508 



the use of (18) and the equations of motion (9)2 3, 

 the expressions for f and £° can be identified in 

 terms of the integrated body force, f*, over the 

 cross-sectional area, a, and specified pressure and 

 surface tension over the boundary, 3a of a [for 

 details, see, for example. Caulk and Naghdi (1978a)]. 

 We observe that since yCt = by (45)2, the equations 

 of motion (30) and (31) assume a slightly simpler 

 form. We do not record here the details of the 

 system of ordinary differential equations which can 

 be obtained from (29) -(33) for both inviscid and 

 linear viscous fluids. They are readily available 

 in the papers cited: see Green and Laws (1968), 

 Green (1975, 1976, 1977), and Caulk and Naghdi 

 (1978a, b) . 



In the rest of this section, we briefly call 

 attention to some available evidence of the relevance 

 and applicability of the direct formulation of the 

 fluid jets. Available solutions obtained to date 

 are limited to those for straight jets and among 

 these most of them deal with jets of circular cross- 

 section. Some general aspects of compressible 

 inviscid jets, including a discussion of ideal gas 

 jets in the context of a thermodynamical theory, 

 have been studied by Green (1975) . Applications to 

 incompressible circular jets for both inviscid and 

 viscous fluids are contained in the papers of Green 

 and Laws (1968) and of Green (1976) . Green (1977) 

 has also studied a steady motion of an incompressible 

 inviscid fluid jet which does not twist along its 

 axis. A more detailed analysis of the motion of a 

 straight elliptical jet of an incompressible inviscid 

 fluid in which the jet is allowed to twist along its 

 axis is contained in a recent paper by Caulk and 

 Naghdi (1978a) . This study, which includes the ef- 

 fects of gravity and surface tension, utilizes the 

 nonlinear differential equations of Section 3 with 

 r and d^ at time, t, specified in the form (44) . A 

 number of theorems are proved in the paper of Caulk 

 and Naghdi (1978a) which pertain to the motion of a 

 twisted elliptical jet and some special solutions 

 are obtained which illustrate the influence of twist. 

 Further, a system of linear equations, derived for 

 small motions superposed on uniform flow of an in- 

 compressible circular jet, is employed by Caulk and 

 Naghdi {1978b) to study the instability of some 

 simple jet motions in the presence of surface ten- 

 sion, i.e., the so-called capillary instability that 

 leads to disintegration of the jet. In particular, 

 they [Caulk and Naghdi {1978b) ] consider the breakup 

 of both inviscid and viscous jets: in the case of 

 an inviscid jet excellent agreement is obtained with 

 the three-dimensional results of Rayleigh (1879a, b) ; 

 and for a viscous jet, through a comparison with 

 available three-dimensional numerical results 

 [Chandrasekhar (1961) ] , the solution obtained is 

 shown to be an improvement over an existing approxi- 

 mate solution of the problem by Weber (1931) . A 

 related study by Bogy (1978) , concerning the insta- 

 bility of an incompressible viscous liquid jet of 

 circular section, partly overlaps with the work of 

 Caulk and Naghdi (1978b) on the temporal instability 

 of a viscous jet, and considers the spatial insta- 

 bility of a semi-infinite jet formulated as a 

 boundary-value problem. 



PART B 



face, we summarize a special case of the theory 

 which is particularly suited for applications to 

 problems of fluid sheets and to the propagation of 

 fairly long water waves. For the sake of simplicity, 

 we confine attention here to homogeneous fluids; but 

 note that, as in Green and Naghdi (1977) , the deriva- 

 tion can be modified to allow for variation of mass 

 density with depth. Although we are concerned mainly 

 with the purely mechanical theory involving appropri- 

 ate forms of the conservation laws for mass, linear 

 momentum, and moment of momentum, we also include 

 the conservation of energy. The latter easily sup- 

 plies motivation for some requirements in the devel- 

 opment of certain solutions. 



5. THE BASIC THEORY OF A COSSERAT SURFACE 



Having introduced the notion of a (three-dimensional) 

 sheet-like body in Section 2, we now formally define 

 a direct model for such a body. Thus, a Cosserat 

 (or directed) surface, C, comprises a material sur- 

 face, 5, (embedded in a Euclidean 3-space) and a 

 single deformable vector, called a director, attached 

 to every point of the surface, 5. The directors 

 which are not necessarily along the unit normals to 

 the surface have, in particular, the property that 

 they remain unaltered under superposed rigid body 

 motions. Let the particles of the material surface 

 of C be identified by means of a system of convected 

 coordinates, 9" (a = 1,2), and let the surface oc- 

 cupied by S in the present configuration of C at 

 time, t, be referred to as J. Let r and d denote 

 the position vector of a typical point of J and the 

 director at the same point, respectively, and also 

 designate the base vectors along the O'^-curves on 

 J hy a . Then, a motion of the Cosserat surface is 

 defined by vector-valued functions which assign posi- 

 tion, r, and director, d, to each particle of C at 

 each instant of time, i.e.. 



r(9",t) 



d(e«,t) 



[aiaad] > (46) 



and the condition (46), ensures that the director, 

 d, is nowhere tangent to«/. The base vectors, a^, 

 and their reciprocals, a", the unit normal, 33, and 

 the components of the metric tensors, a^jg and a'^S^ 

 at each point of a are defined by 



3r 



?r,' 



a = a ■ a 





a a3 = aj X a2 



a = det a „ , a = [aiaoa^] > , 



ag ~-'~^~--' 



(47) 



where Sp is the Kronecker delta in 2-space. The 

 velocity and the director velocity vectors are de- 

 fined by 



v = r , w = d , 



(48) , 



In Part B (Sections 5-8) , after briefly describing 

 the basic theory of a Cosserat (or a directed) sur- 



*For convenience, we adopt the notation for r in (14) and 

 (21) also for the surface (46) 1. This permits an easy 

 identification of the two surfaces, if desired. The choice 

 of positive sign in (46) 3 is for def initeness. Alterna- 

 tively, it will suffice to assume that [aia2d] ^ with the 

 understanding that in any given motion tfie scalar triple 

 product [aia2d] is either > or < 0. 



