ARTICLE

J. Cent. South Univ. (2019) 26: 1146-1160

DOI: https://doi.org/10.1007/s11771-019-4077-8

Magnetohydrodynamic flow and heat transfer impact on ZnO-SAE50 nanolubricant flow over an inclined rotating disk

NAYAK M K¹, MEHMOOD Rashid², MAKINDE O D³, MAHIAN O^{4, 5}, CHAMKHA Ali J^{6, 7}

1. Department of Physics, Radhakrishna Institute of Technology and Engineering,Bhubaneswar-752057, India;

2. Department of Mathematics, Faculty of Natural Science, HITEC University Taxila-47070, Pakistan;

3. Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa;

4. Center for Advanced Technologies, Ferdowsi University of Mashhad, Mashhad, Iran;

5. School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China;

6. Mechanical Engineering Department, Prince Sultan Endowment for Energy and Environment,

Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi Arabia;

7. RAK Research and Innovation Center, American University of Ras Al Khaimah, P.O. Box 10021,Ras Al Khaimah, United Arab Emirates

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: The present article has been fine-tuned with the investigation of mixed convection Darcy-Forchheimer flow of ZnO-SAE50 oil nanolubricant over an inclined rotating disk under the influence of uniform applied magnetic field applied to various industries. The current study has been enriched with additional consideration of slip flow, thermal radiation, viscous dissipation, Joulian dissipation and internal heating. In view of augmentation of thermal conductivity of nanolubricant, a new micro-nano-convection model namely Patel model has been invoked. The specialty of this model involves the effects of specific surface area and nano-convection due to Brownian motion of nanoparticles, kinetic theory based micro-convection, liquid layering and particle concentration. Suitably transformed governing equations have been solved numerically by using Runge-Kutta-Fehlberg scheme. An analysis of the present study has shown that applied magnetic field, porosity of the medium, velocity slip and inertia coefficient account for the slowing down of radial as well as tangential flow of ZnO-SAE50 oil nanolubricant, thereby leading to an improvement in velocity and thermal boundary layers.

Key words: ZnO-SAE50 nanofluid; Darcy-Forchheimer MHD flow; thermal radiation; velocity slip; viscous dissipation; internal heating

Cite this article as: NAYAK M K, MEHMOOD Rashid, MAKINDE O D, MAHIAN O, CHAMKHA Ali J. Magnetohydrodynamic flow and heat transfer impact on ZnO-SAE50 nanolubricant flow over an inclined rotating disk [J]. Journal of Central South University, 2019, 26(5): 1146–1160. DOI: https://doi.org/10.1007/s11771-019-4077-8.

1 Introduction

Have the nanofluids sprung a surprise when researchers least expected to it? We are in the age of smart technology. Our dignified researchers and scientists have used their capabilities for our most pressing research needs in the last quarter century. In their contributions, they have developed different nanofluids with remarkable thermal transport for the purpose of cooling in diversified microfluidic, thermal/heat transfer and medical/bio-medical applications [1–10].

Nanolubricants are nothing but the nanoparticles suspended in base lubricants. These are meant to enhance the performance of machine components at high temperatures. As a consequence, the friction and wear in sliding contact appeared in many vital engineering components, such as navy, aerospace and industrial applications get reduced. Indeed, nanoparticles are heat stable under various thermal conditions and suffer small degradation when exposed to high temperature. Further, the surface morphology of nanolubricant is smooth with small wear rate which is due to the development and effective spreading of the nanoparticles. Such factors are prime responsible for the reduction of friction coefficient and wear rate. The nanolubricants find vital role in automobiles (significant reduction in fuel consumption and hence environmental pollution) and metal-forming applications (deep-drawing, improved tool life). Many investigations regarding the heat transfer characteristics such as viscosity, thermal conductivity, viscosity index and viscous dissipation effects of nanolubricants have been carried out. They found the nanolubricants with improved thermal effect at temperature applications. In view of their use in the compressors of heat pump systems, and advantages such as reduction in friction (improved lubricity, lower energy costs) yielding fuel savings, enhanced efficiency, better thermal dissipation, low wearing of moving parts (leading to high load bearing property), increased horsepower, better gas mileage, lowering operating temperature (because less energy is converted into heat), augmentation in lubricant life and longer changeovers, breaking down of SO_x/NO_x gases in combustion engines, effective corrosion resistance and avoiding clog in filters, nanolubricants are in great demand these days. The nanolubricant approach is implemented to overcome the disadvantages of conventional anti-wear and friction reduction.

