564 VISCOSITY. [CHAP, xi 



and therefore, for the angular velocity (xiv), 



the real part of which is 



a ^ e ~^ (a ~ r) .cos{o*-j8(r-a) + e} .................. (xxi). 



As in the case of laminar motion (Art. 298), this represents a system of waves 

 travelling inwards from the surface with rapidly diminishing amplitude. 



When, on the other hand, the viscosity is very great, /3a is small, and the 

 formula (xiv) reduces to 



o&amp;gt;cos(o- + f) ................................. (xxii), 



nearly, when the imaginary part is rejected. This shews that the fluid now 

 moves almost bodily with the sphere. 



The stress-components at the surface of the sphere are given by (13). In 

 the present case the formula reduce to 



Pry= -p- 



z 



(xxiii). 



If SS denote an element of the surface, these give a couple 

 ^= - JJtePnr -yprx) &amp;lt;*S= CM (ha) JJ(a* + 



9 h*a*fa(ha) , . N 



-a&quot;-&quot;* ^(ll) &quot; ............... (XX1V) 



by (xiii) and Art. 305 (7). 



In the case of small viscosity, where /3a is large, we find, on reference to 

 Art. 267, putting ha = (l-i) /3a, that 



/ fJ \e*f 

 2i+ n (ha} = (-y(Jl^ - ..................... (xxv), 



approximately, where (=(! i) j3a. This leads to 



^V r =-|7r/Lia 3 (l+*)^a ......................... (xxvi). 



If we restore the time-factor, this is equivalent to 



............ (xxvii). 



The first term has the effect of a slight addition to the inertia of the sphere ; 

 the second gives a frictional force varying as the velocity. 



308. The general formulae of Arts. 305, 306 may be further 

 applied to discuss the effect of viscosity on the oscillations of a 



