Abstract
In the production of ethelene glycol, graphene nanoparticles is inevitable and even suggested due to monomolecular layer of carbon atoms which are bounded like honey comb structure is known as graphene due to this structure, graphene has several types of exceptional and unique structural, optical and electronic properties. However, little is known on the enhancement of the transport phenomenon when Joule dissipation, inclined magnetic field, thermal jump and partial slip are apparent. With emphasis to the inherent aforementioned concepts together with heat source/sink and thermal radiation, this paper presents insight into the dynamics of unsteady Ethelene glycol conveying graphene nanoparticles through porous medium. The dimensional governing equation was non-dimenzionalized using fitting similarity variables and solved the dimensionless equations using Runge-Kutta Fehlberg algorithms along with the shooting technique. Also, a statistical method was implemented for multiple quadratic regression estimation analysis on the numerical figures of wall velocity gradient and local Nusselt number to establish the connection among heat transfer rate and physical parameters. Our numerical findings reveal that the magnetic field and porosity parameters boost the graphene Maxwell nanofluid velocity while Maxwell parameter has a reversal impact on it. The regression analysis confers that Nusselt number is more prone to heat absorption parameter as compared to Eckert number. The rate of heat transfer is higher in case of with slip compare to without slip flow in the presence of thermal radiation, viscous dissipation and unsteady parameter. The fluid velocity and temperature distribution is higher in without slip compare to with slip flow.
1 Introduction
In the present scenario, owing to overwhelming significance in industry and public endeavor, the nanofluid problems have received notable attention from researchers working in the domain of nanotechnology. The nanofluid foremost termed by Choi [1] is a kind of fluid that consists of extremely fine metallic or non-metallic particles (up to 50 nm in radius). The addition of extremely fine metallic or non-metallic particles in convectional fluid influences the fluid characteristics like heat transfer rate and thermal conductivity Eastman et al. [2]. For example, in the modern nuclear system addition of nano-scale metal particles in the convectional fluid augments the heat transfer rate characteristics. Nanofluids are enormously useful in various applications including energy conversion, microsystem cooling (due to change in heat capacity of base fluid by adding metal or metal oxide particle of nano-sized), nanomedical industry, etc. The easiest way to enhance the thermophysical properties of base fluid is by adding the nano-sized particle (metal or non-metal). The suspension of nano-sized (less than 100 nanometers) particles (both metal and non-metal) into the base fluid results in the extreme enhancement in the temperature gradient of the fluid Hsiao [3]. The report of Shah et al. [4] shows that optimal Nusselt number is ascertained at a larger value of stretching ratio and suction in the dynamics of water conveying (less dense nanoparticles) multiple wall CNT and silicon dioxide.
Some relevant researches related to nanofluids under various conditions are given by Sheikholeslami [5], Li et al. [6], Mamatha et al. [7], and Patil Mamatha et al. [8]. The carbon atom's monomolecular layer having a similar structure to honeycomb is known as graphene. Owing to the fast mobility of electrons, high thermal conductivity, stability, and expanded surface area the graphene nanoparticles possess novel material, physical, electrical and chemical characteristics Netro et al. [9]. Therefore, it has a wide range of applications including but not limited to electronics, the energy sector, sensing outlets, medical sciences, etc.; see Chung et al. [10]. Upadhya et al. [11] performed the theoretical investigation of magneto-Carreau fluid by adding the graphene and dust particles into it and observed that mixtures of ethylene glycol and graphene nanoparticles are noteworthy in improving the heat transport phenomena. Further, this studywas prolonged by Mahesha et al. [12] assuming the significance of both heat generation and thermal radiation, and found that both temperature and velocity profiles are augmented owing to an increase in unsteadiness parameter values. The hydromagnetic heat transfer flow problems in the presence of porous medium have industrial useful applications, such as plasmas, extraction of energy from the earth, separation of metal from non-metals enclosure, polymer industries, etc. The significance of problems of the hydromagnetic flow of fluid along with heat transfer characteristics in the existence of a porous medium is reported by Shercliff [13], Branover [14], Cramer and Pai [15], Sheremet and Pop [16, 17].
In the aforesaid published studies, the majority of the problems were confined to the flow of Newtonian fluids. In practical situations, it is found that numerous fluids in various engineering and industrial developments are non-Newtonian fluids in characteristics, for instance, molten plastics, polymers, nuclear fuel slurries, pulps, mercury amalgams, liquid metals, lubrication by heavy oil, etc. Owing to the complex characteristics of these fluids one can’t find at least one constitutive equation that reveals all the characteristics of such non-Newtonian fluids. However, complex constitutive models for the non-Newtonian fluid's concept have been proposed by Jayachandra and Sandeep [18], Haroun [19], Sajid et al. [20], and Hayat et al. [21]. The proposed models are concerned with the second grade, third grade, and fourth-grade fluids which can estimate the impact of elasticity while these models are shear independent and unable to envisage the significance of shear relaxation; see Hayat et al. [22, 23]. Moreover, the Maxwell model is proposed for the rate type fluids which can estimate shear stress relaxation and consequently become a popular model among the researchers. These models can also depict the significance of shear-dependent viscosity of boundary layer problems.
In the same direction, Farooq et al. [24], Imran et al. [25], Fetecau [26], and Heyhyat and Khabazi [27] reported their investigations on Maxwell fluid flow problems considering different geometrical configurations to analyze the significance of different germane parameters on the flow field. Heat generation/absorption models of fluid flow driven by permeable stretched sheet have been investigated by many authors because of its numerous industrial applications viz. hot rolling, a chemical reaction that discharges heat, endothermic reactions, heat exchange due to nuclear fuel wastage, glass fiber production, paper production, etc. Following this, the quiescent fluid flow over an exponentially stretched sheet amid the significance of heat source/sink is investigated numerically by Elbashbeshy [28]. Satya et al. [29] solved numerically non-uniform heat source problem fluid flow over a stretching sheet. This study illustrates that suction can be employed to cool down the moving continuous surface. Further, this problem was extended by Elbashbeshy and Bazid [30] considering unsteady state conditions. Subsequently, the influence of suction/blowing on magnetohydrodynamic flow assuming both heat generation and heat absorption over the oblique stretched surface is reported by Eldahab and Aziz [31].