Usually, ZnO-SAE50 nano lubricant is developed by the mechanism of well suspension of zinc oxide (ZnO) nanoparticles with the engine oil SAE50. In the present industrial scenario, SAE50 lubricant is vital because of its significant usage as a working fluid in heat exchange systems, in vehicles to decay friction between several moving parts. Additionally, it improves the efficiency and durability, and reduces the fuel consumption in the way of preventing the corrosion and abrasion of moving parts. In spite of such advantages, SAE50 has low thermal conductivity. Above all, in respect of fulfillment of various industrial needs, for instance, thermal requirements in thermal systems needing more heat transfer rate, ZnO-SAE50 nanolubricant (through addition of nanoparticles ZnO with SAE50 oil) has been developed where the nanoparticles upgrade the heat transfer rate of nano-lubricant (ZnO-SAE50) [11, 12]. It is quite evident that the wear effect of nanomaterials on the engine parts involved in devices like cylinder, shaft, gaskets, valve mechanisms and gear camshaft can be undermined by using the nanolubricant ZnO- SAE50, an anti-wear nanolubricant. The flows and heat transfer over rotating disk associated with nanolubricants are significantly used in numerous industrial devices, geothermal and geophysical systems, computer disk drives and gas turbine rotors. Because of such precious and inevitable industrial advantages dominating over insignificant disadvantages, authors have been motivated to study the flow and heat transfer characteristics of ZnO-SAE50 nanolubricant over industrially used surface namely rotating disk associated with various parametric influence. HAYAT et al [13] explored the effect of double stratification in the magneto hydrodynamic (MHD) flow of a nanofluid over a rotating disk with variable thickness. In their study, they declared that enhanced Prandtl number, thermal and solutal stratification parameters yield augmentation of heat transfer rate.

It is nave to think that the applied magnetic field may play a significant role in developing a controlled cooling system which provides qualities of final products in industrial and manufacturing processes including polymer technology, wire drawing, hot rolling, production of food and papers [14–21]. Researchers are looking forward to large rate of investigation regarding the heat and fluid flow through porous media in view of its numerous applications including hydrology, geothermal energy systems, petroleum reservoirs, crude oil and gas production, heat exchanger design, catalytic reactors, grain storage, fermentation process, water movement in reservoirs, ground water systems/ pollution, recovery systems, beds of fossil fuels, energy storage units, nuclear waste disposal, solar receivers etc [22–24]. Naturally, the non-Darcian porous medium is an extended form of classical Darcy model that accounts for inertia and boundary features. This implicates that classical Darcy’s law is meant for modeling as well as analysis of flow past porous medium. However, the classical Darcian porous medium is valid for small flow velocity and porosity. Further, the classical Darcy’s law is not the most appropriate and well-conceived when fluid flow in porous media is non-linear, namely, inertia and boundary features appear at high flow rate. In such environment, inertia and boundary features cannot be ignored. In the case of high velocities, Reynolds number is greater than unity. In order to exhibit the inertia and boundary features, FORCHHEIMER et al [25] added a square velocity factor in Darcian velocity in momentum equation concerned. Moreover, MUSKAT [26] named this factor as “Forchheimer term” which is always valid for a large Reynolds number. Effect of viscous dissipation as well as thermophoretic force on Darcy-Forchheimer flow through porous media was investigated by SEDDEEK [27]. TIAN et al [28] also studied the related area.

