506 



lateral surface (17) of the jet-like body from the 

 surrounding environment, denoted by r^, , and (ii) an 

 integrated contribution arising from the three- 

 dimensional heat supply denoted by r)-,. Thus, we may 

 write 



~b ~c 



X.^ 



ot a 



C + n 



-b -c 



r, + r 

 b c 



(26) 



The various quantities in (25) are free to be spec- 

 ified in a manner which depends on the particular 

 application in mind and, in the context of the the- 

 ory of a Cosserat curve, the intertia coefficients. 



i,aB 



, and the mass density, p, require constitu- 



tive equations. Indeed, f^, S." and r^, as well as 

 fb, ^? and r^, can be identified with the corre- 

 sponding expressions in a derivation from the three- 

 dimensional equations [see, for example. Green et 

 al. {1974a)]. Likewise, the inertia coefficients, 

 yC, y'^p , and the mass density, p, may be identified 

 with easily accessible results from the three- 

 dimensional theory. 



With the above definitions of the various field 

 quantities and with reference to the present con- 

 figuration, the conservation laws for a Cosserat 

 curve are: 





ds 



52 

 [f(C,t)]^^ = f(C2-t) = f(Ci,t) 



(28) 



The first of (27) is a statement of the conservation 

 of mass, the second is the conservation of linear 

 momentum, the third that of the director momentum, 

 the fourth is the conservation of moment of momentum, 

 and the fifth represents the conservation of energy. 

 Under suitable continuity assumptions , the first 

 four equations in (27) are equivalent to 



3C 



3m 



a3Xn + T-p+Xg = , 

 ~ ~ dt, ~ 



1 3h • 



where 



35 



A(5) = p(a33)^ or pa33 + pa3 



3n 



af" + Af = A(v + y w ) 



S^ +A£ =Tr +A(yv + y w ) , 



(29) 



(30) 



(31) 



(32) 



(33) 



d r^2 „ C2 52 



— p(v + y w )ds = pfds + [n] 



■^ 5i " J 5l ~ 5i 



m = d xp g = d xq 



a „a a* aH 

 q =£ -yv = y w 



(34) 



M 



— J p(y V + y Wg)ds 



r^2 „ _i, „ 52 



(pfc" - (333) =Tr")ds + [pa] 



•^5i ~ ~ 5i 



d r a 



-r- \ p[rxv + y(rxw +d xv) 



at ~ ~ ■' ~ ~a ~a 

 ■^ 5i 



and 



AP 



3v 

 3? 



+ P 



35 



(35) 



is the mechanical power. With the help of (34) , the 

 local form of the moment of momentum equation (31) 

 can be reduced to 



aq X n + d X TT 



3d 

 35 



(36) 



+ d X y°'^Wo]ds 

 -a 6 



M 



f p{r X f + d X {,"') 



ds 



_d 

 dt 



+ [rxn + d xp]^ 



-a ~ 5i 



5? 



r ct ct6 



p[e + ^{v •v + 2yv'W +y w "w^ 



hi ~ ~ ~ -^ ' ~a ~B 



Ids 



.52 



J- p{r + f • V + 



S.a • w )ds 

 -a 



+ n • V + p • w - h' 



52 

 5l 



(27) 



where we have used the notation 



It may be noted that the local field equations 

 in the mechanical theory of a Cosserat curve have 

 the same forms as those that can be derived from the 

 three-dimensional equations; the latter can be de- 

 rived by suitable integration of (9)j,2,3 with re- 

 spect to 9 and 6 and in terms of certain definitions 

 for integrated mass density and resultants of stress 

 [for details, see Green et al . (1974a)]. Moreover, 

 given the approximation (18) , there is a 1-1 corre- 

 spondence between the one-dimensional field equations 

 that follow from the conservation laws of a Cosserat 

 curve and those that can be derived from the three- 

 dimensional equations provided we identify the 

 director d^ in (18) with (22)2 and adopt the defini- 

 tions of the resultants mentioned above. A similar 

 1-1 correspondence can be shown to hold between (33) - 

 and an integrated energy equation derived from the 

 three-dimensional energy equation. 



The above results include the local form of con- 

 servation of energy derived from (27)5. For the 

 purely mechanical theory in which the law of con- 

 servation of energy is excluded, the appropriate 

 conservation laws are the first four of (27) . In 

 the context of the purely mechanical theory , it is 