In the above studies, the effect of viscosity is ignored but in the practical situation its effect has a novel role Mamatha et al. [32] and due to viscosity, viscous dissipation impact has not been considered as the same is supposed to be low but its relevance in food processing, instrumentations, lubrications, polymer manufacturing, etc., is considered as very significant as it enhances the characteristics of temperature distribution which induce the heat transfer rate. Some important studies dealing with the viscous dissipative flow problem induced by stretched sheets are reported by Partha et al. [33], Aziz [34], Rashidi and Erfani [35], Qasim and Noreen [36], Khader and Megahed [37]. Further, Joule dissipation shows the characteristics alike volumetric heat sources in magnetohydrodynamic fluid flows and the collective influence of viscous dissipation and Joule dissipations are imperative in the context of heat-treated materials. Following this, several researchers such as Anjali and Ganga [38], Pal [39], Seth and Singh [40], and Seth et al. [41] modeled their problems considering the impacts of Joule and viscous dissipations. A meticulous review of research papers reported in the literature reveals that the majority of the researchers have considered the no-slip conditions in their investigations and have ignored the slip conditions. But, in many physical problems such as low-pressure flows, micro/nanoscale flow, flow over coated surfaces, etc. Navier's partial slip took place. Owing to this reason, Mukhopadhyay and Gorla [42] Singh and Makinde [43], Seth et al. [44, 45] studied the boundary layer flow problems taking partial slip conditions into account.
In recent days, industries, manufacturing, and design of materials are looking for immediate cooling or heating. Therefore we included the graphene nanoparticles in this study with time-dependent to get the originality of the flow behaviour. This state of the art is an attempt to have an overview of recent research in the field of GR related to its synthesis and scope in multifarious applications in context to electronics and energy in general and related to field emission and sensors in particular. The scope of graphene nanoparticles in tribology and industrial automation has also been explored. Sequel to the aforementioned published facts, it is worth remarking that there is no report on the significance of Joule dissipation, thermal jump, and partial slip on the dynamics of unsteady Ethelene glycol conveying graphene nanoparticles through a porous medium. Hence, this study will provide answers to the following research questions:
In the case of dynamics subject to no-slip and partial slip, how do Maxwell parameter and permeability parameter influences the velocity of unsteady ethelene glycol conveying graphene nanoparticles?
What are the variations in the local skin friction and heat transfer rate in the flow of unsteady Ethelene glycol conveying graphene nanoparticles through porous medium?
How does the aforementioned transport phenomenon (velocity) vary for the case of no-slip and partial slip?
What is the effect of the thermal jump, unsteadiness parameter, viscous dissipation, heat absorption, and heat absorption affect the temperature distribution in the dynamics mentioned above?
2 Governing mathematical equations
The mathematical formulation of the physical problem has been done by considering the flow of an incompressible, electrically conducting, two-dimensional streamline and heat-absorbing non-Newtonian Maxwell graphene nanofluid over a linearly stretching sheet. The nanofluid consists of Ethylene glycol as a base fluid and graphene as a nanoparticle. The flow of fluid is considered on the two-dimensional Cartesian coordinate system in which the stretching sheet is along to x-axis while the y-axis is across it. The fluid flow is limited in the area y > 0. The nanofluid is induced over a stretching sheet owing to the unsteady magnetic field B (t) = B0(1 − αt)−1/2 exerted along the y-axis. The influence of the induced magnetic field is assumed to be ignorable for the applied magnetic field due to the low Reynolds number. Moreover, it is presumed that the considered sheet is shrunk owing to two opposite but equal forces with the unsteady velocity Uw (x, t) = cx(1 − αt)−1, where c and (1 − αt)−1 (here αt < 1) are respectively initial and notable stretching rate. The wall temperature of nanofluid is considered Tw while the temperature outside the boundary regime known as the free stream is denoted by T∞. The geometrical illustration of the problem is presented in Figure 1. The prevailing mathematical equations related to momentum and energy for the Maxwell nanofluids problem by Mukhopadhyay [46] under the assumptions as considered above are expressed as
Geometrical illustration of the problem
boundary conditions for the above mathematical modal are given as:
at
Notations u, v are the graphene Maxwell nanofluid velocity in the x-direction (primary velocity) and y-direction (secondary velocity) respectively. μnf indicates the dynamic viscosity of nanofluid. λ = λ0 (1 − αt), where λ0 is the initial relaxation rate. K = K0 (1 − αt)is the porosity parameter. σnf is a symbol of electrical conductivity of graphene Maxwell nanofluid. y is the angle in anticlockwise sense from stretching surface. T denotes the temperature of graphene Maxwell nanofluid. knf is the symbol of heat conductivity of nanofluid. (ρcp)nf is the symbol of specific heat capacitance of graphene Maxwell nanofluid. Q0 is the heat absorption coefficient.
Here, σ* is the Stefan-Boltzmann constant and k* signifies absorption coefficient.
