Elsevier

Chemical Engineering Science

Volume 173, 14 December 2017, Pages 1-11
Chemical Engineering Science

A review on slip-flow and heat transfer performance of nanofluids from a permeable shrinking surface with thermal radiation: Dual solutions

https://doi.org/10.1016/j.ces.2017.07.024Get rights and content

Highlights

  • Heat transfer performance of nanofluid in the presence of non-linear radiation is investigated.

  • Dual solutions of the flow fields are presented for certain range of shrinking and suction parameters.

  • The effects of velocity slip and magnetic field on the nanofluid flow are further examined.

  • Presence of velocity slip significantly widens the dual solution existence range.

  • The rate of heat transfer was found to be a decreasing function of power-law parameter.

Abstract

A computational study on the stagnation-point flow of an electrically conducting nanofluid over a non-linear stretching/shrinking surface with first-order slip phenomenon is carried out. Because, nanofluids are considered as one of the closest kinds to the practical application of fluid flows owing to its comprehensive properties such as the Brownian motion and thermophoresis. The simulations were performed to understand the heat and mass transfer traits in the presence of non-linear radiation. Moreover, the Rosseland approximation model is incorporated to investigate the mechanisms of radiative heat transfer performance of nanofluids. For all cases, it has been seen that the governing conservation equations of current model possesses a similarity solution. The transformed system of nonlinear ordinary differential equations, is then integrated numerically with a boundary value problem solver, bvp4c in Matlab software. The captured numerical results have been displayed graphically and some interesting features like multiple (upper and lower) solutions are found. The critical values corresponding to the suction parameter fw and the shrinking parameter λ are computed. The rising thermo-physical dimensionless parameters overseeing the flow are power-law parameter, Hartmann number, slip parameter, Eckert number, temperature ratio parameter, radiation parameter, Brownian motion and thermophoresis parameters and Lewis number. It is found that, the slip parameter has a reducing impact on the skin friction coefficient for both upper and lower branch solutions. The major outcome of the present study is that the temperature ratio parameter boosts the temperature profiles for both solutions. While the temperature profiles show a decreasing behavior for higher non-linear radiation parameter. Validation of numerical scheme is accomplished by means of benchmarking with some already reported studies, and a great correlation is illustrated.

Introduction

In modern research era, the convectional heat transfer fluids like water, kerosene, engine oil and acetone assume a crucial part in numerous industrial segments including power generation, manufacturing and transportation, chemical production, air-conditioning and microelectronics. On the other hand, because of their low thermal conductivity they have restricted heat transfer capabilities. Recently, scientists are curious to develop different methods to increase their heat transfer performance. One of such methods to overcome this limitation is to improve thermal conductivity of conventional fluids via suspensions of nanoparticles in base fluids and led us to generate a new composite called “nanofluids”. Moreover, this field is much efficient in terms of heat transfer performance. Technically, these suspensions contain the base fluids and the nanoparticles with a size of (1–100 nm)which are suspended in them. Current works on nanotechnology has proved that nanoparticles with (diameter less than 50 nm) can change properties of the fluid since thermal conductivity of nanoparticles particles was higher than convectional fluids which are widely used as heat transfer fluids in thermal processes. The common nanoparticles those are being used are aluminum, copper, iron and titanium or their oxides. Initially, this idea was given by Choi and Eastman (1995) who concluded that these nanofluids have better conductivity and convective heat transfer coefficient relative to the base fluid. Based on their shape, size, and thermal properties, the thermal conductivity can be enhanced by about 40% with low concentration (1–5% by volume) of solid nanoparticles in the mixture. A broad spectrum of their application includes the sterilization of medical suspensions, cooling of heat sinks, hybrid-powered engines and nuclear reactor coolant.

