^ 











— — - 



■ 





-____________^^ 



" " — — — -. 



__________^ 



— _____ 



~~ 



-_ 



~--~~.__^^ 



y '^^^^^ 



~~~ — - — — — _ 



^CTTv— — — __ 



2b 



i V^-^^^^.,_-__ 



t 



^/\ "^^--— ~-^:!l__ X 



— __^_____p-p°° 



y/ () Axis of Symmetry 



Figure 202 - Streamlines past a semi-infinite solid of revolution obtained 



from a point source at 0, and plot of the pressure along the axis 



and over the solid. See Section 121 



where ds is its width along the tangent to S in a plane through the x-axis. Let the normal to 

 the surface at any point on the ring meet the axis at an angle e . Then the force due to the 

 pressure p on any element of the ring has a component along the axis equal to the force 

 multiplied by cos e , and, since p and e are uniform around the ring, the total component of 

 force due to the ring is dF = p (27Twds) cos « . But d(o'= ds cos e where doTis the element 

 of ST corresponding to ds. IJence the total force on' the solid, measured positively toward 

 negative x, is 



F = 2n \ poidui 



[121i] 



This formula holds for any surface and any pressure distribution which have a common axis 

 of symmetry. 



For steady motion, the excess pressure, p-p^ = p (U^ - g^)/2, may be inserted for p in 

 this formula. It is simpler to change to r as a variable of integration. Substituting 

 ^2 _ ;.2_~2 jj^ Equation [121h], and then differentiating, 



aj^ = 4 6^- , 2(odco= dr 



r2 ,3 



[121j,k] 



Eliminating x-Zr from Equation [121f] by means of [121g] and [121j], 



^2 o a4 



rj^ = U^ [1 + 



\ ,2 ,4 / 



[1211] 



307 



