An integral method is a powerful and flexible technique for the approximate solution of boundary-layer problems. It is based on the integration of the boundary-layer conservation equations over the boundary-layer thickness and the assumption of approximate and well-defined velocity, temperature, and mass-fraction profiles in the boundary layer. In this way, the partial differential conservation equations are replaced with ODEs in which the dependent variable is the boundary-layer thickness. The solution of the ODE derived in this way then provides the thickness of the boundary layer. Knowing the boundary-layer thickness, along with the aforementioned approximate velocity and temperature profiles, we can then easily find the transport rates through the boundary layer. The integral technique is quite flexible and, unlike the similarity solution method, can be applied to relatively complicated flow configurations.

Integral Momentum Equations

Let us first consider the velocity boundary layer on a flat plate that is subject to the steady and uniform parallel flow of a fluid, as shown in Fig. 5.1. We define a control volume composed of a slice of the flow field that has a thickness dx and height Y. We choose Y to be large enough so that it will be larger than the boundary-layer thickness throughout the range of interest. The inflow and outflow parameters relevant to momentum and energy are also depicted in Fig. 5.1.