Natural convection flow from a continuously moving vertical surface immersed in a thermally stratified medium H. S. Takhar, A. J. Chamkha, G. Nath Abstract Natural convection boundary layer ¯ow over a continuously moving isothermal vertical surface immersed in a thermally strati®ed medium has been investigated here. The non-linear coupled partial differential equations governing the non-similar ¯ow have been solved numeri- cally using an implicit ®nite difference scheme. For small values of the streamwise distance the partial differential equations are solved by using a perturbation expansion procedure and also using the Shanks transformation. The results indicate that the thermal strati®cation signi®cantly affects both the surface shear stress and the surface heat transfer. The buoyancy parameter and the Prandtl number increase signi®cantly, both the surface shear stress and heat transfer. Also the buoyancy force gives rise to an overshoot in the velocity pro®le. 1 Introduction Heat transfer processes by natural convection in a ther- mally strati®ed medium frequently occur in the natural environment and in many industrial and technical appli- cations. Strati®cation is important in lakes, rivers and the sea, and in condensers of power plants and various in- dustrial units. The natural convection ¯ow over a heated vertical surface with uniform temperature immersed in an ambient ¯uid whose temperature increases linearly with height has been studied by Eichhorn [1], Chen and Eich- horn [2], and Venkatachala and Nath [3]. They solved the partial differential equations governing the ¯ow by a using series solution method, the local non-similarity method and an implicit ®nite difference scheme, respectively. Kulkarni et al. [4] have obtained a similarity solution of the above problem. The ¯ow and heat transfer in the boundary layer in- duced by a surface moving with uniform or non-uniform velocity in an otherwise ambient ¯uid has many practical applications in manufacturing processes in industry. Sakiadis [5] was the ®rst to study the ¯ow due to a solid surface moving with a constant velocity in an ambient ¯uid. Since then several investigators [6±12] have con- sidered various aspects of this problem such as the heat transfer with prescribed wall temperature or heat ¯ux, mass transfer, non-uniform wall velocity, surface suction or blowing, effect of a magnetic ®eld or and) a parallel free stream velocity etc. In almost all the cases self-similar solutions were obtained. However, Jeng et al. [14] and Chiam [20] have studied the non-similar ¯ow and heat transfer. In all the above studies the buoyancy forces re- sulting from the temperature differences in the ¯uid were neglected. Moutsoglou and Chen [22] have considered the effect of the buoyancy forces on the ¯ow and heat transfer characteristics of the laminar boundary induced by an inclined, continuous ¯at surface that moves with a con- stant velocity in a ¯uid at rest. Subsequently, Ramachan- dran et al. [23] have presented the correlation equations for the local and average Nusselt numbers. In this paper, we have investigated the ¯ow and heat transfer characteristics of the steady laminar boundary layer induced by a vertical ¯at surface that moves with a constant velocity in a stable thermally strati®ed ¯uid at rest. We have considered the effect of the buoyancy forces which arise due to the temperature differences in the ¯uid. The coupled non-linear partial differential equations governing the ¯ow have been solved numerically using an implicit ®nite-difference method [24]. For small values of the streamwise distance, the governing equations have been solved by using a perturbation expansion technique [25] along with the Shanks transformation [26]. For some particular cases, the results have been compared with the theoretical and experimental results of Tsou et al. [8], the theoretical results of Erickson et al. [6] and Moutsoglou and Chen [22] and the experimental results of Grif®n and Throne [7]. 2 Analysis Let us consider a vertical heated ¯at surface with constant wall temperature T w moving with constant velocity U in the x-direction, in a stable thermally strati®ed ambient ¯uid. The ambient temperature T 1 x is assumed to vary linearly with the height x. The buoyancy force arises due to the temperature differences in the ¯uid. We choose a rectangular Cartesian coordinate system with its origin ®xed at the leading edge of the vertical surface, such that Originals Heat and Mass Transfer 38 2001) 17±24 Ó Springer-Verlag 2001 17 Received on 1 February 2000 H. S. Takhar &) Department of Engineering, Manchester Metropolitan University Manchester, M1 5GD, UK A. J. Chamkha Department of Mechanical Engineering, Kuwait University Safat-13060, Kuwait G. Nath Department of Mathematics, Indian Institute of Science Bangalore-560012, India