Renewed interest in the stretching sheet problem was sparked by a realization that some physical problems may be better modeled by a nonlinearly stretching sheet. Researchers who have reported the behaviour of fluid flow due to a nonlinear stretching sheet are Chaim [6], Prasad et al. [18], Ahmad et al. [19], Akyildiz et al. [20] and Kameswaran et al. [21]. Variable thickness of the sheet is useful in the mechanical, civil, marine and aeronautical structures and designs. The use of variable thickness helps to reduce the weight of structural elements and improve the utilization of the material. Sheets with variable thickness are often used in machine design, architecture, nuclear reactor technology, naval structures and acoustical components. With these industrial applications in mind, Lee [22] introduced the idea of variable thickness in theoretical studies. Fang et al. [23] studied the behaviour of boundary layer flow over a stretching sheet with variable thickness and explained the significant effects of the non-flatness of the sheet on the velocity and shear stress profiles by considering a special type of non-linear stretching $u_w(x)=U_0(x+b)^m$ for different values of $m$ being $b$ and $U_0$ constants. Khader et al. [24] extended the work of Fang et al. [23] and obtained the numerical solution for the slip velocity effect. Recently, Prasad et al. [25] and Vajravelu et al. [26], focused on heat transfer characteristics of fluid flow over a stretching sheet with variable thickness and power law velocity in the presence of a variable magnetic field.
Most of the studies above restricted their analysis to the hydromagnetic flow and heat transfer over a horizontal or a vertical plate and assumed the thermo-physical properties of the ambient fluid to be constant. However, it is known that these physical properties may change with temperature, especially the fluid viscosity and fluid thermal conductivity (Prasad et al. [18], Vajravelu et al. [27], Prasad et al. [28], and Hassanien [29]). For lubricating fluids, heat generated by internal friction and the corresponding rise in the temperature affects the physical properties of the fluid, and the properties of the fluid can no longer be assumed to be constant. The increase in temperature leads to an increase in the transport processes including heat transfer at the wall. Therefore, to predict the flow and heat transfer rates, it is necessary to take into account the variable fluid properties. From the literature, we find no evidence of previous studies on the combined effects of variable fluid properties and mixed convection in flow over a slender stretching sheet with variable thickness.
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The problem studied here extends the work of Prasad et al. [18] to the mixed convection flow with variable thickness. The coupled non-linear partial differential equations modeling the flow problem have been transformed to a system of coupled non-linear ordinary differential equations. These equations have been solved numerically using the Keller-box method, which is essentially a second order finite difference method. Computed numerical results for the flow and heat transfer characteristics are found to be in good agreement with experimental results in the literature (Fang et al. [23] and Khader et al. [24]. It is expected that the obtained results will not only provide useful information for industrial applications but would also serve to compliment and validate previous works.
The rate of heat transfer increases with increasing magnetic parameter and the Prandtl numbers. Hence, the effect of Prandtl number is to decrease the thermal boundary layer thickness and the wall-temperature gradient. In addition to this, the effect of the variable thermal conductivity parameter is to enhance the temperature field. 2ff7e9595c
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