An Improvement of the Differential Transformation Method and Its Application for Boundary Layer Flow of a Nanofluid

The main feature of the boundary layer flow problems of nanofluids or classical fluids is the inclusion of the boundary conditions at infinity. Such boundary conditions cause difficulties for any of the series methods when applied to solve such a kind of problems. In order to solve these difficulties, the authors usually resort to either Padé approximants or the commercial numerical codes. However, an intensive work is needed to perform the calculations using Padé technique. Due to the importance of the nanofluids flow as a growing field of research and the difficulties caused by using Padé approximants to solve such problems, a suggestion is proposed in this paper to map the semi-infinite domain into a finite one by the help of a transformation. Accordingly, the differential equations governing the fluid flow are transformed into singular differential equations with classical boundary conditions which can be directly solved by using the differential transformation method. The numerical results obtained by using the proposed technique are compared with the available exact solutions, where excellent accuracy is found. The main advantage of the present technique is the complete avoidance of using Padé approximants to treat the infinity boundary conditions. 1. Introduction Nanotechnology is an advanced technology, which deals with the synthesis of nanoparticles, processing of the nano materials and their applications. It is well known that 1？nm (nanometer) = 10？9 meter. Normally, if the particle sizes are in the 1–100？nm range, they are generally called nanoparticles. Nanotechnology has been widely used in industry since materials with sizes of nanometers possess unique physical and chemical properties. Nanoscale particle added fluids are called as nanofluid. The term “nanofluid” was first used by Choi [1] to describe a fluid in which nanometer-sized particles are suspended in conventional heat transfer basic fluids. Fluids such as oil, water, and ethylene glycol mixture are poor heat transfer fluids, since the thermal conductivity of these fluids plays important role on the heat transfer coefficient between the heat transfer medium and the heat transfer surface. Numerous methods have been taken to improve the thermal conductivity of these fluids by suspending nano/micro or larger-sized particle materials in liquids. An innovative technique to improve heat transfer is by using nanoscale particles in the base fluid [1]. Therefore, the effective thermal conductivity of nanofluids is expected to enhance heat transfer compared with conventional heat transfer