# Stability Analysis of Mixed Convection Flow towards a Moving Thin Needle in Nanofluid
## 1 Introduction  
Along with the development of technology, most engineers have ventured to create a new system of application. Previously, most of the systems used in industries considered the conventional heat transfer fluid such as oil, water and ethylene glycol. Unfortunately, these types of heat transfer fluids did not show good performance in a cooling system. [PERSON] [1] found an alternative way to overcome this disadvantage by introducing the term nanofluid in fluid dynamics. The usage of the nanofluid as a cooling device will intrinsically enhance the thermal performance of the manufacturing processes and also reduce the operating costs. In fact, nanofluid is formed by adding some of the nanometer-sized particles (called nanoparticles), the diameter of which is less than 100 nm, into the base fluid. There are a few types of nanoparticles such as metal, oxide, carbides and carbon nanotubes. Nanofluid possesses some special behaviors, by which it is very stable and does not have any additional problems such as erosion, sedimentation, non-Newtonian behavior or additional pressure drop. This happens due to the tiny size of the nano-elements and low volume fraction of nano-elements needed for the thermal conductivity improvement. In view of its applications, nanofluid has the potential of being a new generation of coolants in automotive applications. It tends to cool down in the electronic application effectively by removing the high heat flux such as liquid cooling, air cooling and two-phase cooling. It is also helpful in cancer imaging and drug delivery for cancer therapeutics in biomedical industries [2; 3; 4]. The comprehensive literature on the mathematical models of nanofluid is reported by some researchers such as [PERSON] et al. [5], [PERSON] [6], [PERSON] and [PERSON] [7], [PERSON] and [PERSON] [8], as well as [PERSON] [9]. The effectiveness of the thermal conductivity of nanofluid among the models is studied in order to compare the theoretical data with the experimental data. There are two common types of nanofluid models that have been considered in fluid dynamics, namely the model proposed by [PERSON] [6] and [PERSON] and [PERSON] [7]. [PERSON]'s model is a two-phase model in which the slip velocity between base fluid and nanoparticles is not equal to zero. This model takes into account seven slip mechanisms, which produce a relative velocity between the base fluid and nanoparticles such as inertia, thermophoresis, diffusiophoresis, Brownian motion, fluid drainage, gravity and the Magnus effect. In contrast with [PERSON]'s model, the [PERSON] and [PERSON] model is a single-phase model that considers the viscosity model proposed by [PERSON] [10]. Since this model is a phase model, base fluid and nanoparticles are said to be in thermal equilibrium, flowing at a uniform velocity, and a no-slip condition occurs between them. This model considers the effect of nanoparticles volume fraction. There are several articles regarding these two models that can be found in the literature [11; 12; 13; 14]. Due to the importance of several industrial and technological applications, researchers are now grabbing the opportunity to investigate the boundary layer flow and heat transfer analysis induced by a thin needle in various fluid. The term thin needle simply means a parabolic revolution about its axes direction in addition to the variable thickness. The motion of the thin needle distracts from the free stream direction, and this situation is the main concern in experimental studies for the flow and heat transfer analysis in order to measure the velocity and temperature profiles of the system. It has many practical applications especially in the medicine and engineering industries. Some of the applications are a hot wire anemometer for measuring the velocity of the wind, the blood flow problem, transportation, coating of wires, lubrication and geothermal power generation. The boundary layer flow past a thin needle in viscous fluid have been studied by [PERSON] [15].