CHAPTER IV 

 CASES OF THREE-DIMENSIONAL FLOW 



118. INTRODUCTION 



In this chapter the principal cases of three-dimensional potential flow that have been 

 worked out will be described, or at least listed. The mathematical solution of problems is 

 much more difficult in three dimensions than in two, since the mjethod of complex variables is 

 no longer available. The usual procedure is to obtain solutions of the Laplace equation by 

 any means whatever, and then by superposing solutions to work out proper combinations for 

 certain specified boundary conditions. A stream function ijj can be defined only for the case 

 of axial symmetry, as described in Section 16. 



For the components of the particle velocity of the fluid in the directions of Cartesian 

 or a;, y, z axes the symbols u, v, u: will be written without repetition of their definition and, 

 as usual, a will denote the speed, so that 



q= + (u^ + v^ + w2)y2 [118a] 



Confusion with the use of «■ for the complex potential in the last chapter should not arise, 

 since in two-dimensional motion the third velocity component v is always zero. The com,- 

 ponents of the velocity are understood to be calculated from the velocity potential (f> by means 

 of the usual equations 



dip dcf) d4> 



u = , V = - — , u = - Lll8b,c,dJ 



dx dy dz 



If polar coordinates r, d, a> are used, or cylindrical coordinates x, Tj, w as described 

 in Section 6, the components of the velocity in the corresponding coordinate directions are, 

 as in Equations [6k, 1, m, p, q, r], 



q = , qn = , q, = L118e,f,g] 



^' dT ' '^ r dd '^ 7- sin e dco 



d4> d<f> 1 dd, r . ■ .. 



"/hen the motion is steady, the pressure is given by the Bernoulli equation, which may 

 be written as in [34h] or 



298 



