Streamline singular

As Re2 decreases, the streamlines are slightly deformed and the singular streamline (on which the fluid velocity is zero), lying in the meridian plane inside the drop, slightly moves toward the front surface. 

The streamlines corresponding to (7 79) are shown in Fig. 7 5. Although the velocity components ur and tie are perfectly well behaved, the shear stress Tr is singular in the limit r - 0. Indeed if we calculate tr0 0 0, we find that... 

Comparison of Equations 4-66 and 4-69 shows that 9p(x,y)/9x = 9 F(x,y)/9y, while Equations 4-65 and 4-68 show that 9p(x,y)/9y = -9 F(x,y)/9x. Hence, the Cauchy-Riemann conditions are identically satisfied, so that the streamfunction in Equation 4-67 is complementary to the vortex pressure in Equation 4-64. These results provide expressions for streamline tracing in the presence of distributed singularities. In general, as we will show in Chapter 5, the flow past (or from) an arbitrary entity can be represented by superpositions of line sources and vortexes, respectively, having strengths f(x) and g(x). The net pressure field... 

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