

y 





~ 





-— — ________^ 





^— -~..,_^_~~ 





-~-,^^^^]~~ — — 



•f---^^ 



\ ^ — 



A B 



\^~--.-_____ 





h. -- 



X 



Figure 177 — Streamlines for the situation in Figure 176. 

 See Section 107. 



and, from Equation [107d] and hyperbolic formulas listed in Section 32, 



= V 



(cosh y. COS V - 1) + sinh jx sin v 



(cosh y. cos V + 1) + sinh y. sin v 



1/2 



q^^V^ 



cosh a - cos V 



cosh /i + cos t^ 

 On the positive a;-axis, fi > 0, !/ = 0, g = \u\ and 



X = — (sinh /x + fi), u = - V tanh — 



77 2 



On the end OB of the obstacle, where y - (), < v <n, q = \v\ and 



V = — (sin V + v)\ V = U tan — . 



On the face AB, ii.>_Q, y = v = tt, q - \u\ and 



[107j] 



[107k, 1] 



[107m, n] 



h . /x 



a; = (sinh y - fj.), u = - U coth — 



77 2 



[107o,p] 



These velocities are most easily calculated thus: on OB, for example, dy = (h/Tr) (cos ^+l)(/i^, 

 d cf> = - {hU/n) sin i^c^i/, hence -y = - dcfe./dy = - U sin u/(cos i/ + 1) = - f tan (u/2). 



If the total force on OB is calculated by integrating the Bernoulli pressure and evalu- 

 ating the improper integral in the usual way, the force is found to be zero provided the pres- 

 sure in the undisturbed stream is zero. This result is correct, as will be shown in the next 

 section. It is unsafe, however, to integrate the pressure up to a point at which q^^^; see 

 Section 85. 



(For notation and method; see Section 34; Reference 2, Section 10.6.) 



265 