There are numerous instances of fluid dynamic systems with micro-scale dimension where the flow field obeying the no-slip condition at the boundary cannot be applied. This is because the flow behavior of the associated fluids is mostly responsive to slip flow regime which is quite different from traditional flow. Besides, no slip boundary is not quite enough for the situations viz. polymer solutions, several suspensions, different kinds of emulsions and foams where the partial slip is the most appropriate boundary condition. On account of significant roles in polymer and electrochemical industry, many authors have been motivated to investigate the several flows in association with partial slip condition. The combined effects of slip and convective boundary conditions on CNT suspended nanofluid flow where an increase in slip parameter undermines the fluid flow and upsurges the fluid temperature developing an ascending thermal boundary layer [29], for a fixed non-zero slip, magnetic field belittles the heat transfer rate [30], influence of natural convection as well as thermal radiation on a stretched flow over a cylinder [31] and effects of partial slip and non-linear thermal radiation on stretched flow of nanofluid [32] have been analyzed.

Thermal radiation is very much essential for the devices designed for space applications where they provide high thermal efficiency operating at high absolute temperatures. Thermal radiation should be preferably kept low for obtaining significant heat transfer rate [33]. The flow as well as heat transfer gets boosted by thermal radiation in the span-wise direction confined in three- dimensional cavity which has been discussed recently by ZHANG et al [34]. Further, effect of thermal radiation in heat transfer associated with CNTs between two rotating disks has been analyzed by MOSAYEBIDORCHEH et al [35].

Convective boundary condition augments the thermal conductivity of the nanofluids due to the heat transfer through the surface of the disk more suitably compared to isothermal conditions. The behavior of convective flow of Jeffrey nanofluid subjected to two stretchable rotating disks [36], minimum steady state rotation subjected to Fe₃O₄ nanoparticles [37] and the larger convection at the surface produces greater thermal penetration [38] have been explored. In fact, viscous dissipation is associated with stronger gravitational fields, massive planets and heavier gases in space. Casson parameter undermines the wall shear stress and heat transfer rate while suction improves the heat transfer rate [39]. Heat generation produces thermal energy in the boundary layer, thereby upsurging the fluid temperature. That is why heat generation is an instrument of heat transfer rate in thermal boundary layer. MHD flow of nanofluid subjected to internal heat generation/absorption using Boungiorno model has been studied by GANGA et al [40]. Greater heat source accounts for an ascending thermal boundary layer [41].

Motivated by the above studies, the authors were inspired to investigate the impact of velocity slip, thermal radiation, viscous dissipation and internal heating on mixed convection Darcy- Forchheimer flow of ZnO-SAE50 nanolubricant over an inclined rotating disk under the influence of uniform applied magnetic field. To our mind, such investigation has not been done yet.

The objective of the present study is to explore the influence of velocity slip, thermal radiation, viscous dissipation, Julian dissipation and internal heating on magneto-hydrodynamic Darcy- Forchheimer flow of ZnO-SAE50 nanolubricant past an inclined rotating disk. In a fairly reasonable manner, we declare that the novelty of the present investigation includes the introduction of ZnO-SAE50 nanolubricant as flow fluid in an inclined rotating disk and implementation of Patel model (a micro-nano-convection model) for thermal conductivity enhancement of nanolubricant. In the present study, Runge-Kutta-Fehlberg scheme has been implemented to devise the required numerical solution. The effects of significant parameters on the dimensionless radial as well as tangential velocity and temperature are discussed.

2 Model development

In the present investigation, we deal with a steady Darchy-Forchheimer magneto- hydrodynamic flow of incompressible viscous ZnO- SAE50 nanofluid over an inclined rotating disk subjected to partial slip condition, thermal radiation and Joule heating. The disk at z=0 spins with constant angular velocity Ω. The velocity components (u, v, w) are in the directions of increasing (r, f, z) respectively. A magnetic field of uniform strength B₀ is applied in the axial direction (see Figure 1). Assume that small magnetic Reynolds number neglects the magnetic field. Because of axial symmetry of the problem, the derivatives with respect to the coordinate f are omitted.

The resulting boundary layer equations governing the flow are [35, 42]:

                           (1)

Figure 1 Flow geometry

      (2)

      (3)

      (4)

                (5)

with appropriate boundary conditions [37]:

       (6)

where

The effective density and heat capacitance of the nano-lubricant (ZnO-SAE50) [43] are defined as

 (7)

The effective dynamic viscosity of the ZnO-SAE50 nano-lubricant [12] is defined as

 (8)

                     (9)

where f is the solid volume fraction (%) and T is the temperature (°C).

