Fluid Jets and Fluid Sheets: 

 A Direct Formulation 



p. M. Naghdi 

 University of California 

 Berkeley , California 



ABSTRACT 



The object of this paper is to present an account 

 of recent developments in the direct formulation of 

 the theories of fluid jets and fluid sheets based 

 on one and two-dimensional continuum models origi- 

 nating in the works of Duhem and E. and'F. Cosserat. 

 Following some preliminaries and descriptions of 

 (three-dimensional) jet-like and sheet-like bodies, 

 the rest of the paper is arranged in two parts , 

 namely Part A (for fluid jets) and Part B (for fluid 

 sheets) , and can be read independently of each other. 

 In each part, after providing the main ingredients 

 of the direct model and a statement of the conserva- 

 tion laws, appropriate nonlinear differential equa- 

 tions are derived which include the effects of 

 gravity and surface tension. Application of these 

 theories to various one and two-dimensional fluid 

 flow problems, including water waves, are discussed. 



1 . INTRODUCTION 



Jets and sheets are a class of three-dimensional 

 bodies whose boundary surfaces have special charac- 

 teristic features. To this extent they are, respec- 

 tively, similar to another class, namely that of 

 rods and shells (or plates) , although the nature of 

 the specified surface (or boundary) conditions in 

 the two classes may be different. Moreover, the 

 kinematics of jets and rods are identical, as are 

 the kinematics of sheets and shells. Indeed, it is 

 only through their constitutive equations that a 

 distinction appears between rods and jets on the 

 one hand, and shells and sheets on the other. It 

 is natural to inquire as to the possible utility of 

 methods of approach in the construction of theories 

 in the class of rods and shells for that of jets 

 and sheets and vice versa. The main purpose of this 

 paper is to call attention to the possible utility 

 of a direct approach for jets and sheets, an approach 



which has met with considerable success in the case 

 of rods and shells. The direct approach for fluid 

 jets is based on a one-dimensional model, called a 

 Cosserat (or a directed) curve which is defined in 

 Section 3 ; and the direct approach for fluid sheets 

 is based on a two-dimensional model , called a Cos- 

 serat (or a directed) surface which is defined in 

 Section 5. It should be emphasized that a Cosserat 

 curve and a Cosserat surface are not, respectively, 

 just a one-dimensional curve and a two-dimensional 

 surface; but are, in fact, endowed with some struc- 

 ture in the form of additional primitive kinematical 

 vector fields. 



The concept of 'directed' or 'oriented' media 

 originated in the work of Duhem (]893) and a first 

 systematic development of theories of oriented media 

 in one, two, and three dimensions was carried out by 

 E. and F. Cosserat (1909). In their work, the Cos- 

 serats represented the orientation of each point of 

 their continuum by a set of mutually perpendicular 

 rigid vectors. The purely kinematical aspects of 

 oriented bodies characterized by ordinary displace- 

 ment and the independent deformation of N deformable 

 vectors in N-dimensional space has been discusssed 

 by Ericksen and Truesdell (1958) , who also intro- 

 duced the terminology of directors . 



A complete general theory of a Cosserat surface 

 with a single deformable director given by Green 

 et al. (1955) was developed within the framework of 

 thermomechanics; and their derivation (Green et al . 

 1965) is carried out mainly from an appropriate 

 energy equation, together with invariance require- 

 ments under superposed rigid body motions. A re- 

 lated development utilizing three directors at each 

 point of the surface, in the context of a purely 

 mechanical theory and with the use of a virtual work 

 principle, is given by Cohen and DeSilva (1966) . A 

 further development of the basic theory of a Cosserat 

 surface along with certain general considerations re- 

 garding the construction of nonlinear constitutive 

 equations for elastic shells is given by Naghdi 



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