This paper presents semi-analytical solution of natural convection flow of non-newtonian fluid with temperature-dependent viscosity in pipe. The governing equations were solved using perturbation technique. The results obtained were analyzed for various thermo-solutal parameters involved in the dimensionless equations. Results within the constant viscosity show that increase in these parameters increases the velocity of the fluid flow as well as the temperature of the cylindrical pipe. It is observed that increase in the Reynold’s viscosity indices increases the temperature of the cylindrical pipe greatly.
Natural convection problems involved in non-Newtonian fluids are very significant in many engineering applications. The equations are very complex because of the nature of the fluids. Such fluid include oil, greases etc. Because of the compleity of these fluids, it is difficult to suggest a single model that will handle the problems involved. As such researchers in the past decade have done sme considerable research in the area of non-Newtonian fluid of the differential type. Amongst the earliest researchers are Fosdic and Rajagopal . They examined the thermodynamic stability of fluid of third grade. Massoudi and Christie  dealt with the effcts of variable viscosity and viscous dissipation on the flow of third grade fluid. The boundary layer equations of third grade fluid was treated by Pakdemirli .
Bejan  studied entropy generation in fundamentally convective heat transfer. Johnson etal  investigated a fluid flow which was infused with solid particles in a pipe, while approximate analytical solutions for flow of third grade fluid was examined by Yurusoy and Pakdemirli . Okedayo etal  studied the effects of viscous dissipation, constant wall temperature and a periodic field on unsteady flow through a horrizontal channel. Okedayo etal  analyzed the magnetohydrdynamic (MHD) flow and heat transfer in cylindrical pipe filled with porous media. They applied the Galerkin weighted residual method for the solution of momentum equation and semi- implicit finite differece method for the energy equation. They found that an increase in Darcy number leads to an increase in the velocity profiles, while increase in Brinkman number enhances the temperature of the system.
Obi  on approximate analytical solution of natural convection flow of non-Newtonian fluid through parallel plates , solved the coupled momentum and energy equations using the regular perturbation methd. He treated cases of constant and temperature-dependent viscosities in which Reynold’s and Vogel’s models were considered to account for the temperature-dependent viscosity case, while third grade fluid was introduced to account for the non-Newttonian effects. Obi  numerically analyzed the reactive third grade fluid in cylindrical pipe. He observed that the non-Newtonian parameters considered in the analysis: third grade parameter ( ), magnetic field parameter ( ), Eckert number ( ) and the Brinkman number ( ) had psitive effects on the velocity and temperature profiles.
Aksoy and Pakdemirli  examined the flow of a non-Newtonian fluid through a porous medium between two parallel plates. They involed Reynold’s and Vogel’s models viscosity and derived the criteria for validity for the approximate solution. Yurusoy etal  analyzed the flow of third grade fluid between parallel plates at different temperatures. Narges and Mahmood  investigated the effects of thin film flow of third grade fluid for a class of nonlinear second order differential equations. Ayub etal  examined the exact flow of third grade fluid. They used homotopy perturbation method for the analysis.
The distribution of this paper is in six sections. Section 1 was the intrduction of the research and the literature review. Section 2 was the problem formulation. In section 3, the analytical solutions of the constant viscosity case was presented and velocity and temperature-dependent viscosity considered in Reynold’s model. Section 4 was the discussion of the results obtained in section 3. Section 5 was the concluding remarks while section 6 was the reference.
This paper presents semi-analytical solution of natural convection flow of non-newtonian fluid with temperature-dependent viscosity in pipe. The governing equations were solved using perturbation technique. The results obtained were analyzed for various thermo-solutal parameters involved in the dimensionless equations. Results within the constant viscosity show that increase in increases the velocity of the fluid flow as well as the temperature of the cylindrical pipe. Results further indicate that increase in Brinkman number increases the temperature of the system. It is observed that increase in the Reynold’s viscosity indices n and M increases the temperature of the cylinder greatly.
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