force F) ; and (c) the fluid properties (density p, specific weight Yj 

 viscosity y, surface tension o, and elastic modulus E) . If it is 

 assumed that, for the flow problem being investigated, all of these 

 variables are involved in the prototype or model flow, or both, 



f (a, b, c, d, V, Ap, p, 7, n, a, E) = '. (2-21) 



Since there are 3 fundamental dimensions (L, T, M or L, T, F) and 11 

 variables, the final functional relationship (according to Buckingham, 

 1914) must contain 8 ir terms, with 3 variables common to each. The 

 three common variables must also contain each of the fundamental dimen- 

 sions at least once. Selecting a characteristic length a, density p, 

 and velocity V as the three common variables, with the remaining eight 

 terms appearing singly in each group with a negative exponent (the selec- 

 tion of a, p, and V, and the use of a negative exponent for the re- 

 maining eight variables in the ir terms were selected in such a way as 

 to obtain it terms that are in the form of the different types of niom- 

 bers obtained previously by dynamical reasoning; other forms of the it 

 terms would have been correct, but would not have been as useful as model 

 scale ratios or for experimental analysis) the functional equation is 



in which 



0(7rj, 7r2, 7:3, 7:4,. . .TTg) = (2-22) 



TTj = a ' V 1 p J b"i , 



and 



ir^ = a ■^ V ^ p ^ c~^ ' 



7r3 = a^3 V^3 p^3 ^- 1 ^ 



Xc yc Zr 1 



7r5 = aV-^p-' 7"' , 



nj = a^ V^^ p^^ a" ^ , 



7r« = a'^Sv^S/SE-l 



Expressing the quantities in each tt term in their respective dimensional 

 units (L, T, and M) , and placing the sum of their exponents equal to zero 

 (which assures that each it term will be dimensionless) , there will be 

 three dimensionless unknowns and three simultaneous linear equations for 



33 



