NO. 15.] APPLICATION OF SMALL SCALE EXPERIMENTS. 49 



E. ON THE APPLICATION OF SMALL SCALE EXPERIMENTS. 



It is obvious that experimental investigations of the phenomenon of dead- 

 water cannot be made except on a considerably diminished scale. It is of 

 importance, therefore to see under what circumstances we may draw trust- 

 worthy conclusions from such small-scale experiments. 



The possibility of answering a question in hydrodynamics by means of 

 experiments on a reduced scale, depends upon the possibility of so arranging 

 the circumstances that the motion in the two cases — the real, full scale case, 

 and the small scale experiment — become "geometrically similar"; that is to 

 say, that all linear dimensions in the one case have a constant ratio to the 

 corresponding lengths in the other case, that the velocities in corresponding 

 points are in a constant ratio to one another, and similarly for the other 

 quantities concerned. The conditions for such "geometrically similar motions" 

 may — by a simple method given by Helmholtz — be found directly from 

 the general hydrodynamical equations 1 . 



Let the ratio of the linear dimensions in the two cases, be I 



— » — velocities — » — v 



— » — densities of the fluids — » — 6 



— » — fluid pressures — » — n 



— » — extraneous forces, e. g. gravity — » — y 



— » — coefficients of viscosity — » — x 



Then the ratio of the time-intervals — » — is l/v 



The dynamical equations to be satisfied in water, which may be regarded as 

 incompressible, are three, of the type 



du dp . „ , . / d 2 u , d*u . 3% \ 



(8) 



where q is the density, k the (dynamic) coefficient of friction (not the kine- 

 matic coefficient, used by Helmholtz), u, v, w the components of velocity 



1 H. v. Helmholtz. Wissenschaftliche Abhandlungen, Vol. I, p. 158. "Ueber ein Theorem, 

 geometrisch fihnliche Bewegungen flussiger Korper betreffend, nebst Anwendung auf das 

 Problem, Luftballons zu lenken". 



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