In terras of these quantities the von Ka'rman momentum equation for two- 

 dimensional flow becomes 



dx" + (H + 2) TT d3T ~ -3" [6a] 



p\i 



and for axisymmetric flow it becomes for 6 « r 



w 



d( v ] , ,„ , ,; v du r w r , hl 



pU 



where r is the shearing stress at the wall. 



Physically the von Karman momentum equation is interpreted as repre- 

 senting the rate of change of the momentum of the fluid within the boundary 

 layer as a function of the frictional resistance and of the pressure gradient. 

 When solving a particular problem, U, the velocity at the outer edge of the 

 boundary layer, and dU/dx, representing the pressure gradient, are given by 

 measurements from pressure taps on the body or by potential-flow calculations. 

 Quantities such as the shape parameter H and the local-skin-friction coeffi- 

 cient t^/oU 2 , have to be determined by other means. By assuming flat-plate 

 values for H and r /oU 2 , the momentum equation can be integrated to give an 

 approximate solution which in many cases is not adequate for the problem in 

 question. The more exact solution depends on acquiring auxiliary relations 

 for the variation of H and r w /pU 2 in a pressure gradient. The detailed con- 

 sideration of H and r /0U 2 in a pressure gradient is the major theme of this 

 paper and will be discussed fully in the subsequent sections. 



As an illustration of the flow idealized by boundary-layer theory, 

 Figure 2 shows the flow pattern around a body of revolution. Essentially, 

 there is potential flow outside the boundary layer and viscous flow within. 

 The viscous boundary-layer flow is laminar from the nose up to the transition 

 zone where it develops into a turbulent flow. Beyond the tail, the flow 

 through the boundary layer merges to form a wake trailing behind the body. 



The velocity U at the outer edge of the boundary layer is that of 

 potential streamline and varies from zero at the stagnation point at the nose, 

 through a range of values along the body to the free-stream velocity Uoo in 

 the wake at infinity. Although the pressure p at the outer edge of the bound- 

 ary layer is essentially the same as that across the boundary layer, it varies 

 along the body with x and thus produces a pressure gradient. The pressure p 

 and the velocity U of the potential streamline are related by the Bernoulli 

 equation as follows 