3 Numerical solution of the problem
By making use of stream functions
The dimensionless form of boundary conditions is given as:
as
where
In above expressions ρnf represents the graphene Maxwell nanofluid's density, ρf is the base fluid's density, μnf denotes dynamic viscosity of graphene Maxwell nanofluid, μf is the density of Maxwell fluid (base fluid), σs represents the electrical conductivity of graphene nanoparticle, σf represents the electrical conductivity of ethylene glycol, ks represents heat conductivity of graphene particles, kf represents the heat conductivity of Maxwell fluid (base fluid),
Thermophysical properties of Ethelene glycol and graphene nanoparticles; sources Asim et al. [47] and Elbashbeshy and Bazid [48].
| Thermo-physical properties | ρ (kg/m3) | cp (J/kgK) | K (W/mK) | α (S/m) |
|---|---|---|---|---|
| Ethelene glycol | 1114 | 2415 | 0.252 | 5.5 × 10−6 |
| Graphene | 2250 | 2100 | 2500 | 1 × 10−7 |
From the engineering applications viewpoint, the expressions for Sf characterizes skin friction coefficient and local Nusselt number Nux are defined as:
where τw and qw are
Applying the similarity transformation (9), we obtained
where λ1 = λ0c is the fluid relaxation parameter and
3.1 Implementation of the numerical method
The complexity of the mathematical model reported in segment 3 is very high, so the analytical solution of the model can’t be obtained. Therefore, to analyze the solution of highly non-linear ordinary differential equations mentioned by equations (7) and (8) along with conditions (9) and (10), a numerical solution is analyzed employing the Runge-Kutta Fehlberg scheme with shooting technique. At first, equations (7) and (8) are converted to the form of five first-order differential equations. After this, the Runge-Kutta Fehlberg method is used to solve this set of five differential equations in which the initial functional values of f″ (0) and θ′ (0) are obtained with the help of shooting technique. The value of similarity variable η is chosen η = 7 as aimed at infinity boundary conditions whereas the step size is considered as 0.001 during the numerical computation. To achieve a precise result the tolerance error is taken as 10−6. The complete method is repeated until the anticipated correctness in the results is attained.
3.2 Validation of numerical solution
For the validation of the numerical solution obtained in this paper, we have done a comparison of −fJ (0) for various values of Prandtl number Pr and unsteadiness parameter A and with that of Elbashbeshy and Bazid [48] when β = M = K1 = Q = Ec = A1 = € = 0 and are presented in Table 2. Table 2 shows excellent conformity of numerical results which validates the correctness of our numerical solution.
Comparison of −f″ (0) with the previously published paper
| A Pr | 0.8 | 1.2 | 2 | |||
|---|---|---|---|---|---|---|
| Elbashbeshy and Bzid [48] | Present result | Elbashbeshy and Bzid [48] | Present result | Elbashbeshy and Bzid [48] | Present result | |
| 0.01 | 1.3321 | 1.3382 | 1.4691 | 1.4637 | 1.7087 | 1.6979 |
| 0.1 | 1.3321 | 1.3382 | 1.4691 | 1.4637 | 1.7087 | 1.6979 |
| 1 | 1.3321 | 1.3382 | 1.4691 | 1.4637 | 1.7087 | 1.6979 |
| 10 | 1.3321 | 1.3382 | 1.4691 | 1.4637 | 1.7087 | 1.6979 |
From an engineering interest viewpoint, the numerical findings of the coefficient of skin friction and Nusselt number are also presented in Tables 3 and 4. This is in good agreement with the observation reported by Elbashbeshy and Bazid [48] that unsteady stretching affects the transfer of heat across the fluid flow. For this, values of nondimensional parameters are considered as Prandtl number Pr=150, unsteadiness parameter A = 0.1, thermal slip parameter ɛ = 0.1, Eckert number, magnetic parameter M = 0.7, Maxwell parameter β = 0.5, heat absorption parameter Q = 3, and porosity parameter K1 = 0.1. These values remain unchanged throughout the analysis. It is obvious from Table 3 that skin friction is getting enhanced due to an increase in Magnetic field, porosity, and unsteadiness parameters for both conditions (slip and without slip flow). It is also evident from Table 3 that skin friction is getting reduced due to an increase in Maxwell parameter for both conditions (slip and without slip flow). It is noticed from Table 4 that the Nusselt number gets enhanced owing to an increase in heat absorption and thermal slip parameters for both conditions (slip and without slip flow) while it gets reduced due to an increase in viscous dissipation, radiation parameter, and unsteadiness parameters for both conditions (slip and without slip flow). Notably, the rate of heat transfer is higher in the case of slip compare to without slip flow in the presence of thermal radiation, viscous dissipation, and unsteady parameter.