The flow and heat transfer phenomena for nanofluids has been a topic of much research over the past two decades. In recent years, numerous analysts have assessed the properties and impact of nanofluids on the heat transfer change in thermal systems. After the work of Choi and Eastman (1995), numerous endeavors in this field have been accomplished to formulate the heat and transfer characteristics of nanofluid flows. In 2006, Buongiorno (2006) presented a comprehensive study concerning the heat transport in nanofluids and in his work he found an extraordinary rise in the thermal conductivity of nanofluids. After that, Khan and Pop (2010) have broken down the boundary layer flow of a nanofluid over a stretching surface. This was probably the first attempt to ponder the flow of nanofluids over stretching sheet by utilizing a model in which the Brownian motion and thermophoresis impacts were considered. A theoretical analysis has been done by Makinde and Aziz (2011) to investigate the impact of convective heat transfer on the flow of nanoparticle past a stretching sheet. Sheikholeslami (2014) scrutinized the hydrothermal characteristics of nanofluid flow and heat transfer between two parallel plates. The MHD thin film flow and heat transfer of pseudo-plastic nanofluids over a unsteady stretching surface is analyzed by Lin et al. (2015). Furthermore, Hashim and Khan (2016) numerically investigated the heat and mass transfer analysis in the flow of Carreau nanofluids. In this article, they utilized the revised model for nanofluids and solutions are obtained with the help of Runge-Kutta numerical technique. The influence of particle shape on Marangoni convection boundary layer flow of nanofluid is deliberated by Ellahi et al. (2016). They implemented the convective boundary and nanoparticles mass flux conditions in this analysis. Sheikholeslami (2016) reported the impact of variable magnetic field on the flow of Fe3O4-H2O nanofluid in a cavity with circular hot cylinder. Innovative numerical method, namely CVFEM is selected to perform the numerical computations. Further, Akbarzadeh et al. (2016) presented a study that concentrate on the sensitivity analysis of the thermal and hydraulic managements simultaneously for nanofluid flow inside a wavy channel. Recently, a number of investigations have been carried out to highlights the nanofluids transport by various authors, for instance, Bhatti et al., 2017, Esfahani et al., 2017, Ellahi et al., 2017, Shirvan et al., 2017, Sheikholeslami, 2017b, Sheikholeslami, 2017c, Sheikholeslami and Bhatti, 2017.

In recent times, heat transfer problem with the impact of non-linear thermal radiation is one of the thrust fields of contemporary research by reason of their tremendous applications in the field of Engineering and Physics. For example, in space technology such as comical flight aerodynamics rocket, in high-temperature processes such as plasma physics and space craft reentry aerodynamics. Furthermore it assumes a key part to enhance the heat transfer properties in polymer processing industry. Further, the MHD flow problems have attracted the interest of many researchers due to their wide appearances in technological process, such as, in MHD accelerator, nuclear fusion device, astrophysics, aeronautics and aerospace. In some conditions, the thermal radiation assumes prominent part on MHD flow and heat transfer with specific parameters, which has been verified in lots of research works. There are right now some delightful works on the fluid flows with non-linear radiation. Impact of thermal radiation on mixed convection flow over a vertical surface in a porous medium was studied by Bakier (2001). He employs the fourth-order Runge–Kutta method to obtain the numerical solutions of the governing equations. Cortell (2008) presented an endeavors to study the flow of viscous fluid over a nonlinear stretched surface by encountering the effects of thermal radiation in the energy equation. Later on, Lin et al. (2014) performed a numerical computation to discuss the radiation effects on Marangoni convection flow driven by a power-law temperature gradient for pseudo-plastic nanofluids. Again, the radiative heat transfer analysis of nanofluids against a flat plate in the presence of first order chemical reaction is investigated by Zhang et al. (2015). In this study, they employed the DTM-BF method to obtain approximate analytic solutions of their problem. Recently, Hashim et al. (2017) obtained the dual solutions in heat transfer analysis of a non-Newtonian Carreau fluid flow by encountering the non-linear thermal radiation. In their study, they utilized the non-linear Rosseland approximation for thermal radiation and noticed that the dual solutions exist for different values of shrinking parameter. Further, Sheikholeslami (2017a) investigated the influence of Lorentz forces on Fe3O4-water nanofluid by taking into account the radiation source term in energy equation.