The thermal expansion of nanolubricant [43] is

    (10)

Invoking Ref. [44], the effective thermal conductivity of nanolubricant is

 (11)

The transformations necessitated for the present problem include [37]:

 (12)

where η is the non-dimensional distance along the axis of rotation. Further, F, G and θ are functions of η.

Taking help of Eqs. (7)–(12), Eqs. (2), (3), (5) and (6) take the forms:

                            (13)

                            (14)

              (15)

        (16)

where

,

                       (17)

here

            (18)

With the help of Newtonian formulae, the quantities of engineering interest such as the radial wall stress τ_r and the circumferential wall shear stress τ_φ are expressed as

              (19)

              (20)

That results in the skin friction coefficient as

                       (21)

The non-dimensional local skin friction coefficient can be formed as

          (22)

Further, torque required to turn a disk of radius r₀ can be obtained using radial strip integration:

          (23)

Employing Fourier law, local Nusselt number Nu_r can be expressed as

                  (24)

where is the heat flux due to the disk.

The non-dimensional local Nusselt number reads

                    (25)

3 Method of solution

The governing boundary value problem consisting of system of Eqs. (13)–(15) along with boundary conditions (16) is coupled and nonlinear in nature. Therefore, Runge-Kutta-Fehlberg quadrature is applied to obtaining numerical solution. The higher boundary value problem is transformed into an initial value of first order system of equations using suitable substitutions.

Introducing the following substitutions in system of Eqs. (13)–(16):

;

;

              (26)

we have the following system of first order ordinary differential equations along with Initial conditions:

 (27)

            (28)

where a₁, a₂, a₃ are the shooting parameters. The convergence criterion of 10^–5 is set to obtain reliable results.

4 Results and discussion

As such, the article is more a reflection of the authors’ ideals and wishes. In the present study we have investigated the influence of partial slip, thermal radiation, Joule heating and porous matrix on steady Darchy-Forchheimer flow of ZnO-SAE50 nanofluid over an inclined rotating disk in the presence of uniform magnetic field. The current study espouses the use of Runge-Kutta-Fehlberg scheme to have an appropriate numerical solution of the transformed differential equations. The behavior of diversified and associated pertinent parameters has been ferreted out in the present analysis through appropriate graphs and well discussion. The thermophysical properties of ZnO nanoparticle and SAE50 are incorporated in Table 1.

With the introduction, Figures 2 and 3 portray the effect of wall slip (Γ) on radial velocity F′(η) and tangential velocity G(η). The observed behavior is that enhancement of Γ slows down the fluid motion (radial and tangential) contributing to improvement of respective VBLs. Also it observed that for large Γ (Γ→∞), the rotating disk does not rotate fluid particles. The rationally behind it is that the flow becomes entirely potential, leading to no motion of fluid in this range of Γ. In addition, the centrifugal force acting on the rotating disk will inject the attached fluid. On the other hand, the axial flow will definitely compensate this injected fluid. However, augmenting the slip on the surface of the disk peters out the amount of fluid stuck to it. As a consequence, the efficiency of the rotating disk fritters away substantially and hence is unable to deliver its circumferential momentum to the fluid particles concerned. Figures 4 and 5 together reveal the radial and tangential velocity profiles for different Gr respectively. The observation is that the increasing values of Gr upsurges the radial flow of fluid along with multiple overshoots in the neighborhood of the surface of the disk. It observes the marvel of absolute maximum of velocity profiles that have been ballooned towards the surface of the disk. This result is coincident with that achieved by SIBANDA et al [45]. Meanwhile,the tangential velocity exhibits the reverse nature (increasing trend) in response to rising Gr.