Skin friction coefficient under both slip (A1 = 0) and no-slip (A1 = 0.3) conditions
| M | K1 | β | A | A1 = 0 Skin friction coefficient | A1 = 0.3 Skin friction coefficient |
|---|---|---|---|---|---|
| 1 | −1.550280 | −1.091902 | |||
| 2 | −1.795996 | −1.224990 | |||
| 3 | −2.018683 | −1.336651 | |||
| 0.1 | −1.464016 | −1.055576 | |||
| 0.3 | −1.550862 | −1.10089 | |||
| 0.5 | −1.629970 | −1.142603 | |||
| 0.1 | −1.504320 | −1.062202 | |||
| 0.3 | −1.493485 | −1.059860 | |||
| 0.5 | −1.464016 | −1.055576 | |||
| 0.1 | −1.550280 | −1.091902 | |||
| 0.9 | −1.751600 | −1.201253 | |||
| 1.7 | −1.938760 | −1.297035 |
Heat transfer rate at the stretched sheet under both no-slip (A1 = 0) and slip (A1 = 0.3) conditions
| Nr | Ec | ɛ | A | Q | −Nux(Nusselt Number) | |
|---|---|---|---|---|---|---|
| A1 = 0 | A1 = 0.3 | |||||
| 0.1 | 1.793817 | 1.782474 | ||||
| 0.3 | 1.609518 | 1.569714 | ||||
| 0.5 | 1.352808 | 1.286409 | ||||
| 0.10 | 1.493379 | 1.439736 | ||||
| 0.25 | 0.680038 | 0.656444 | ||||
| 0.40 | 0.580038 | 0.473152 | ||||
| 0.10 | 1.024231 | 1.030984 | ||||
| 0.25 | 1.215093 | 1.201549 | ||||
| 0.40 | 1.493379 | 1.439736 | ||||
| 0.1 | 1.493379 | 1.439736 | ||||
| 0.4 | 1.341298 | 1.253572 | ||||
| 0.7 | 1.157093 | 0.995805 | ||||
| 1 | 0.805981 | 0.813852 | ||||
| 3 | 0.935618 | 0.944211 | ||||
| 5 | 1.027387 | 1.035966 | ||||
3.3 Multiple regression analysis: Estimation of Nusselt number and skin friction coefficients
Multiple regression analysis is a statistical tool that is used to estimate the relationship between two or more variables. Generally, regression analysis is beneficial to distinguish in what way the specific value of dependent variable changes owing to variation in independent variable though other independent variables are assumed constant. In this portion, quadratic regression estimation analysis for Nusselt number and skin friction coefficients has been presented. The model of multiple quadratic regression estimation for the coefficient of skin friction is mentioned for 100 different values of M and K1, obtained arbitrarily from the intervals [1,7] and [0.1, 0.5] respectively for two different values of β = 0.1 and β = 0.5. Apart from this, a model of multiple quadratic regression estimation for the Nusselt number is provided for a set of 100 values of Ec and Q analyzed arbitrarily from the intervals [0.1, 0.3] and [1, 3] for two different values of Pr = 5 and Pr = 10. During the estimation process, the remaining parameters are kept constant as stated. The estimated quadratic regression model for
While the regression formula for
Tables 5 and 6 present the coefficients of multiple quadratic regression estimated findings of skin friction coefficients and Nusselt number for different parameters. The maximum relative error bound for skin friction € = |Cfest − Cf | /Cf and Nusselt number €1 = |Nuest − Nu| /Nu are also obtained and presented in Tables 5 and 6 respectively. From the tabulated values, it is evident that the coefficient of the porous medium parameter is higher than that of the magnetic field parameter, which suggests that the variation in skin friction is more sensitive for the values of K1 than M for both values of β. In the same way, we can observe that the Nusselt number is more prone to heat absorption parameter Q for both values of Pr as compared to Eckert number Ec.
Error bound ɛ and multiple quadratic regression coefficients for the estimated
| B | Cf | b1 | b2 | b3 | b4 | b5 | ɛ |
|---|---|---|---|---|---|---|---|
| 0.1 | −0.9310 | −0.2160 | −0.3164 | 0.0070 | 0.0417 | 0.0227 | 0.0042 |
| 0.5 | −0.8742 | −0.2352 | −0.3806 | 0.0089 | 0.0655 | 0.0317 | 0.0066 |
Error boundand multiple quadratic regression coefficients for the estimated
| Pr | Nu | c1 | c2 | c3 | c4 | c5 | ɛ1 |
|---|---|---|---|---|---|---|---|
| 5 | −1.3931 | −0.7258 | 1.6353 | 0.0768 | 0.4318 | −0.2417 | 0.000024 |
| 10 | −1.7862 | −0.9962 | 2.6242 | 0.1319 | 0.6564 | −0.4804 | 0.000028 |
4 Result and discussions
In this section, numerical computation is executed and the results are reported to demonstrate a comparative analysis revealing the impacts of several parameters on the flow field. Throughout the numerical computation, values of various non-dimensional parameters are assumed as Prandtl number Pr=150, unsteadiness parameter A = 0.1, thermal slip parameter ɛ = 0.1, Eckert number Ec = 0.1, magnetic parameter M = 0.7, Maxwell parameter β = 0.5, heat absorption parameter Q = 3, radiation parameter Nr = 0.2, and porosity parameter K1 = 0.1. These values remain unchanged throughout the analysis while the varying values of parameters are specified in respective figures. In all the figures, solid lines indicate the impact of pertinent flow parameters under no-slip conditions while dashed lines denote the impact under slip conditions.
Figures 2–5 illustrate the effects of the magnetic field, porous medium, Maxwell, and unsteadiness parameters on the velocity of graphene Maxwell nanofluid. Figures 6–10 describes the significance of thermal radiation, heat absorption, viscous dissipation, thermal slip, and unsteadiness parameter on the temperature of graphene Maxwell nanofluid. Figure 2 shows that graphene Maxwell fluid velocity decreases due to magnetic field parameter (M) enhancement under both slip (A1 = 0.3) and no-slip (A1 = 0) velocity conditions. This type of behaviour is happened due to resistive Lorentz force; which enhances sowing to increase the strength of the magnetic field. Figure 3 explores the effect of porosity medium (K1) on nanofluid velocity. It is seen that due to an increase in porosity the fluid velocity is decreasing. The reason behind this nature of velocity is that resistance opposes the fluid flow is experienced due to increment in a porous medium. Figure 4 shows that the velocity of graphene Maxwell fluid increases due to an increase in the Maxwell parameter (β). Since Maxwell fluid is non-Newtonian fluid, therefore, it generates repulsion forces between molecules, due to this, there is a velocity increment. Figure 5 exhibits that graphene Maxwell fluid velocity is getting reduced owing to an increase in unsteadiness parameter (A). As increasing unsteady parameter, improve the unsteadiness of the flow. Further, it is to be noted from Figures 2–5 that graphene Maxwell nanofluid velocity is higher in case of the no-slip condition than that of slip condition.
Velocity profiles due to variation in the magnetic parameter.
Velocity profiles due to variation in the permeability parameter.
Velocity profiles due to variation in Maxwell parameter.
Velocity profiles due to variation in the unsteadiness parameter.
