layer profiles for numerous values of Eckert number $(Ec)$ . From figure, we examined that the temperature profiles increased when we give rise to the value of Eckert number $Ec$ . If $Ec = 0$ , it corresponds to the flow without viscous dissipation as well as Ohmic dissipation. Figure 8 highlights the deviations of local skin friction coefficient $f ″ (1)$ versus electric field parameter $(E_{1})$ for numerous values of magnetic number $(M)$ . This figure shows that the skin friction coefficient $f ″ (1)$ rises as we increase electric parameter $(E_{1})$ and magnetic number $(M)$ . Figure 9 illustrates that the influence of Biot number $(Bi)$ on Nusselt number $(Nu)$ for several values of electric parameter $(E_{1})$ . Nusselt number $(Nu)$ shows increasing behavior with the increase in Biot number $(Bi)$ and electric parameter $(E_{1})$ . Figure 10 shows the behavior of Nusselt number $(Nu)$ with radiation number $(N_{r})$ for different values of electric parameter $(E_{1})$ . From figure, we got that Nusselt number $(Nu)$ rises by growing radiation number $(N_{r})$ , but by growing electric parameter $(E_{1})$ , the Nusselt number $(Nu)$ drops.

Figure 2.

Impact of magnetic number M on velocity distribution $f' (η)$ .

Figure 3.

Impact of magnetic number M on temperature distribution $θ (η)$ .

Figure 4.

Impact of electric parameter $E_{1}$ on velocity distribution $f' (η)$ .

Figure 5.

Impact of electric parameter $E_{1}$ on temperature distribution $θ (η)$ .

Figure 6.

Impact of radiation number $N_{r}$ on temperature distribution $θ (η)$ .

Figure 7.

Impact of Eckert number $Ec$ on temperature distribution $θ (η)$ .

Figure 8.

Impact of electric parameter $E_{1}$ on skin friction coefficient $f ″ (1)$ .

Figure 9.

Impact of Biot number $Bi$ on Nusselt number $Nu$ .

Figure 10.

Impact of radiation parameter $N_{r}$ on Nusselt number $Nu$ .

Concluding remarks

The current investigation has many useful applications in various fields such as piping, wire drawing, metal spinning, flow meter design, solar collectors, casting systems, manufacturing of rubber sheets, crude oil refinement, electronics cooling, drilling operations, commercial refrigeration, geothermal power generation, cylindrical wires coating, and polymer fiber coating. In this study, the problem of hydromagnetic transport of $A l_{2} O_{3}$ nanoparticles in EG over a linearly stretching cylinder along with a convective heating condition is analyzed. The effect of thermal radiation and Ohmic heating is also considered in this article. The governing flow problem is highly nonlinear system of differential equations, which is solved by RKF45 numerical scheme with shooting algorithm in MATLAB software. The utilization of a convective heating condition instead of commonly employed conditions makes this article extra innovative. The key outcomes are written as follows:

The velocity and temperature distributions depend on parameters as follows: Reynolds number $Re$ , magnetic number $M$ , Prandtl number $\Pr$ , Eckert number $Ec$ , electric parameter $E_{1}$ , radiation number $N_{r}$ , and the Biot number $Bi$ ;

For extremely large Biot number $(Bi \to \infty)$ illustrating the convective condition resembles to isothermal condition;

By increasing the Biot number $Bi$ the Nusselt number $Nu$ rises for various values of electric parameter $E_{1}$ ;

Skin friction coefficient enhanced as we increase electric parameter $E_{1}$ and magnetic number $M$ ;

With the increasing magnetic number $M$ , the flow profiles fall but thermal boundary layer profiles grow up;

Both the flow and thermal profiles boosted with the increase in electric parameter $E_{1}$ .

Footnotes

Appendix 1 Acknowledgements

The authors wish to express their sincere thanks to the anonymous reviewers for their essential suggestions and comments to enhance the superiority of this manuscript.

Handling Editor: Moran Wang

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.

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Electromagnetohydrodynamic transport of Al 2 O 3 nanoparticles in ethylene glycol over a convectively heated stretching cylinder

Abstract

Keywords

Introduction

Convective transport equations

Numerical solution

Discussion of results

Concluding remarks

Footnotes

Appendix 1

Acknowledgements

Declaration of conflicting interests

Funding

References