the Method of Dimensions, 703 



6. Mistakes which arise from the use of Incomplete Equations; 

 flow of liquid along a smooth pipe. 



Examples have just been given of equations which are 

 at once seen to be incomplete from the fact that no 

 (Hmensionless product can be formed. Such cases are 

 simple, because the number of quantities in the initial 

 equation is too small. But in more complicated problems, 

 in which the number of quantities which must obviously, 

 from the Plrysics of the case, be included in the initial 

 equation is greater than the number of fundamental units 

 needed, the foregoing test is often not applicable. There 

 is then no recourse but to compare the results with the 

 observed facts ; and such a case will now be illustrated. 



Let it be required to find the relation between the 

 pressure gradient G and the constant speed S of a liquid 

 of density p flowing through a smooth straight pipe of 

 diameter D. We write down, as the initial equation, 



F (a, S, p, D) = ; .... (15 N 



and since three fundamental units suffice for the fou> 

 quantities, we have n — k = l and the II theorem gives us 

 the equation 



/Q-.O ...... (16, 



or o 2 



G=N?g-, ...... (17) 



where N is a pure number, — a root of equation (16). 



But equation (17) is not generally correct ; for it does 

 not agree with the facts observed at low speeds, Avhich are 

 described by Poiseuille's law. The source of the difficulty 

 is that in writing down equation (15) the fact was over- 

 looked that viscosity is one of the things which may 

 influence the flow of a fluid. Equation (15) is correct, 

 i. e. complete, only when the circumstances are such that 

 the resistance is independent of the value of the viscosity. 

 It is not sufficient that the considerations be limited to 

 liquids which all have the same viscosity ; it is required 

 that the course of the phenomenon remain unchanged 

 even if the viscosity be varied. A dimensional constant. 

 the viscosity //,, has been omitted from equation (15) : the 

 equation is not complete and the result expressed by 

 equation (17) is not generally correct. 