Figure 6 shows that the temperature of graphene Maxwell nanofluid is boosting due to an increase in a radiation parameter (Nr). As raising values of thermal radiation generates collision between molecules of the fluid. Figure 7 depicts that graphene Maxwell nanofluid temperature is reduced due to enhancement in the heat absorption parameter (Q). This is owing to the reason that increment in heat absorption parameter results in the heatabsorbing capacity of fluid throughout the boundary layer region. It is apparent from Figure 8 that by enhancing the Eckert number (Ec), graphene Maxwell nanofluid temperature can also be increased. The reason behind this behaviour of graphene Maxwell nanofluid is that Eckert number signifies the ratio of kinetic energy to enthalpy and the total work is done against viscosity where the kinetic energy is converted into internal energy. Hence, viscous dissipation has a nature to increase the temperature of the fluid in the whole boundary layer. Figure 9 presents that graphene Maxwell fluid temperature is reducing due to enhancing the thermal slip parameter ɛ. The reason for this tendency of fluid temperature presented by Figure 9 is that due to increment in thermal slip parameter less heat is flowing from surface to fluid. Hence, the temperature is reducing. It can be noticed from Figure 10 that, due to an increase in unsteadiness parameter, graphene Maxwell temperature is also increasing. Increasing unsteady parameters, improve the chaotic behaviour of the flow. Hence that will help in increment in the collision between molecules. It is to be noted from Figures 6–10 that graphene Maxwell nanofluid temperature is higher in the case of the no-slip condition than that of slip conditions.
Temperature profiles due to variation in heat radiation parameter.
Temperature profiles due to variation in heat absorption parameter.
Temperature profiles due to variation of viscous dissipation.
Temperature profiles due to variation in thermal slip parameter.
Temperature profiles due to variation in the unsteadiness parameter.
5 Concluding remarks
In this report, the significance of Joule dissipation, thermal jump, and partial slip on the dynamics of unsteady ethylene glycol conveying graphene nanoparticles through porous medium had been examined. Based on the outcome of the analysis, it is worth concluding that the graphene Maxwell nanofluid velocity gets reduced owing to enhancement in a magnetic field, the inclination angle of the magnetic field, the porosity of porous medium, and unsteadiness parameters whereas the behaviour of fluid velocity gets reversed due to Maxwell parameter. The heat absorption and thermal slip parameters reduce the temperature of graphene Maxwell nanofluid in the boundary layer regime while the temperature of Maxwell nanofluid is increased owing to an increase in inclination angle of the magnetic field, viscous dissipation, and unsteadiness parameters. The magnetic field, porosity, angle of inclination of the magnetic field, and unsteadiness parameters improve the shear stress at the stretched sheet. The rate of heat transfer of nanofluid is augmented owing to heat absorption and thermal slip parameters while it is reduced due increase in viscous dissipation and unsteadiness parameters.
The extension of this study to the cases of nanofluid flow due to radially elongated disk with Coriolis and Lorentz forces by Mahanthesh et al. [49], Quadratic convective flow of Casson fluid in a micro-channel by Kunnegowda et al. [50], dynamics of ethylene glycol conveying graphene nanoparticles rotating stretchable disk by Mahanthesh et al. [51], increasing effects of the haphazard motion of graphene nanoparticles in ethylene glycol by Animasaun et al. [52] and dynamics of blood - graphene nanoparticles Carreau nanofluid induced by partial slip and buoyancy by Koriko et al. [53] are recommended as a future study.
Funding information: One of the authors (Rohit Sharma) is thankful to the University Grant Commission-New Delhi, Government of India for supporting this work under research project No. F.30-457/2018(BSR).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
References
[1] Choi SU, Eastman JA. Enhancing thermal conductivity of fluids with nanoparticles. Argonne National Lab., IL (United States); 1995 Oct 1.Search in Google Scholar
[2] Eastman JA, Choi SUS, Li S, Yu W, Thompson LJ. Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Applied Physics Letters [Internet]. AIP Publishing; 2001 Feb 5;78(6):718–20. Available from: http://dx.doi.org/10.1063/1.134121810.1063/1.1341218Search in Google Scholar
[3] Hsiao K-L. Stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet. Applied Thermal Engineering [Internet]. Elsevier BV; 2016 Apr;98:850–61. Available from: http://dx.doi.org/10.1016/j.applthermaleng.2015.12.13810.1016/j.applthermaleng.2015.12.138Search in Google Scholar
[4] Sheikholeslami M. Numerical investigation of MHD nanofluid free convective heat transfer in a porous tilted enclosure. Engineering Computations [Internet]. Emerald; 2017 Aug 7;34(6):1939–55. Available from: http://dx.doi.org/10.1108/ec-08-2016-029310.1108/EC-08-2016-0293Search in Google Scholar
