Flow and heat transfer of nanofluids at a stagnation point flow over a stretching/shrinking surface in a porous medium with thermal radiation

Highlights

• Momentum boundary layer thickness decreases with an increase in ϕ, when $a/c>1$.

• Cu–water velocity is higher than ${\text{Al}}_2{\text{O}}_3$–water and ${\text{TiO}}_2$–water for stretching and shrinking sheets.

• Skin-friction coefficient increases with increasing the value of volume fraction ϕ.

• Decrease in suction parameter $s>0$ decreases velocity profiles for Cu–water.

• Temperature for Cu–water increase with decreasing the suction parameter $s>0$.

Abstract

In this paper, the effects of thermal radiation and viscous dissipation on a stagnation point flow and heat transfer over a flat stretching/shrinking surface in nanofluids are analyzed. The effects of suction/injection are also considered. Using a similarity transformation, the governing equations are transformed into a system of nonlinear ordinary differential equations. The resulting system is then solved numerically by Runge–Kutta–Fehlberg method with shooting technique. It is observed that the local Nusselt number increases with increment in the suction/injection parameter for stretching sheet whereas reverse effect is observed for shrinking sheet. It is found that skin-friction coefficient increases for both stretching/shrinking sheet with increase in volume fraction of the nanoparticles.

Introduction

Studies on boundary layer flow and heat transfer characteristics at a stagnation point considering nanofluids have received great attention during recent times. A new breakthrough has been made by thermal scientists and they are discovering unexpected thermal properties of nanofluids, and proposing new mechanisms behind the hidden thermal properties of nanofluids. Nanofluid is a suspension of solid nanoparticles (1–100 nm diameters) in conventional liquids like water, oil or ethylene glycol. The concept of a nanofluid, which is a new kind of fluid, has been advanced by Choi [1] who showed substantial augmentation of heat transported in suspensions of copper or aluminum nanoparticles in water and other liquids. By using different solvents and particles it is hoped to create composite fluids with completely new properties. Nanofluids may be used as next generation coolants for computers and safe coolants for nuclear reactors. Nanofluids are now being developed for medical applications, including cancer therapy and safer surgery by cooling. Other possible areas for the application of nanofluid technology includes cooling of a new class of super powerful and small computers and other electronic devices for use in military systems, vehicle cooling and transformer cooling. Nanofluids could be used in major process industries, including materials and chemicals, food and drink, oil and gas, paper and printing, and textiles. Nanofluids, when used as coolants can provide dramatic improvements in the thermal conductivity of host fluids compared to that of the traditional fluids. By using nanofluids highest possible thermal properties at the smallest possible concentrations by uniform dispersion and stable suspension of nanoparticles in base fluids can be achieved.

Xuan and Li [2] used pure copper particles in the study of convective heat transfer and flow features of nanofluids. They found that not only the volume fraction but also the particle dimensions and material properties are important in order to achieve a substantial rise in heat transfer coefficient. Rapid advances in nano manufacturing, many inexpensive combinations of liquid/particle are now available. These include particles of metals such as aluminum, copper, gold, iron and titanium or their oxides. The base fluids normally used are water, ethylene glycol, toluene and oil. The main features of using nanofluids in such systems are to enhance the heat transfer and improving the viscosity. Considering these advantages, research in different areas using nanofluids has attracted more and more interests theoretically and experimentally. A comprehensive survey of convective transport in nanofluids was made by Buongiorno [3], who gave an explanation for the abnormal increase of the thermal conductivity of nanofluids. Therefore, by mixing the nanoparticles in the fluid, thermal conductivity of the fluids improve the heat transfer capability. Pantzali et al. [4] presented an experimental study that showed the role of CuO–water nanofluids as coolant in the plate heat exchangers. The experimental results confirmed that the type of flow inside the heat exchanging equipment also affects the efficiency of a nanofluid as coolant, besides its physical properties. Experimental studies of pressure drop and convective heat transfer of ${\text{TiO}}_2$–water nanofluid in a double pipe heat exchanger were reported by Duangthongsuk et al. [5]. They analyzed the effect of mass flow rate of hot and cold fluids, and nanofluid temperature on heat transfer coefficient. The results showed that the convective heat transfer coefficient of nanofluid is slightly higher than that of the base liquid by about 6–11%. The problems related to boundary layer flows in nanofluids have been considered by several researchers in recent years and a good amount of work can be found in the literature. Recently, an analysis has been carried out by Vajravelu et al. [6] to study the convective heat transfer in a nanofluid flow over a stretching surface. In particular, they have focused on Ag–water and Cu–water nanofluids to investigate the effects of the nanoparticle volume fraction on the flow and heat transfer characteristics in the presence of thermal buoyancy and temperature-dependent internal heat generation/absorption.