Table 1 Thermo-physical properties of nanoparticles (ZnO) and base fluid (SAE50) (SEPYANI et al [12])

Figure 2 Influence of Γ on F′(η)

Figure 3 Influence of Γ on G(η)

Figure 4 Influence of Gr on F′(η)

Figure 5 Influence of Gr on G(η)

The nature of radial and tangential velocity profiles for different magnetic field strength is noticed from Figures 6 and 7, respectively. It has indicated that the deceleration of the fluid has been attained around the disk due to enhancement of magnetic field strength for both the cases. One can see that the reason for slowing down of fluid motion is due to the restraining nature of Lorentz force created as a result of interaction between conducting fluid and applied magnetic field. Such drag force yields ascending momentum boundary layer in the entire flow field. The current outcome is an excellent agreement with that obtained by SIBANDA et al [45]. In an ironic way, Figures 8 and 9 convey exactly the same behavior of F′(η) and G(η) in response to increase in porosity parameter (K). Resistance offered by the porous matrix is the prime factor for such fashion. As a way to reduce the velocity, the fluid motion is encountered by the resistive force due to porous medium. This outcome is exactly merged with that obtained by HAYAT et al [46]. Variations of fluid velocity along radial and tangential directions for different inertia coefficient (F_r) are shown in Figures 10 and 11. These figures indicate that the fluid velocities F′(η) and G(η) follow descending trend with increase in F_r. This well agrees with the report declared by HAYAT et al [46]. Interestingly, the decelerating fluid motions along radial and tangential directions in response to augmented Γ, G_r, M, K and F_r yield velocity boundary layers of diminishing thickness. Augmented Reynolds number Re has shown a reasonable decline F′(η) with multiple overshoots near the surface of the disk, indicating shift of the maximum radial velocity towards the surface of the disk (Figure 12). The fundamental cause behind such velocity diminution is that the dominating inertial force compared to weaker viscous force opposes to the flow along radial direction. This behavior is coincident with that reported by HAYAT et al [47]. On the other hand, azimuthal velocity G(η) upsurges due to enhancement in Re throughout the flow domain (Figure 13). The rationally behind such velocity augmentation is that the dominating inertial force favors the flow along tangential direction.

Figure 6 Influence of M on F′(η)

Figure 7 Influence of M on G(η)

Figure 8 Influence of K on F′(η)

Figure 9 Influence of K on G(η)

Figure 10 Influence of F_ron F′(η)

Figure 11 Influence of F_r on G(η)

Figure 12 Influence of Re on F′(η)

However, it is important to note that augmented magnetic field strength upsurges the fluid temperature, leading to ascending thermal boundary layer. This means that increase in M hikes the area under θ(η) vs η curve, yielding more temperature of the fluid in association with Joule heating (Figure 14). This noble feature is in consent with the report addressed by HAYAT et al [48]. Meanwhile, multiple overshoots appear near the surface of the disk, implicating that the absolute maximum temperature approaches towards the surface of the disk concerned. The reality behind such hike in temperature is that greater strength of the impressed magnetic field yielding more drag coefficient at the surface due to strong Lorentz force imparts considerable resistance to the fluid particles there by restraining the flow around the disk, which is why more heat is generated leading to rise in temperature. As a result, less hot fluid is transferred from the surface of the disk. Figure 15 exudes the temperature profiles for different Bi. Convective heating yields considerable change in the fluid temperature in the disk. It is from Figure 16 that higher Prandtl fluids having lower thermal diffusivity account for diminishing heat propagation followed by a transition at η=0.5 (merged with HAYAT et al [48] and HAYAT et al [47]). It is often the root cause of alteration of the structure of thermal boundary layer. Significant enhancement in heat generation develops ascending thermal boundary layer (Figure 17).

Figure 13 Influence of Re on G(η)

Figure 14 Influence of M on θ(η)

Figure 15 Influence of Bi on θ(η)

Figure 16 Influence of Pr on θ(η)

Figure 17 Influence of Q on θ(η)

Physically, by strengthening the radiation parameter provides more heat into the liquid, and subsequently, the thickness of the thermal boundary layer is intensified. Thus, the radiation plays a key role in boosting the rate of heat transfer from the surface of the disk in the high focus flow domain (Figure 18).Increase in Eckert number Ec corresponds to more kinetic energy compared to enthalpy produced. As a result, the fluid temperature and hence surface temperature gradient get augmented in the entire flow domain(Figure 19). This behavior is the same as that accomplished by SIBANDA et al [45].