[5] Shah NA, Animasaun IL, Wakif A, Koriko OK, Sivaraj R, Adegbie KS, et al. Significance of suction and dual stretching on the dynamics of various hybrid nanofluids: Comparative analysis between type I and type II models. Physica Scripta [Internet]. IOP Publishing; 2020 Aug 17;95(9):095205. Available from: http://dx.doi.org/10.1088/1402-4896/aba8c610.1088/1402-4896/aba8c6Search in Google Scholar
[6] Li Z, Sheikholeslami M, Mittal AS, Shafee A, Haq R. Nanofluid heat transfer in a porous duct in the presence of Lorentz forces using the lattice Boltzmann method. The European Physical Journal Plus [Internet]. Springer Science and Business Media LLC; 2019 Jan;134(1), 30. Available from: http://dx.doi.org/10.1140/epjp/i2019-12406-810.1140/epjp/i2019-12406-8Search in Google Scholar
[7] Mamatha Upadhya S, Mahesha, Raju CSK. Cattaneo - Christov heat flux model for magnetohydrodynamic flow in a suspension of dust particles towards a stretching sheet. Nonlinear Engineering [Internet]. Walter de Gruyter GmbH; 2018 Sep 25;7(3):237–46. Available from: http://dx.doi.org/10.1515/nleng-2017-016210.1515/nleng-2017-0162Search in Google Scholar
[8] Patil M, Mahesha, Raju CSK. Convective conditions and dissipation on Tangent Hyperbolic fluid over a chemically heating exponentially porous sheet. Nonlinear Engineering [Internet]. Walter de Gruyter GmbH; 2019 Jan 28;8(1):407–18. Available from: http://dx.doi.org/10.1515/nleng-2018-000310.1515/nleng-2018-0003Search in Google Scholar
[9] Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, Geim AK. The electronic properties of graphene. Reviews of Modern Physics [Internet]. American Physical Society (APS); 2009 Jan 14;81(1):109–62. Available from: http://dx.doi.org/10.1103/revmodphys.81.10910.1103/RevModPhys.81.109Search in Google Scholar
[10] Chung C, Kim Y-K, Shin D, Ryoo S-R, Hong BH, Min D-H. Biomedical Applications of Graphene and Graphene Oxide. Accounts of Chemical Research [Internet]. American Chemical Society (ACS); 2013 Mar 12;46(10):2211–24. Available from: http://dx.doi.org/10.1021/ar300159f10.1021/ar300159fSearch in Google Scholar PubMed
[11] Mamatha Upadhya S, Mahesha, Raju CSK. Unsteady Flow of Carreau Fluid in a Suspension of Dust and Graphene Nanoparticles With Cattaneo–Christov Heat Flux. Journal of Heat Transfer [Internet]. ASME International; 2018 May 22;140(9):092401. Available from: http://dx.doi.org/10.1115/1.403990410.1115/1.4039904Search in Google Scholar
[12] Mahesha, Santhosh HB, Raju CSK. Unsteady Carreau Radiated Flow in a Deformation of Graphene Nanoparticles with Heat Generation and Convective Conditions. Journal of Nanofluids [Internet]. American Scientific Publishers; 2018 Dec 1;7(6):1130–7. Available from: http://dx.doi.org/10.1166/jon.2018.154510.1166/jon.2018.1545Search in Google Scholar
[13] Shercliff JA, A Textbook of Magnetohydrodynamics (Pergamon Press, Oxford, 1965).Search in Google Scholar
[14] Branover GA, AB, Tinober, Magnetohydrodynamics of Incompressible Media (Nauka, Moscow, 1970).Search in Google Scholar
[15] Cramer KR, Pai SI. Magnetofluid dynamics for engineers and applied physicists. (McGraw-Hill Book Company, Washington, D.C., 1973).10.1002/eej.4390930120Search in Google Scholar
[16] Sheremet MA, Pop I. Natural convection in a horizontal cylindrical annulus filled with a porous medium saturated by a nanofluid using Tiwari and Das’ nanofluid model. The European Physical Journal Plus [Internet]. Springer Science and Business Media LLC; 2015 Jun;130(6), 107. Available from: http://dx.doi.org/10.1140/epjp/i2015-15107-410.1140/epjp/i2015-15107-4Search in Google Scholar
[17] Sheremet MA, Pop I, Shenoy A. Natural convection in a wavy open porous cavity filled with a nanofluid: Tiwari and Das’ nanofluid model. The European Physical Journal Plus [Internet]. Springer Science and Business Media LLC; 2016 Mar;131(3). Available from: http://dx.doi.org/10.1140/epjp/i2016-16062-210.1140/epjp/i2016-16062-2Search in Google Scholar
[18] Jayachandra Babu M, Sandeep N. MHD non-Newtonian fluid flow over a slendering stretching sheet in the presence of cross-diffusion effects. Alexandria Engineering Journal [Internet]. Elsevier BV; 2016 Sep;55(3):2193–201. Available from: http://dx.doi.org/10.1016/j.aej.2016.06.00910.1016/j.aej.2016.06.009Search in Google Scholar
[19] Haroun MH. Effect of Deborah number and phase difference on peristaltic transport of a third-order fluid in an asymmetric channel. Communications in Nonlinear Science and Numerical Simulation [Internet]. Elsevier BV; 2007 Dec;12(8):1464–80. Available from: http://dx.doi.org/10.1016/j.cnsns.2006.03.00210.1016/j.cnsns.2006.03.002Search in Google Scholar
[20] Sajid M, Hayat T, Asghar S. Non-similar analytic solution for MHD flow and heat transfer in a thirdorder fluid over a stretching sheet. International Journal of Heat and Mass Transfer [Internet]. Elsevier BV; 2007 May;50(9–10):1723–36. Available from: http://dx.doi.org/10.1016/j.ijheatmasstransfer.2006.10.01110.1016/j.ijheatmasstransfer.2006.10.011Search in Google Scholar
[21] Hayat T, Abbas Z, Sajid M. MHD stagnation-point flow of an upper-convected Maxwell fluid over a stretching surface. Chaos, Solitons & Fractals [Internet]. Elsevier BV; 2009 Jan;39(2):840–8. Available from: http://dx.doi.org/10.1016/j.chaos.2007.01.06710.1016/j.chaos.2007.01.067Search in Google Scholar
[22] Hayat T, Siddiqui AM, Asghar S. Some simple flows of an Oldroyd-B fluid. International Journal of Engineering Science [Internet]. Elsevier BV; 2001 Jan;39(2):135–47. Available from: http://dx.doi.org/10.1016/s0020-7225(00)00026-410.1016/S0020-7225(00)00026-4Search in Google Scholar
