CHAPTER III 

 CASES OF TWO-DIMEMSIONAL FLO»y 



The principal known types of tvvo-dimensional flow, including all that are treated in 

 Lamb's Hydrodynamics, '^ will be described or listed in this chapter. The important formulas 

 will be deduced and plots of the streamlines or sometimes the flow net will be shown. 



As the theory of complex variables is particularly suited for two-dimensional problems, 

 it will be used consistently. Acquaintance with the theory will be assumed, to the extent of 

 the summary in Chapter II, and also with hyperbolic functions, for which some formulas are 

 listed in Section 32. As a rule the standard formulation described in the next section will 

 be adopted. 



34. NOTATION AND FORM OF PRESENTATION 



The given boundaries pertaining to a particular problem are assumed to be drawn on the 

 plane of the complex variable z = x + iy, on which x and y are real Cartesian coordinates. Dia- 

 grams on this plane will be labeled indiscriminately with symbols representing geometrical 

 magnitudes, suci) as points or distances, and with symbols representing complex numbers. 



The appropriate mathematical transformation is represented in each case by 



w{2) = <^(a?,2/) + i iIj ix,y) . 



where (p and ip are real functions of a; and y. Except as stated, the fluid is assumed to be at 

 rest at infinity. For simplicity, each transformation is regarded as giving rise to two conjugate 

 types of flow. 



In one type of flow, cp represents the velocity potential and tl/ the stream function; the 

 equipotential curves are given by </> = constant, and the streamlines by i/r = constant. The 

 X and y components of the velocity are then 



[34a,b] 



In the conjugate type of flow, the velocity potential <;6 ' and the stream function i//'are 

 related to <^ and il/ as follows: 



0'= "A. !'/'=-</) 



The equipotential curves and the streamlines are interchanged; and the velocity components w' 

 V ' are : 



[34c,d] 





(90 



di'' 





d<{> diJ' 





dx 



"^' 



V — 



dy dx 





d4>' 







defy' 



u' = 



dx 



- w. 



v' = 



"dy ^" 



70 



