122 Proceedings of the Royal Irish Academy. 



CHAPTER III. 



Applications of the Method of Osboene Eeynolds. 



Akt. 28. Explanation of Osborne Reynolds' Method. 



Professor Osborne Eeynolds* has discussed the question of the stability of 

 flow from a point of view very different from that adopted by Lord Kelvin. 

 He supposes the turbulent or unstable motion to be already in existence, and 

 seeks to determine a criterion as to whether the relative kinetic energy of the 

 disturbed motion will increase, diminish, or remain stationary. In case the 

 disturbance is regarded as finite, i.e. if, in the expressions for the velocities, 

 terms of higher order than the first in small quantities are retained, the 

 magnitudes of the velocities enter into the determining condition ; but if only 

 terms of the first order are taken into account, the criterion does not involve 

 the scale of the disturbance, and moreover gives a lower limit than is obtain- 

 able when the disturbance is finite, for the slowest steady motion, under 

 assigned conditions, for which a disturbance of assigned type could possibly 

 increase. Thus the discussion of infinitesimal disturbances would appear in 

 reality as important as that of finite ones, and is moreover considerably 

 simpler. For infinitesimal disturbances, considering only the case in which 

 the velocity in the steady motion is in the aj-direction, and is independent 

 of X, the criterion may be obtained as follows. Let the velocity in the steady 

 motion be U, and that in the disturbed U ^ u, v, tv, let the stress- components 

 in the steady motion be P.^x, Pxy, etc., and those in the disturbed be P^x + Pxx> 

 Pxy + Pxtj, etc. By writing down the fundamental equations for the disturbed 

 and for the steady motions, and subtracting, we evidently obtain the equations 

 dujdt + Udujdx + vdUldy + wdUjdz = p~^{dpxx/dx + dp^y/dy + dpy;zldz], 

 dvjdt + Udvldx = p'^{dpscy/dx + dpyyjdy + dpyjdz}, 



dwldt+ Udiv/dx = p~^[dp^^/dx + dpyjdy +dpsz/dz}. (1) 



Multiplying by pu, pv, pw, respectively, and integrating throughout any 

 volume, we have 



d/dt . ^ p(u~ + v'^ + W') d . vol = - pu(vd IT/dy + wd U/dz) d . vol. 



1 r f 



- - p U'd/dx{u'^ + v'^ + w^) + u [dpxx/dx + dp^y/dy + dp^z/dz} d.Yol 

 + two terms similar to the last. (2) 



* For reference, see Introduction, p. 7o. An excellent resume of Reynolds' method is 

 contained in Lamb's "Hydrodynamics," 3rd Edition, Art. 346, from wliich I have paraplirased 

 a few sentences. 