[23] Hayat T, Noreen S, Sajid M. Heat transfer analysis of the steady flow of a fourth grade fluid. International Journal of Thermal Sciences [Internet]. Elsevier BV; 2008 May;47(5):591–9. Available from: http://dx.doi.org/10.1016/j.ijthermalsci.2007.05.00510.1016/j.ijthermalsci.2007.05.005Search in Google Scholar
[24] Farooq U, Lu D, Munir S, Ramzan M, Suleman M, Hussain S. MHD flow of Maxwell fluid with nanomaterials due to an exponentially stretching surface. Scientific Reports [Internet]. Springer Science and Business Media LLC; 2019 May 13;9(1), 7312. Available from: http://dx.doi.org/10.1038/s41598-019-43549-010.1038/s41598-019-43549-0Search in Google Scholar
[25] Imran MA, Riaz MB, Shah NA, Zafar AA. Boundary layer flow of MHD generalized Maxwell fluid over an exponentially accelerated infinite vertical surface with slip and Newtonian heating at the boundary. Results in Physics [Internet]. Elsevier BV; 2018 Mar;8:1061–7. Available from: http://dx.doi.org/10.1016/j.rinp.2018.01.03610.1016/j.rinp.2018.01.036Search in Google Scholar
[26] Fetecau C, Athar M, Fetecau C. Unsteady flow of a generalized Maxwell fluid with fractional derivative due to a constantly accelerating plate. Computers & Mathematics with Applications [Internet]. Elsevier BV; 2009 Feb;57(4):596–603. Available from: http://dx.doi.org/10.1016/j.camwa.2008.09.05210.1016/j.camwa.2008.09.052Search in Google Scholar
[27] Heyhat MM, Khabazi N. Non-isothermal flow of Maxwell fluids above fixed flat plates under the influence of a transverse magnetic field. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science [Internet]. SAGE Publications; 2011 Apr;225(4):909–16. Available10.1243/09544062JMES2245Search in Google Scholar
[28] Elbashbeshy EM. Heat transfer over an exponentially stretching continuous surface with suction. Archives of Mechanics. 2001 Jan 11;53(6):643–51.Search in Google Scholar
[29] Mishra SR, Baag S, Dash GC, Acharya MR. Numerical approach to MHD flow of power-law fluid on a stretching sheet with nonuniform heat source. Nonlinear Engineering [Internet]. Walter de Gruyter GmbH; 2019 Sep 25;9(1):81–93. Available from: http://dx.doi.org/10.1515/nleng-2018-002610.1515/nleng-2018-0026Search in Google Scholar
[30] Elbashbeshy EM., Bazid MA. Heat transfer over an unsteady stretching surface with internal heat generation. Applied Mathematics and Computation [Internet]. Elsevier BV; 2003 Jun;138(2–3):239–45. Available from: http://dx.doi.org/10.1016/s0096-3003(02)00106-610.1016/S0096-3003(02)00106-6Search in Google Scholar
[31] Abo-Eldahab EM, El Aziz MA. Blowing/suction effect on hydromagnetic heat transfer by mixed convection from an inclined continuously stretching surface with internal heat generation/absorption. International Journal of Thermal Sciences [Internet]. Elsevier BV; 2004 Jul;43(7):709–19. Available from: http://dx.doi.org/10.1016/j.ijthermalsci.2004.01.00510.1016/j.ijthermalsci.2004.01.005Search in Google Scholar
[32] MAMATHA E, REDDY C, SHARMA R. Effects of viscosity variation and thermal effects in squeeze films. Annales de chimie Science des Matériaux [Internet]. International Information and Engineering Technology Association; 2018 Mar 28;42(1):57–74. Available from: http://dx.doi.org/10.3166/acsm.42.57-7410.3166/acsm.42.57-74Search in Google Scholar
[33] Partha M, Murthy P, Rajasekhar G. Effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface. Heat and Mass Transfer [Internet]. Springer Science and Business Media LLC; 2004 Sep 10;41(4):360–6. Available from: http://dx.doi.org/10.1007/s00231-004-0552-210.1007/s00231-004-0552-2Search in Google Scholar
[34] Abd El-Aziz M. Viscous dissipation effect on mixed convection flow of a micropolar fluid over an exponentially stretching sheet. Canadian Journal of Physics [Internet]. Canadian Science Publishing; 2009 Apr;87(4):359–68. Available from: http://dx.doi.org/10.1139/p09-04710.1139/P09-047Search in Google Scholar
[35] Rashidi MM, Erfani E. Analytical method for solving steady MHD convective and slip flow due to a rotating disk with viscous dissipation and Ohmic heating. Engineering Computations [Internet]. Emerald; 2012 Aug 17;29(6):562–79. Available from: http://dx.doi.org/10.1108/0264440121124628310.1108/02644401211246283Search in Google Scholar
[36] Qasim M, Noreen S. Heat transfer in the boundary layer flow of a Casson fluid over a permeable shrinking sheet with viscous dissipation. The European Physical Journal Plus [Internet]. Springer Science and Business Media LLC; 2014 Jan;129(1). Available from: http://dx.doi.org/10.1140/epjp/i2014-14007-510.1140/epjp/i2014-14007-5Search in Google Scholar
[37] Khader MM, Megahed AM. Numerical solution for boundary layer flow due to a nonlinearly stretching sheet with variable thickness and slip velocity. The European Physical Journal Plus [Internet]. Springer Science and Business Media LLC; 2013 Sep;128(9). Available from: http://dx.doi.org/10.1140/epjp/i2013-13100-710.1140/epjp/i2013-13100-7Search in Google Scholar
[38] Anjali Devi SP, Ganga B. Effects of Viscous and Joules Dissipation on MHD Flow, Heat and Mass Transfer past a Stretching Porous Surface Embedded in a Porous Medium. Nonlinear Analysis: Modelling and Control [Internet]. Vilnius University Press; 2009 Jul 20;14(3):303–14. Available from: http://dx.doi.org/10.15388/na.2009.14.3.1449710.15388/NA.2009.14.3.14497Search in Google Scholar
