574 VISCOSITY. [CHAP, xi 



in Poiseuille's experiments, but when the irregular mode of flow 

 has set in, dp/dz varies more nearly as w<?. 



The practical formula adopted by writers on Hydraulics, for 

 pipes whose diameter exceeds a certain limit, is 



where R is the tangential resistance per unit area, w is the mean 

 velocity relative to the wetted surface, and / is a numerical 

 constant depending on the nature of the surface. As a rough 

 average value for the case of water moving over a clean iron 

 surface, we may take /= "005*. A more complete expression for 

 R y taking into account the influence of the diameter, has been 

 given by Darcy, as the result of his very extensive observations on 

 the flow of water through conduits f. 



The resistance, in the case of turbulent flow, is found to be 

 sensibly independent of the temperature, and therefore of the 

 viscosity of the fluid. This is what we should anticipate from 

 considerations of ' dimensions/ if it be assumed that R oc w 2 {- 



If we accept the formula (2) as the expression of observed 

 facts, a conclusion of some interest may be at once drawn. 

 Taking the axis of z in the general direction of the flow, if w 

 denote the mean velocity (with respect to the time) at any point 

 of space, we have, at the surface, 



dw 



if w denote the general velocity of the stream, and Bn an element 

 of the normal. If we take a linear magnitude I such 



w /l = dw/dn, 



then I measures the distance between two planes moving with a 

 relative velocity W Q in the regular ' laminar' flow which would give 

 the same tangential stress. We find 



(3). 



* See Kankine, Applied Mechanics, Art. 638; Unwin, Encyc. Britann., Art. 

 " Hydromechanics." 



t Recherches experimentales relatives au mouvement de Veau dans les tuyaux, 

 Paris, 1855. The formula is quoted by Kankine and Unwin. 



J Lord Bayleigh, " On the Question of the Stability of the Flow of Fluids," 

 Phil. Mag., July 1892. 



