538 VISCOSITY. [CHAP, xi 



in which the kinetic energy (T] and the dissipation-function (F) can be 

 expressed as quadratic functions of the generalized velocities, with constant 

 coefficients. 



If the extraneous forces have not a single-valued potential, or if instead of 

 given velocities we have given tractions over the boundary, the theorems 

 require a slight modification. The excess of the dissipation over double the 

 rate at which work is being done by the extraneous forces (including the 

 tractions on the boundary) tends to a unique minimum, which is only 

 attained when the motion is steady*. 



Periodic Motion. 



298. We next examine the influence of viscosity in various 

 problems of small oscillations. 



We begin with the case of laminar motion, as this will enable 

 us to illustrate some points of great importance, without elaborate 

 mathematics. If we assume that v = 0, w=Q, whilst u is a 

 function of y only, the equations (4) of Art. 286 require that 

 p = const., and 



du d 2 u 



This has the same form as the equation of linear motion of 

 heat. In the case of simple-harmonic motion, assuming a time- 

 factor e i(&amp;lt;Tt+f&amp;gt; , we have 



d?u ia- 



-j-- u ........................... (2), 



ajf v 



the solution of which is 



u = Ae (l+i)f * y + Be~ (l+i} M ...................... (3), 



provided 0=(&amp;lt;r/2v)* ........................... (4). 



Let us first suppose that the fluid lies on the positive side of 

 the plane xz t and that the motion is due to a prescribed oscillation 



w =rae* (&amp;lt;rt+e) ........................... (5) 



of a rigid surface coincident with this plane. If the fluid extend 

 to infinity in the direction of y-positive, the first term in (3) is 

 excluded, and determining B by the boundary-condition (5), we 

 have 



* Cf. Helmholtz, Z.c. 