[39] Pal D. Mixed convection heat transfer in the boundary layers on an exponentially stretching surface with magnetic field. Applied Mathematics and Computation [Internet]. Elsevier BV; 2010 Nov;217(6):2356–69. Available from: http://dx.doi.org/10.1016/j.amc.2010.07.03510.1016/j.amc.2010.07.035Search in Google Scholar
[40] Seth GS, Singh JK. Mixed convection hydromagnetic flow in a rotating channel with Hall and wall conductance effects. Applied Mathematical Modelling [Internet]. Elsevier BV; 2016 Feb;40(4):2783–803. Available from: http://dx.doi.org/10.1016/j.apm.2015.10.01510.1016/j.apm.2015.10.015Search in Google Scholar
[41] Seth GS, Sarkar S, Makinde OD. Combined Free and Forced Convection Couette-Hartmann Flow in a Rotating Channel with Arbitrary Conducting Walls and Hall Effects. Journal of Mechanics [Internet]. Cambridge University Press (CUP); 2016 Aug 17;32(5):613–29. Available10.1017/jmech.2016.70Search in Google Scholar
[42] Singh G, Makinde O. Mixed convection slip flow with temperature jump along a moving plate in presence of free stream. Thermal Science [Internet]. National Library of Serbia; 2015;19(1):119–28. Available from: http://dx.doi.org/10.2298/tsci120718110s10.2298/TSCI120718110SSearch in Google Scholar
[43] Mahanthesh B, Mabood F, Gireesha BJ, Gorla RSR. Effects of chemical reaction and partial slip on the three-dimensional flow of a nanofluid impinging on an exponentially stretching surface. The European Physical Journal Plus [Internet]. Springer Science and Business Media LLC; 2017 Mar;132(3). Available from: http://dx.doi.org/10.1140/epjp/i2017-11389-810.1140/epjp/i2017-11389-8Search in Google Scholar
[44] Seth GS, Sharma R, Mishra MK, Chamkha AJ. Analysis of hydromagnetic natural convection radiative flow of a viscoelastic nanofluid over a stretching sheet with Soret and Dufour effects. Engineering Computations [Internet]. Emerald; 2017 Apr 18;34(2):603–28. Available from: http://dx.doi.org/10.1108/ec-10-2015-029010.1108/EC-10-2015-0290Search in Google Scholar
[45] Seth GS, Tripathi R, Mishra MK. Hydromagnetic thin film flow of Casson fluid in non-Darcy porous medium with Joule dissipation and Navier's partial slip. Applied Mathematics and Mechanics [Internet]. Springer Science and Business Media LLC; 2017 Jul 18;38(11):1613–26. Available from: http://dx.doi.org/10.1007/s10483-017-2272-710.1007/s10483-017-2272-7Search in Google Scholar
[46] Mukhopadhyay S. Heat Transfer Analysis of the Unsteady Flow of a Maxwell Fluid over a Stretching Surface in the Presence of a Heat Source/Sink. Chinese Physics Letters [Internet]. IOP Publishing; 2012 May;29(5):054703. Available from: http://dx.doi.org/10.1088/0256-307x/29/5/05470310.1088/0256-307X/29/5/054703Search in Google Scholar
[47] Asim MN, Wasim M, Khan MUG, Mahmood W, Abbasi HM. A survey of ontology learning techniques and applications. Database [Internet]. Oxford University Press (OUP); 2018 Jan 1;2018. Available from: http://dx.doi.org/10.1093/database/bay10110.1093/database/bay101Search in Google Scholar PubMed PubMed Central
[48] Elbashbeshy EMA, Bazid MAA. Heat transfer over an unsteady stretching surface. Heat and Mass Transfer [Internet]. Springer Science and Business Media LLC; 2004 Oct 8;41(1):1–4. Available from: http://dx.doi.org/10.1007/s00231-004-0520-x10.1007/s00231-004-0520-xSearch in Google Scholar
[49] Mahanthesh B, Lorenzini G, Oudina FM, Animasaun IL. Significance of exponential space- and thermal-dependent heat source effects on nanofluid flow due to radially elongated disk with Coriolis and Lorentz forces. Journal of Thermal Analysis and Calorimetry [Internet]. Springer Science and Business Media LLC; 2019 Nov 21;141(1):37–44. Available from: http://dx.doi.org/10.1007/s10973-019-08985-010.1007/s10973-019-08985-0Search in Google Scholar
[50] Kunnegowda T, Mahanthesh B, Lorenzini G, Animasaun I. Significance of Induced Magnetic Field and Exponential Space Dependent Heat Source on Quadratic Convective Flow of Casson Fluid in a Micro-channel via HPM. Mathematical Modelling of Engineering Problems [Internet]. International Information and Engineering Technology Association; 2019 Sep 15;6(3):369–84. Available from: http://dx.doi.org/10.18280/mmep.06030810.18280/mmep.060308Search in Google Scholar
[51] Mahanthesh B, Gireesha BJ, Animasaun IL, Muhammad T, Shashikumar NS. MHD flow of SWCNT and MWCNT nanoliquids past a rotating stretchable disk with thermal and exponential space dependent heat source. Physica Scripta; 2019 May 29;94(8):085214. http://dx.doi.org/10.1088/1402-4896/ab18ba10.1088/1402-4896/ab18baSearch in Google Scholar
[52] Animasaun IL, Ibraheem RO, Mahanthesh B, Babatunde HA. A meta-analysis on the effects of haphazard motion of tiny/nano-sized particles on the dynamics and other physical properties of some fluids. Chinese Journal of Physics; 2019 Aug;60:676–87. http://dx.doi.org/10.1016/j.cjph.2019.06.00710.1016/j.cjph.2019.06.007Search in Google Scholar
[53] Koriko OK, Animasaun IL, Mahanthesh B, Saleem S, Sarojamma G, Sivaraj R. Heat transfer in the flow of blood-gold Carreau nanofluid induced by partial slip and buoyancy. Heat Transfer-Asian Research; 2018 Jun 20;47(6):806–23. http://dx.doi.org/10.1002/htj.2134210.1002/htj.21342Search in Google Scholar
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