Further thought about the thickening process will make it evident that the increase in velocity that takes place along a normal to the surface must be continuous. Let у be the perpendicular distance from the surface at any point and let и be the corresponding velocity parallel to the surface. If и were to increase discontinuously with у at any point, then at that point du/dy would be infinite. This would imply an infinite shearing stress [since the shear stress r = fi(du/dy)] which is obviously untenable
Consider again a small element of fluid (Fig. 7.2) of unit depth normal to the flow plane, having a unit length in the direction of motion and a thickness by normal to the flow direction. The shearing stress on the lower face AB will be r = /r(du/dy) while that on the upper face CD will be т + (дт/ду)ёу, in the directions shown, assuming и to increase with y. Thus the resultant shearing force in the x-direction will be [t + (дт/ду)ёу] – t = (дт/ду)ёу (since the area parallel to the x-direction is unity) but r = fi(du/dy) so that the net shear force on the element = ^(сРи/ду^ёу. Unless fi be zero, it follows that d2u/dy2 cannot be infinite and therefore the rate of change of the velocity gradient in the boundary layer must also be continuous.
Also shown in Fig. 7.2 are the streamwise pressure forces acting on the fluid element. It can be seen that the net pressure force is -(dp/dx)8x. Actually, owing to the very small total thickness of the boundary layer, the pressure hardly varies at all normal to the surface. Consequently, the net transverse pressure force is zero to a very good approximation and Fig. 7.2 contains all the significant fluid forces. The
effects of streamwise pressure change are discussed in Section 7.2.6 below. At this stage it is assumed that dp/dx = 0.
If the velocity и is plotted against the distance у it is now clear that a smooth curve of the general form shown in Fig. 7.3a must develop (see also Fig. 7.11). Note that at the surface the curve is not tangential to the и axis as this would imply an infinite gradient du/dy, and therefore an infinite shearing stress, at the surface. It is also evident that as the shearing gradient decreases, the retarding action decreases, so that
at some distance from the surface, when dujdy becomes very small, the shear stress becomes negligible, although theoretically a small gradient must exist out to у = oo.