Figure 18 Influence of R_d on θ(η)

Figure 19 Influence of Ec on θ(η)

At the moment, Table 2 oozes the variation of skin friction in response to K for three different values of f. It is understood from the analysis of data incorporated in the table that as K increases, wall shear stress upsurges whatever the solid volume fraction may be. However, at fixed K (for instance, K=0.1), rise in f augments wall shear stress. Table 3 conveys us the variation of skin friction in response to slip parameter Γ for three different values of M. One has to now ensure from table that enhanced Γ fritter away the wall shear stress irrespective the value of M chosen. However, at fixed Γ (for instance Γ=0.1), augmentation of magnetic field strength has improved the wall shear stress. This behavior is exactly the same as that envisioned by HAYAT et al [48] and SIBANDA et al [45] in their study.

Table 2 Skin friction against K for three different values of φ when ψ=π/6, Γ=0.2, Re=2.0, Gr=0.1, M=0.3, R_d=1.0, γ=0.5, Bi=0.1, F_r=0.1, Q=0.1, Ec=0.1

Table 3 Skin friction against Γ for three different values of M when ψ=π/6, K=0.3, Re=2.0, Gr=0.1, f=0.01, R_d=1.0, γ=0.5, Bi=0.1, Fr=0.1, Q=0.1, Ec=0.1

Table 4 accounts for the variation of heat flux in response to f for different values of R_d. With increase of f heat flux gets enhanced significantly whatever the value of R_d may be. However, for fixed volume fraction (for instance f=0.006) augmented thermal radiation upsurges the heat flux from the surface of the disk. Table 5 reveals the influence of convective heating and magnetic field strength on heat flux. Rise in convective heating offers diminution of heat flux from the surface of the disk which is due to the up in temperature under the influence of same condition in the presence of definite strength of magnetic field. However, at fixed Bi (for instance, Bi=0.1), hike in magnetic field strengthens up the fluid temperature thereby belittling the heat flux from the concerned surface. This variation agrees well with that obtained by SIBANDA et al [45] in their investigation.

Table 4 Heat flux against f for three different values of R_d when ψ=π/6, K=1.0, Re=2.0, Gr=0.1, Γ=0.2, γ=0.5, Bi=0.1, F_r=0.1, Q=0.1, Ec=0.1

Table 5 Heat flux against Bi for three different values of M when λ=0.1, Re=0.5, n=2, Gr=0.2, R_d=0.5, β=0.3, f=0.005

5 Conclusions

The present investigation plays a major role in exploring the influence of partial slip, thermal radiation, Joule heating and porous matrix on steady Darchy-Forchheimer flow of ZnO-SAE50 nanolubricant due to an inclined rotating disk in the presence of uniform magnetic field appearing in diversified industries. One of the strengthening accomplishments of the present study is that the presence of velocity slip accounts for the slowing down of radial and tangential flow of ZnO-SAE50 nanofluid. Applied uniform magnetic field and porous matrix contribute to enhance the velocity and thermal boundary layer thickness. Further, a descending trend of radial as well as azimuthal flow of ZnO-SAE50 nanofluid has been achieved due to augmentation of F_r. Furthermore, radial and tangential velocities exhibit diametrically opposite trends in response to increase in Reynolds number Re. Augmented Ec leads to significant surface temperature gradient, there by reducing heat transfer rate from the surface of the disk. The next outcome of the investigation is that wall shear stress is an increasing function of f, K and Γ while it is a decreasing function of M. Finally, enhanced f and R_d upsurge the heat flux while B_i and M reduced it.

