D,7 • CLASSICAL RESULTS 



The initial condition is, as usual, T{r, 0) = 0. The dimensionless varia- 

 bles appropriate to the cylinder problem are 



9 = ^^-1, . = ^, ., = ^, (7-2) 



by which Eq. 7-1 and its initial and boundary conditions become 



^20 1 ae ^ ae 



d(ji^ CO doi dTa 



, +-^ = ^ (7-3) 



= (t) 



l^ ^[■-]Q at w = 1 



-— = at CO = - 



oco a / 



(7-4) 



e = -1 at Ta = (7-5) 



From this nondimensional form it is seen that the solution depends 

 essentially on only two dimensionless parameters, namely a type of Biot 

 number, Na = ha/k and the ratio h/a, which we shall designate as 0, 

 the thickness number. When the system is solved by separation of varia- 

 bles as discussed in Art. 2, the results are expressed in terms of the 

 following combinations of Bessel functions J and Neumann functions Y: 



Riinco) = 



JiifjiQ) Yiiix^) 



JlilJLUl) Fl(/XC0) 



^1(^1 12) Fi(iufi) 

 where the general eigenvalue fx satisfies the characteristic equation [9], 



-11 = ^" (^-«) 



the eigenf unction corresponding to the nth root Hn of Eq. 7-6 is: 



^„(co) = jRo(MnCo) 



with the orthogonaUty condition in this case (i.e. polar coordinates) 



/ (pnio})<pm{o})iodo} = {m ^ fl) (7-7) 



The complete solution in nondimensional variables is: 



00 



0(co, T„) = |^= 1 + e(co, r„) = 1 + ^ AnRoilJ.nC»)e-'^> (7-8) 



* n = l 



< 267 ) 



