507 



worth recalling that the rate of work by all contact 

 and assigned forces acting on the curve, H, and its 

 end points minus the rate of increase of the kinetic 

 energy can be reduced to: 



52 



hi 



p(f 



V + 8, 



w )ds + [n • V + p 



t,2 



where p^ is an arbitrary scalar function of E,,t. 

 For an incompressible viscous jet, the constraint 

 response, n,Tra,pC, ^j-g determined similarly with the 

 use of the constraint condition (39), but constitu- 

 tive equations are required for n,Tr",pC in (40) . We 

 do not record here the results for~a viscous jet and 

 refer the reader to Green (1976) and Caulk and 

 Naghdi (1978b) . 



_d 

 dt 



r 



.52 



5i' 



%P (v 



v+2yv'w +y w 



w„)ds 



pPds 



(37) 



where P is defined by (35) . 



Before closing this section, we note that the 

 restriction imposed on the motion of the medium by 

 the condition of incompressibility reduces to |||| 

 [see Green (1976) ] 



— [did2a3] = 



(38) 



and can alternatively be expressed in the form 



X aq • w + di X do 

 -' ~a -^ -^ 



as 



(39) 



_ae 



where e is the permutation symbol in 2-spaces. To 

 complete the theory of a Cosserat curve under the 

 constraint condition (39) , we assume that each of 

 the functions, n,Tr°', and p'*, is determined to within 

 an additive constraint response so that 



n = n + n 



-a ~a 



TT + IT 



-a "a 

 p + p 



,(40) 



where n, ft™, and g™ are determined by constitutive 

 equations and the functions n(5,t), Tf"(g,t), and 

 pO'(C,t) are the response due to the constraint; the 

 latter quantities are arbitrary functions of 5,t and 

 do no work. For an incompressible inviscid fluid 

 jet, which models the properties of the three- 

 dimensional inviscid fluid at constant temperature, 

 we introduce the constitutive assumption that n,Troi, 

 p*^ do not depend explicitly on the kinematic quan- 

 tities, 3v/3£;,W(jj,3w^/3£;, and are furthermore work- 

 less , i.e. , 



3v 



3w 



(41) 



provided w^, 3y/3£ satisfy the constraint condition 

 (39) . It can then be shown that [Green and Laws 

 (1968) and Green (1976)] 



4. STRAIGHT FLUID JETS. ADDITIONAL REMARKS 



We now specialize the results of the previous sec- 

 tion to straight jets of elliptical cross-section. 

 In order to display some details of the kinematics 

 of a straight jet, including the rotation of the 

 directors in a plane normal to the jet axis, it is 

 convenient to introduce a fixed system of rectangular 

 Cartesian coordinates (x,y,z) with the z-axis paral- 

 lel to the jet. Further, let the unit base vectors 

 of the rectangular Cartesian axes be denoted by 

 (i,j,k) and introduce, for later convenience, the 

 additional base vectors 



ej = i cos S + j sin 9 , 



e2 = -i sin 9 + j cos 9 , 63 = k , 



(43) 



where 9 is a smooth function of z and t. We assume 

 that the directors are so restricted that they de- 

 scribe an elliptical cross-section of smoothly vary- 

 ing orientation along the length of the jet and that 

 at each z = const., the base vectors, ej and e2, 

 lie along the major and minor axes of the ellipse, 

 respectively. Then, the angle, 9, called the 

 sectional orientation, specifies the orientation of 

 the cross-section as a function of position. With 

 this background, henceforth we restrict motions of 

 the directed curve, R, such that in the present con- 

 figuration at time, t. 



= z(5,t)e3 



>1 



hgi 



l'2g2 



(44) 



where <))i and (i>2 measure the semiaxes of the ellipti- 

 cal cross-section. In the case of a circular jet, 

 ^l = (j)2. 



The complete theory also requires the specifica- 

 tion of explicit values for A,y'*,y"P,f and S.o'. In 

 particular, the values for A,y",ycB may be obtained 

 by an appeal to certain results from the three- 

 dimensional description of the jet. Thus, recall- 

 ing (18) and the remark made following (17) , here 

 we choose the curve, 9™ = 0, as the line of centroids 

 of the jet-like body and identify this curve with 

 the curve, I, in the theory of a Cosserat curve. 

 This leads to the identification 



-K'^i '^ ^2 





"" §3 



X = p(a33)'^ = rp*g^d9ld92 , 



p« = 



(42) 



nil In, general , there are three conditions of incompressibility 

 in the theory of incompressible directed fluid jets; for a 

 discussion of these, see Caulk and Naghdi (1978a, Appendix) . 

 In restricted forms of the theory discussed in the next sec- 

 tion, two of the three conditions are satisfied identically. 

 The specification (38) is motivated from an examination of 

 the incompressibility condition in the three-dimensional 

 theory when the position vector is approximated by (18). 



Xy" = p g^9 d9 d8 = , 



/„ 



Ay = -' p g 9 9 d9 d9 , 

 a 



(45) 



where p is the three-dimensional mass density in 

 (9) and the determinant g defined by (3) 3 is cal- 

 culated from the approximation (18) . Again, with 