Nomenclature

(u, v, w)

Velocity components in increasing (r, f, z) directions, m/s

ρ_ZnO-SAE50

Effective density of the nano- lubricant(ZnO-SAE50), kg/m³

(ρC_p)_ZnO-SAE50

Heat capacitance of the nano- lubricant(ZnO-SAE50), J·kg²·m³/K

(ρC_p) _SAE50

Heat capacitance of base fluid (SAE50), J·kg²·m³/K

β_SAE50

Thermal expansion of base fluid capacitance of base oil (SAE50), K^–1

(ρC_p)_ZnO

Heat capacitance of ZnO nanoparticles, J·kg²·m³/K

ρ_ZnO

Density of base fluid (SAE50), kg/m³

ρ_SAE50

Density of ZnO nanoparticles, kg/m³

μ_ZnO-SAE50

Effective dynamic viscosity of nano-lubricant (ZnO-SAE50),

kg/(m·s)

μ_SAE50

Effective dynamic viscosity of (SAE50), kg/(m·s)

β_ZnO

Thermal expansion of nanoparticle (ZnO), K^–1

k_ZnO-SAE50

Thermal conductivity of nano- lubricant (ZnO-SAE50), W/(m·K)

k_SAE50

Thermal conductivity of base fluid (SAE50), W/(m·K)

k_ZnO

Thermal conductivity of ZnO nanoparticle, W/(m·K)

σ_ZnO-SAE50

Electrical conductivity of nano- lubricant(ZnO-SAE50), Ω^–1·m^–1

σ_ZnO

Electrical conductivity of ZnO nanoparticles, Ω^–1·m^–1

σ_SAE50

Electrical conductivity of base fluid (SAE50), Ω^–1·m^–1

A_ZnO, A_SAE50

Heat transfer area of particles and fluid medium respectively, m²

u_ZnO

Brownian motion velocity of ZnO nanoparticles, m/s

T_w

Surface temperature, K

T_f

Temperature of heated fluid, K

T

Fluid temperature in the boundary layer, K

T_∞

Ambient fluid temperature, K

k^*

Mean absorption coefficient

L

Wall slip coefficient

σ^*

Stefan Boltzmann constant

h_f

Heat transfer coefficient, W/(m²·K)

K

Porosity parameter

p

Pressure, Pa

Pe

Peclet number

c

A constant

d_SAE50

Molecular size of the fluid (SAE50), m

d_ZnO

Diameter of ZnO nanoparticle, m

α_SAE50

Thermal diffusivity of the fluid (SAE50), m²/s

Gr

Thermal Grassof number

R_d

Radiation parameter

Permeability of porous medium

f

Solid volume fraction

Pr

Prandtl number

τ_r

Radial wall stress, Pa

τ_φ

Circumferential shear stress, Pa

Re

Rotational Reynolds number

M

Hartmann number

Ec

Eckert number

Q

Heat generation parameter

γ

Stretching strength parameter

Γ

Wall slip parameter

F_r

Inertia coefficient

Bi

Biot number

Non-uniform inertia coefficient

C_d

Drag coefficient

Subscripts

s

Surface

w

Stands for wall

∞

Stands for ambient fluid

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(Edited by YANG Hua)

中文导读

倾斜旋转盘对ZnO-SAE50纳米磁流体流动和传热的影响

摘要：考虑工业中外加磁场对倾斜旋转盘的影响，对氧化锌-SAE50纳米润滑剂的Darcy-Forchheimer混合对流进行了详细研究。研究中同时考虑了滑移流动、热辐射、黏性耗散、Joulian耗散和内部加热的影响，对当前研究进行了改进。考虑到纳米润滑剂热导率的增大，提出了一种新的微纳米对流模型，即Patel模型。该模型的特点是由于纳米粒子的布朗运动、基于动力学理论的微对流、液体分层和粒子聚集而引起的比表面积和纳米对流的变化。利用Runge-Kutta-Fehlberg方法，对适当变换的控制方程进行了数值求解。分析表明，外加磁场、介质的孔隙度、速度滑移和惯性系数减慢了ZnO-SAE50纳米润滑剂的径向和切向流动，从而导致速度和热边界层的改善。

关键词：ZnO-SAE50纳米流体；Darcy-Forchheimer磁流体(MHD)；热辐射；速度滑移；黏性耗散；内部加热

Received date: 2018-07-28; Accepted date: 2018-11-08

Corresponding author: NAYAK M K, PhD, Associate Professor; E-mail: mkn2122@gmail.com, manoj.nayak@riteindia.in; ORCID: 0000-0003-0529-5525