The studies on stagnation-point flow have been important in the boundary layer flows and their relevance in many manufacturing processes such as boundary layer along material handling conveyers, aerodynamic extension of plastic sheet and petroleum industries. Bachok et al. [7] analyzed three-dimensional stagnation-point flow and heat transfer in a nanofluid. In the past, suction/injection effects in Newtonian fluid was considered by several authors. Vajravelu [8] studied convection heat transfer at a stretching sheet with suction or blowing in Newtonian fluid. Gorla and Sidawi [9] studied flow and heat transfer characteristics from a continuous surface with suction and blowing. Watanabe [10] studied the forced and free convection boundary layer flows with uniform suction or injection on a vertical flat plate. Viscous dissipation changes the temperature distributions by playing a role like an energy source, which affects heat transfer rates considerably. In fact, the shear stresses can induce a considerable amount of heat generation. This effect is usually neglected which is considered as an important parameter in the present study. Earlier, Murthy and Singh [11] have studied effects of viscous dissipation on the flow of an incompressible fluid in a saturated porous medium. Al-Hadhrami et al. [12] discussed a new model for viscous dissipation in a porous medium across a range of permeability values. Exact modeling of the internal heat generation/absorption is quite difficult. However, some simple mathematical models can express its average behavior for most physical situations. Heat generation/absorption has been assumed to be constant, space-dependent or temperature dependent by many previous investigators. Sparrow and Cess [13] considered temperature-dependent heat absorption to analyze steady stagnation-point flow and heat transfer. Chamkha [14] discussed the effects of magnetic field on mixed convection flow in a saturated porous medium with heat generation/absorption. Fang [15] analyzed boundary layer flow over a shrinking sheet with power-law velocity and found that dual solutions exist for such type of flows. Later, Fang et al. [16] examined viscous flow over a shrinking sheet with a second order slip flow model and shown the dual solutions do exists for the problem under consideration. We have found the dual solutions for the problem under consideration for various values of suction and stretching parameters on velocity and temperature profiles.

The thermal radiation effects becomes intensified at high absolute temperature levels due to basic difference between radiation and the convection and conduction energy-exchange mechanisms. In the context of space technology, some devices for space applications are designed to operate at high temperature levels in order to achieve high thermal efficiency. Hence, the effects of radiation are of vital importance when calculating thermal effects in the processes involving high temperatures. Thus Akbar et al. [17] studied radiation effects on MHD stagnation point flow of nanofluid towards a stretching surface with convective boundary condition. Abdul Hakeem et al. [18] analyzed the effect of heat radiation in a Walter’s liquid B fluid over a stretching sheet with non-uniform heat source/sink and elastic deformation. Nadeem et al. [19] examined heat transfer analysis of water-based nanofluid over an exponentially stretching sheet. In all these works, the linear approximation to the thermal radiation model is used based on the assumption that the temperature differences within the flow are so small that $T^4$ can be expressed as a linear function of $T_{\infty }$, which is obtained by expending $T^4$ in Taylor series about $T_{\infty }$ and neglecting the higher order terms. Whereas, Elbashbeshy [20] examined the thermal radiation effect on heat transfer over a stretching surface by taking into account of the full form of radiation term. Swati Mukhopadhyay [21] investigated the effect of boundary layer flow and heat transfer over a porous moving plate in the presence of thermal radiation, taking into account of the full form of radiation term. Gururaj and Devi [22] analyzed the effect of magnetohydrodynamic boundary layer flow with forced convection past a nonlinearly stretching surface in the presence of variable temperature and nonlinear radiation effects. Recently, Pal and Mondal [23] studied the effects of temperature-dependent viscosity and variable thermal conductivity on MHD non-Darcy mixed convective diffusion of species over a stretching sheet.

In view of all the above mentioned applications, the present work deals with steady stagnation-point flow of nanofluids in a porous medium in the presence of thermal radiation and viscous dissipation for three different types of nanoparticles, namely copper (Cu), alumina (${\text{Al}}_2{\text{O}}_3$), titanium dioxide (${\text{TiO}}_2$) by considering the nanofluid model proposed by Tiwari and Das [24]. In this paper, we study the effects of effective Prandtl number, heat source, permeability of the porous medium, Eckerts number, solid volume fraction of the nanofluid, suction/injection parameter and the stretching/shrinking parameter on velocity and temperature fields for nanofluids using the thermophysical properties nanoparticles of the base fluid (water) (see Table 1).