## 1 Introduction  
A better understanding of heat transfer and boundary layer flow through a permeable surface can benefit several important areas, especially in technology and engineering fields, in such applications as wire drawing, glass-fiber and paper production, and insulation design, to name a few. Further, an innovative technique to improve heat transfer known as nanofluid has been extensively used during these last decades because the traditional heat transfer fluids are known to behave poorly in terms of heat transfer. The nanofluid term was first introduced by [PERSON] [1], and is defined as a fluid capable of suspending nanoscale particles in the base fluid. Basically, the nanoparticles are constructed from carbon, aluminium, copper, and generally come from chemically stable materials. The nanofluid concept has been introduced and improvised as a method of enhancing the heat transfer performance rates in liquids. Materials in the size of nanometers possess unique chemical and physical properties, since nanofluids make it possible for these materials to flow smoothly through microchannels without clogging because of their small size, as they behave similarly to liquid molecules [2]. This fact has attracted researchers to perform investigations on heat transfer characteristics through nanofluids. Several authors [3; 4; 5; 6] have extensively studied and analyzed the effects of nanofluids on heat transfer  
In an area of fluid dynamics, the flow of potential around a circular cylinder is a solution to allow the flow of an incompressible and inviscid flow around a cylinder that is transverse to the flow itself. A boundary layer is created where the surface of a velocity is at zero value and transforms to the value of the free stream some distance away from the cylinder surface. When the boundary layer thickness is less than the radius of cylinder, then the flow is considered to be two-dimensional. To the contrary, the thickness of the boundary layer may be the same order of the cylinder radius in the case of a thin or slender cylinder. Therefore, instead of considering the flow as two-dimensional, it is more accurate that the flow be considered as an axisymmetric flow (see [7]). Due to this reason, the research of mixed convection through a cylinder has lead to an excellent collection of articles such as by [8; 9; 10; 11]. At the same time, considerable progress on the understanding of radiative heat transfer in flow processes in industry and engineering fields is very important since it is one of the three fundamental modes of heat transfer. Thermal radiation can be explained as electromagnetic radiation generated by charged particles of thermal motion in matter. The effects of thermal radiation are of considerably interest in flow processing at a high temperature, because the radiation can significantly influence the participating fluids' heat transmit rate, as well as the temperature distribution in the flow of the boundary layer when temperatures are high. Considerable research in the area of thermal radiation have been proposed recently, and the concept of thermal radiation is still of significant interest, as it may provide better results to affect the kinematic flows and achieve a suitable control on the cooling rate in such a way as to ensure the solidification at a slower rate than other well-known methods. Here we study the steady, two-dimensional mixed convection boundary layer flow, with thermal radiation filled with a nanofluid along a permeable vertical cylinder lodged in a porous medium, where we apply the mathematical model as presented by [PERSON] and [PERSON] [12], which is also extended by [PERSON] et al. [13]. Copper (Cu), aluminium (Al\({}_{2}\)O\({}_{3}\)) and titanium (TiO\({}_{2}\)) are among the nanoparticles reviewed in this study, and by performing a similarity transformation method, a set of ordinary differential equations has been reduced from the governing partial differential equations system.