Thermal boundary-layer problems in a semi-infinite flat plate by the differential transformation method
Introduction
A common area of interest in aerodynamics is the analysis of thermal boundary-layer problems for two-dimensional steady and incompressible laminar flow. Accordingly, it is necessary to develop numerical methods capable of providing accurate solutions for problems of this type. In his pioneering work of 1904, Prandtl established that the boundary-layer momentum equation can be transformed into an ordinary differential equation for flows passing over a flat plate with a uniform free-stream velocity. In 1908, Blasius obtained the solution of the boundary-layer momentum equation by using a series expansion method. In 1938, Howarth employed numerical methods to obtain precise solutions of the Blasius equation [1]. Na [2] introduced a group of transformations to reduce third-order boundary value problems to a pair of initial value problems, and solved them by using a forward integration scheme. Subsequently, Yu and Chen [3] combined this reduction method with the differential transformation concept introduced by Zhou [4] to calculate the numerical solutions of the Blasius equation.
In the present study, the differential transformation method is employed to obtain the solutions of thermal boundary-layer problems for flow passing a flat plate. The study begins by using a group of transformations to reduce the third-order nonlinear boundary-layer equation and the second-order linear thermal boundary-layer equation to a pair of initial value problems. The differential transformation method is then employed to solve this system of initial value problems. Finally, the current numerical results are compared carefully with those given by other integral approximation methods to verify the accuracy of the proposed method.
Section snippets
Mathematical formulation––differential transformation method
Consider the flow of a viscous fluid over a semi-infinite flat plate, as shown in Fig. 1. The temperature of the wall, Tw, is uniform and constant and is greater than the free stream temperature, T∞. It is assumed that the free stream velocity, U∞, is also uniform and constant. Further, assuming that the flow in the laminar boundary layer is two-dimensional, and that the temperature changes resulting from viscous dissipation are small, the continuity equation and the boundary-layer equations
Numerical results and discussion
The numerical calculation procedure adopted in the current investigation of the boundary-layer energy equation in a flat plate flow may be described as follows:
(1) The boundary value problems (Eqs. , , ) are reduced to a pair of initial value problems (Eqs. , , , ), and are then solved by using the differential transformation method. The detailed procedural steps are as follows. Firstly, the value of dF(∞)/dξ is computed by solving Eq. (18) with the initial conditions given in Eq. (19).
Conclusion
This paper has discussed the applicability of the differential transformation method to thermal boundary-layer problems for a flow passing over a semi-infinite flat plate. Some numerical results of the thermal boundary-layer problems have been presented in order to demonstrate the accuracy and adaptability of the differential transformation method. It has been demonstrated that the results for the velocity and temperature field obtained by the present method are in good agreement with those
Acknowledgements
The current author wish to thank Professors C.K. Chen of Department of Mechanical Engineering of National Cheng Kung University and L.T. Yu of Department of Mechanical Engineering of Chengshiu Institute of Technology for their kind assistance and support provided during this research study.
References (4)
- et al.
Convective Heat Transfer
(1995) Computational Methods in Engineering Boundary Value Problems
(1979)