505 



t, which we assume to be smooth and nonintersecting. 

 Every point of this surface has a position vector, 

 r, specified by (14) . Let the boundary of the three- 

 dimensional continuum be specified by the material 

 surfaces 



Si(ei,e2) 



iii' 



ii 



-) 



?i < ?2 



(19) 



with the surface, 5=0, lying either on one of the 

 two surfaces (19) 12°^ between them (see, for ex- 

 ample, Figure 2), and a material surface 



f (9' 



'-) = 



(20) 



which is chosen such that C = const, forms closed 

 smooth curves on the surface (20) . As pointed out 

 previously [Naghdi (1975)], in the development of 

 a general theory, it is preferable to leave unspeci- 

 fied the choice of the relation of the surface, s, 

 (5=0) to the major surfaces, s and s~. In spe- 

 cial cases of the general theory or in specific ap- 

 plications, however, it is necessary to fix the 

 relation of s to the surfaces (19) i 9 . 



Suppose now that r in (1) is a continuous func- 

 tion of 6''-,t, and has continuous space derivatives 

 of order 1 and continuous time derivatives of order 

 2 in the bounded region, Ci=C=C2- Hence, to any 

 required degree of approximation, r may be repre- 

 sented as a polynomial in 5 with coefficients which 

 are continuously dif ferentiable functions of 9°',t. 

 However, instead of considering a general represent- 

 ation of this kind, we restrict attention here to 

 the approximation 



= r + 5d 



(21) 



curve occupied by L in the present configuration of 

 R at time, t, be referred to as I. Let r and d(j 

 (a = 1,2) denote the position vector of a typical 

 point of I and the directors at the same point, 

 respectively, and also designate the tangent vector 

 to the curve, £, by a3. Then, a motion of the Cos- 

 serat curve is defined by vector-valued functions 

 which assign a position, r, and a pair of directors, 

 dp,, to each particle of R at each instant of time, 

 i?e.§§ 



r(C,t) 



% = ?a(5't' 



f?1^2?3 



] > (22) 



and the condition (22)3 ensures that the directors, 

 dj(, are nowhere tangent to I and that di,d2 never 

 change their relative orientation with respect to 

 each other and §3 . The velocity and the director 

 velocities are defined by 



= d 



and from (23) j and (11) we have 



^3 = 



35 



(23) 



(24) 



where a superposed dot denotes material time dif- 

 ferentiation with respect to t holding 5 fixed. 



Consider an arbitrary part of the material curve, 

 L, in the present configuration, bounded by 5 = 5i 

 and 5 = S2 (?1 < C2)' and let 



ds = (a3 3)^d5 



333 = 33 



33 



(25) 



where r is defined by (14) and d = d(9'^,t) 



PART A 



In Part A (Sections 3-4) , we summarize the basic 

 theory of a Cosserat (or a directed) curve and then 

 briefly discuss a restricted form of the theory ap- 

 propriate for straight fluid jets. Although we are 

 concerned here mainly with the purely mechanical 

 theory involving appropriate forms of the conserva- 

 tion laws for mass, linear momentum, and moment of 

 momentum, we also include the conservation of energy. 

 The latter is useful in some applications and sup- 

 plies motivation for some requirements in the de- 

 velopment of certain solutions. 



3. THE BASIC THEORY OF A COSSERAT CURVE 



Having defined a (three-dimensional) jet-like body 

 in Section 2, we now formally introduce a direct 

 model for such a body. Thus, a Cosserat (or a 

 directed) curve, R, comprises a material curve, L, 

 (embedded in a Euclidean 3-space) and two deformable 

 directors attached to every point of the curve, L. 

 The directors which are not necessarily along the 

 unit principal normals and the unit binormals of 

 the curve have, in particular, the property that 

 they remain unaltered under superposed rigid body 

 motions. Let the particles of L be identified by 

 means of the convected coordinate, 5j and let the 



be the element of the arc length along the curve, 

 H- . It is convenient at this point to define the 

 following additional quantities: The mass density, 

 P = P(S,t), of the space curve, t; the contact 

 force, n = n(5,t), and the contact director couples 

 p" = p°'{£;,t), each a three-dimensional vector field 

 in the present configuration; the assigned force, 

 f = f(5,t), and the assigned director couples, 

 Ja = £°'(5,t), each a three-dimensional vector field 

 and each per unit mass of the curve, i; the intrin- 

 sic (curve) director couples, tt^^ = Tr™(5,t), per unit 

 length of H which make no contribution to the supply 

 of momentum; the inertia coefficients, y™ = y°'(5) 

 and y^^B = y'^'^ (.^) , with y™P being components of a 

 symmetric tensor, which are indenpendent of time; 

 the specific internal energy, £ = e(C,t); the spe- 

 cific heat supply, r = r(5,t), per unit time; and 

 the heat flux, h = h(£;,t), along I, in the direction 

 of increasing 5/ per unit time. The assigned field, 

 f, represents the combined effect of (i) the stress 

 vector on the lateral surface (17) of the jet-like 

 body denoted by f^, and (ii) an integrated contri- 

 bution arising from the three-dimensional body force 

 denoted by fj^, e.g. , that due to gravity. A parallel 

 statement holds for the assigned fields, £". Sim- 

 ilarly, the assigned heat supply, r, represents the 

 combined effect of (i) heat supply entering the 



For convenience, we adopt the notation for r in (10) and 

 (18) also for the surface (22) j . This permits an easy iden- 

 tification of the two curves, if desired. The choice of 

 positive sign in (22)3 is for def initeness . Alternatively, 

 it will suffice to assume that [did2a3] / with the under- 

 standing that in any given motion the scalar triple product 

 [did2a3] is either > or < 0. 



