Nanofluids are the suspension of metallic, nonmetallic or polymeric nano-sized powders in base liquid which are employed to increase the heat transfer rate in various applications. The term nanofluid, first introduced by Choi [1], refers to the fluids with suspended nanoparticles. Most of the convectional heat transfer fluids such as water, ethylene glycol and mineral oils have low thermal conductivity and thus are inadequate to meet the requirements of today’s cooling rate. An innovative way of improving the thermal conductivities of such fluids is to suspend small solid particles in the base fluids to form slurries. An industrial application test was carried out by Liu *et al.* [2] and Ahuja [3], in which the effect of particle volumetric loading, size and flow rate on the slurry pressure drop and heat transfer behavior was investigated (Xuan and Li [4]). Experimental results by Eastman *et al.* [5] showed that an increase in thermal conductivity of approximately 60% is obtained for the nanofluid consisting of water and 5% volume fraction of CuO nanoparticles. The procedure for preparing a nanofluid is given in the paper by Xuan and Li [4].

Many of the publications on nanofluids are about understanding of their behaviors so that they can be utilized where straight heat transfer enhancement is paramount as in many industrial applications, nuclear reactors, transportation, electronics as well as biomedicine and food (see Ding *et al.* [6]). Nanofluid is a smart fluid, where the heat transfer capabilities can be reduced or enhanced at will. These fluids enhance thermal conductivity of the base fluid enormously, which is beyond the explanation of any existing theory. They are also very stable and have no additional problems, such as sedimentation, erosion, additional pressure drop and non-Newtonian behavior, due to the tiny size of nanoelements and the low volume fraction of nanoelements required for conductivity enhancement. Much attention has been paid in the past to this new type of composite material because of its enhanced properties and behavior associated with heat transfer, mass transfer, wetting and spreading as well as antimicrobial activities, and the number of publications related to nanofluids increases in an exponential manner. The enhanced thermal behavior of nanofluids could provide a basis for an enormous innovation for heat transfer intensification, which is of major importance to a number of industrial sectors including transportation, power generation, micro-manufacturing, thermal therapy for cancer treatment, chemical and metallurgical sectors, as well as heating, cooling, ventilation and air-conditioning. Nanofluids are also important for the production of nanostructured materials, for the engineering of complex fluids, as well as for cleaning oil from surfaces due to their excellent wetting and spreading behavior (Ding *et al.* [6]).

There are some nanofluid models available in the literature. Among the popular models are the model proposed by Buongiorno [7] and Tiwari and Das [8]. Buongiorno [7] noted that the nanoparticle absolute velocity can be viewed as the sum of the base fluid velocity and a relative velocity (that he calls the slip velocity). He considered in turn seven slip mechanisms: inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage and gravity settling (Nield and Kuznetsov [9]). The nanofluid mathematical model proposed by Buongiorno [7] was very recently used by several researchers such as, among others, Nield and Kuznetsov [9, 10], Kuznetsov and Nield [11, 12], Khan and Pop [13], Khan and Aziz [14], Makinde and Aziz [15], Bachok *et al.* [16, 17], *etc*. On the other hand, the Tiwari and Das model analyzes the behavior of nanofluids taking into account the solid volume fraction of the nanofluid. In the present paper, we study the flow and heat transfer characteristics near a stagnation region of a permeable stretching/shrinking sheet immersed in a Cu-water nanofluid using the Tiwari and Das model. It is worth mentioning that this model was recently employed in Refs. [18–33], and the flow over a shrinking sheet was considered in Refs. [34–44]. The velocity distribution of the two-dimensional stagnation flow was first analyzed by Hiemenz (see White [45]) who discovered that this flow can be analyzed exactly by the Navier-Stokes equations. Homann (see White [45]) extended this problem to the axisymmetric stagnation flow and found that the solution differs a little from the plane flow, where the displacement and boundary layer thicknesses are slightly smaller and the wall shear stress is slightly larger. On the other hand, the temperature distributions of the Hiemenz and Homann flows were given by Goldstein [46] and Sibulkin [47], respectively. The governing partial differential equations are first transformed into a system of ordinary differential equations before being solved numerically. We study the effects of suction and injection at the boundary. Suction or injection of a fluid through the bounding surface, as, for example, in mass transfer cooling, can significantly change the flow field and, as a consequence, affect the heat transfer rate at the surface. In general, suction tends to increase the skin friction and heat transfer coefficients, whereas injection acts in the opposite manner (Al-Sanea [48]). Injection of fluid through a porous bounding heated or cooled wall is of general interest in practical problems involving film cooling, control of boundary layer, *etc*. This can lead to enhance heating (or cooling) of the system and can help to delay the transition from laminar flow (see Chaudhary and Merkin [49]). We mention to this end that studies of the boundary layer flows of a Newtonian (or regular) fluid past a permeable static or moving flat plate have been done by Merkin [50], Weidman *et al.* [51], Ishak *et al.* [52], Zheng *et al.* [53] and Zhu *et al.* [54, 55], while Bachok *et al.* [32] have considered the boundary layers over a permeable moving surface in a nanofluid.