With the deepening of the studies, researchers started to find that boundary slip condition portraying the relative motion between the solid surface and the fluid adjacent to the solid surface is an imperative interfacial property to affect the fluid flow characteristics. There is a finite velocity of the fluid-solid interface and such type of boundary condition for velocity is the so-called boundary slip, and it can be characterized by slip length. Possibly, Navier (1827) was the first who proposed the velocity slip boundary condition in which the tangential slip velocity uw is linearly related to the wall shear stress τw, in the form uw=Lτw, where L is the slip coefficient varies with temperature, pressure, normal stress, molecular parameter, and the characteristic of the fluid solid interface. It may be pointed out that there are several physical situations for which slip on a solid surface occurs. For instance, it happens in flow of rarefied gas (Sharipov and Seleznev, 1998), in flow over lubricated or coated surfaces (Teflon), rough or striated surfaces (Wang, 2003) and most recently, superhydrophobic nano-surfaces (Choi and Kim, 2006). Further, no-slip phenomenon arises in various industrial processes at the boundary of pipes, walls, and curved surfaces. The fluids displaying boundary slip find attention in technological problems like polishing of artificial heart valves and internal cavities. For this reason, researchers and scientists have given considerable attention to incorporate the slip condition at wall rather than no slip condition. For example, one of the earlier study that took into account the slip boundary condition for the boundary layer flow over stretching surface was conducted by Wang (2002). He obtained the numerical solutions for the governing problem by employing Runge-Kutta technique. Zheng et al. (2013) explored the radiative heat transfer problem for the flow of nanofluids past a stretching sheet by incorporating the velocity slip and temperature jump in porous medium. Afterward, Mukhopadhyay (2013) made an attempt to consider the partial slip on boundary layer flow of an incompressible viscous fluid with suction and injection. Khan and Hashim (2016) imparted an article to study the effects of velocity, temperature and solutal slip on the heat and mass transfer analysis of Carreau rheological model. In this analysis, they assumed the flow to be generated by a stretching wedge and achieved the numerical solutions for governing problem.

In perspective of all the previously mentioned applications, the novelty of the current investigation incorporates the following aspects:

  • 1.

    The impact of non-linear thermal radiation plays a key role in engineering and space technology in order to get high thermal efficiency.

  • 2.

    Nanofluids are essential in view of their substantial engineering applications for instance better coolants in computers and nuclear reactors, heat exchangers and cancer therapy.

  • 3.

    The velocity-slip at the walls is necessary to take into consideration for a reliable design and operation of microfluidic devices made of hydrophobic devices.

In order to speak about these issues, in this article, we furnished a theoretical model of magnetohydrodynamic, stagnation-point, slip-flow of Newtonian nanofluids towards a non-linear stretching/shrinking sheet along with heat and mass transfer. It is very much clear from the review of relevant literature that there does not exist too many papers in which the multiple solutions for such type of physical model are presented. Consequently, we are mainly focusing on how the dual solutions for the flow, heat and concentration field are obtained with respect to involved physical parameters. In addition, we try to determine the condition for which the dual solutions exist and where they don’t. In this regard, the proper non-dimensional similarity variables are utilized to change leading partial differential equations into a set of ordinary differential equations. An efficient Matlab package bvp4c based on finite difference scheme has been adopted for solving these non-linear ordinary differential equation systems emerging. Graphs and tables are presented to see and discuss the important hydrodynamic and thermal features of the flow characteristics of the nanofluids.

Section snippets

Mathematical formulation of the model

We suppose a steady two-dimensional flow of an incompressible and electrically-conducting nanofluid in the vicinity of stagnation-point from a stretching/shrinking sheet. The two slip effects of Brownian motion and thermophoresis are considered in this investigation. A locally orthogonal set of coordinates x,y is taken with the origin O to be fixed. The x-axis is guided along the continuous stretching/shrinking sheet and y-axis is in the transverse direction and the flow takes place at y0. A

Numerical approach

The self-similar ordinary differential Eqs. (10), (11), (12) are highly non-linear in nature. Therefore, the non-linear ordinary differential Eqs. (10), (11), (12) together with associated boundary conditions (13), (14) are solved numerically. In order to do this, we convert the current governing problem to a set of first-order equations. Here, we denotef=y1,f=y2,f=y3,θ=y4,θ=y5,ϕ=y6,ϕ=y7.Hence, the system of first order equations becomesy1=y2,y2=y3,y3=βy22-β+Ha2y2-Ha2-y1y3,y4=y5,y5=-3Nr

Results and discussion

In the following section, our main goal is to understand the physics of the mathematical model through graphical and tabular forms. We express skin-friction coefficient and local Nusselt to analyze the impacts of power-law parameter β, the Hartmann number Ha, the suction parameter s, the shrinking parameter λ, the slip parameter α, the Eckert number Ec, the temperature ratio parameter θw, the non-linear radiation parameter Nr, the Brownian motion Nb, the thermophoresis parameters Nt, the Lewis

Main findings

This perusal presents a theoretical and numerical study for the existence of dual similarity solutions for flow on a moving plate in nanofluids. The effects of slip-flow in the vicinity of stagnation-point along with non-linear radiative heat transfer have been further investigated. For the computational intent, Matlab function bvp4c is utilized to achieve the dual solutions. In comparability with the stretching sheet, the phenomenon of shrinking sheet has some interesting results. At last,

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