In these equations 
K K 
a La y =—2 {12] 
PA. 9) iy 
In the foregoing set of equations it will be convenient to express each of the time- 
dependent quantities as the sum of its steady-state and time-dependent increment. Thus 
U(t)=U+AU(t) 1(¢)=1+ AT(¢) 
[13] 
T(¢)=T, +AT_@) EA(r,¢) = T*(r) + h(n, t) 
where U, T., I, etc., are to be considered time-independent parameters unless they are written 
explicitly as functions of time. If the time-dependent increments are assumed to be small com- 
pared with their steady-state values, products and squares of these increments may be neg- 
lected and 
AUC 
2O)= PAN Oe DAL’ maWUAG) 
4a Uy U 
{14] 
2p? 
py? (¢) = wp? + Ap? (t) = pw? + ar AT(t) 
If substitutions are made into Equation [11] and if second-order small quantities are neglected, 
the new set of equations will contain steady-state terms and first-order small quantities. If 
the steady-state equations from Equations [3], [5], and [7] are subtracted, the set of equations 
for the temperature increments ¢, (r,t) and ¢, (7,2) becomes 
it @ [, Cen). 2 2p? ea 1]_1 9%? 
1 Bl Shep fp, (r,t) + So Be) T*(r) 1 Maar 
il at 5) ull db, 
r or OR Wy t 
“20( sel +P, (a,t)=PAT, (t) -(T*(a) - T,) AP(t) [15] 
r 
co) 
n?( $22) = a> ( “1) ; f, (b,t) = , (6,2) 
or b 
f , (0,2) # 
¢ (7,0) = 0 
