The Boundary Layer Equations for Laminar Flow

The flow in the boundary layer can be either laminar or turbulent. In the present section, attention is being restricted to those cases in which the flow in the boundary layer is laminar. 

in general, the effects of viscosity and heat transfer do not extend to the same distance from the surface. For this reason, it is convenient to define both a velocity boundary layer thickness and a thermal or temperature boundary layer thickness as shown in Fig. 2.14. The velocity boundary layer thickness is a measure of the distance from the surface at which viscous effects cease to be important while the thermal boundary layer thickness is a measure of the distance from the wall at which heat transfer effects cease to be important. 

For the flow of gases, the two boundary layer thicknesses are generally, approximately equal. For the flow of liquids, however, the two thicknesses are generally 

In order to illustrate how the boundary layer equations are derived, [7],[9],[13], [ 14],[ 15], consider two-dimensional constant fluid property flow over a plane surface which is set parallel to the x axis. The following are then defined  

8 = Some measure of the order of magnitude of the boundary layer thickness. 

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Governing Differential Equations. For a fluid as general as a gas in chemical equilibrium, the boundary layer equations for laminar flow over a flat plate are ... 

The physical situation where the fluid stream parallel to the moving plate is in an opposite direction to the motion of the plate (Fig. 18.11c) is also encountered in materials processing. Using a similarity method, Klemp and Acrivos [75] found that a critical value of the moving surface to the free stream velocity ratio (A/ = USIUJ) was 0.3541. The inability to obtain similarity solutions of the boundary layer equations for laminar flow was attributed to the boundary layer separation from the moving plate. Similarity and integral solutions for fluid friction... 

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