T^S' ""&> '^T ^^^ ■'^8 ^^^ ■"•" ^^® form o£ the Froude, Reynolds, Weber, 

 and Mach-Cauchy numbers, respectively. Thios, 



Ap/p 



CQ'(f' f' !' Pn> Rn> W^' M„) (2-25) 



in which C is a constant independent of the choice of dimensional imits 

 and of the variation of the tt terms. The function Q' is also free 

 from dimensional influences, provided that all variables in the relation- 

 ship are expressed in units of the same dimensional system. If necessary 

 to study the variation of velocity in a fluid-flow problem, equation 2-25 

 can be converted to the form: 



V = C'Q"(f , f , % F„, R„, W„, M„)(^)'^' . (2-26) 



The above exercise in dimensional analysis has not given any informa- 

 tion as to the form of the function Q" or the value of C. However, 

 the TT theorem has reduced the number of essential terms and produced 

 parameters independent of dimensional units. The form of the function 

 and values of the constant, for different types of flow conditions, must 

 be determined by analytical reasoning, experiment, or a combination of 

 reasoning and experiment. Equation 2-25 can also be used to establish 

 the requirements of similitude for scale models of fluid- flow problems 

 and to indicate the terms that must be investigated to determine scale 

 effects. 



3. Similarity by Differential Equations . 



If the differential equations that govern a phenomenon are known they 

 may give more insight into the laws of similarity than the use of dimen- 

 sional analysis of the variables that are known to influence or are sus- 

 pected of influencing the phenomenon. This is especially true if it is 

 desired to ascertain the laws of similarity for models with distorted 

 scales. If a phenomenon can be described with sufficient accuracy by 

 differential equations, the equations, after being converted to dimension- 

 less form, provide the basis of determining transfer parameters between 

 model and prototype. The transfer parameters obtained in this manner 

 also show, automatically, whether scale distortion can or cannot be used. 

 The equations arising in physical investigations can be reduced to dimen- 

 sionless forms in several ways, as in the following example. The equation 

 of motion of a simple pendulum is 



e^ + g sin = (2-27a) 



dt^ 



where I is the length of pendulum, g the acceleration of gravity, and 

 6 the angle of the pendulum with the vertical in circular measure. If 



35 



