502 



more space is devoted to fluid sheets and water 

 waves. This is partly due to the fact that, in the 

 context of the direct formulation, the theory of 

 fluid sheets has to date received more attention 

 than that of fluid jets. Thus, in Part A (Sections 

 3-4) , we summarize the basic theory of a Cosserat 

 curve and briefly discuss a restricted form of the 

 theory for straight jets which are not necessarily 

 circular. The resulting system of nonlinear ordi- 

 nary differential equations includes the effects of 

 surface tension and gravity and has been derived for 

 both inviscid and viscous jets. We do not record 

 these here,- but we call attention in Section 4 to a 

 number of existing solutions, which serve as evidence 

 of the relevance and applicability of the direct 

 formulation of the theory of fluid jets. 



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

 ing the basic theory of a Cosserat surface in Sec- 

 tion 5, we present in outline a derivation of a 

 restricted theory in Section 6 , and then obtain a 

 system of nonlinear partial differential equations 

 for the propagation of fairly long waves in a homo- 

 geneous stream of variable depth (Section 7) . This 

 system of differential equations, which includes 

 the effects of surface tension and gravity, is de- 

 rived for incompressible inviscid fluids. Some ex- 

 tensions of these results to nonhomogeneous and 

 viscous fluids are available but these are not dis- 

 cussed here. In the final section of the paper we 

 make a comparison between the differential equations 

 derived in Section 7 and the systems of equations 

 for water waves that are often used in the litera- 

 ture; and, on the basis of compelling physical con- 

 siderations, argue as to why the system of equations 

 of the direct formulation should in general be pre- 

 ferred to others. In Section 8, we also call at- 

 tention to a number of existing solutions , which 

 serve as further evidence of the relevance and ap- 

 plicability of the direct formulation of the theory 

 of fluid sheets. 



In the course of our development, sometimes the 

 same symbol is utilized in Parts A and B to denote 

 different quantities; but this should not give rise 

 to confusion, as the two parts can be read indepen- 

 dently of each other. Throughout the paper, Latin 

 indices (subscripts or superscripts) take the values 

 1, 2, 3, Greek indices take the values 1, 2 only, 

 and the usual convention for summation over a re- 

 peated index is employed. 



signs position, r*, to each particle of g at each 

 instant of time, i.e.. 



r* = v"* 



= r*(e 



\t) 



(1) 



We assume that the vector function, f * , — a 1- 

 parameter family of configurations with t as the 

 real parameter — is sufficiently smooth in the sense 

 that it is differentiable with respect to e-"- and t 

 as many times as required. In some developments, 

 it may be more convenient to set 

 the notation 



)^ = C and adopt 



11 = 



a 



= (6 ,?) , 9^ = C . 



We recall the formulas 



3r* 

 <3i = -^ ' gii = 9i • gj ' g = det(gij) , 

 ■36 



t2) 



,1 = ai3 



g -'gj / g 



dv = g^de^de^de^ 



i . aJ = a^^ 



and further assume that 



[g g g ] > 

 -1-2-3 



(3) 



(4) 



(5) 



In (4), g. and g are the covariant and the contra- 

 variant base vectors at time, t, respectively, g^^^ 

 is the metric tensor, g^3 is its conjugate. Si- is 

 the Kronecker symbol in 3-space and dv the volume 

 element in the present configuration. 



The velocity vector, y* > of a particle of the 

 three-dimensional body in the present configuration 

 is defined by 



V* = r* , (6) 



where a superposed dot denotes material time dif- 

 ferentiation with respect to t holding Q-'- fixed. 

 The stress 'vector t across a surface in the present 

 configuration with outward unit normal y* is given 

 by 



* ik 

 ^^ !k 



(7) 



2. GENERAL BACKGROUND 



In this section, we provide appropriate definitions 

 for jet-like and sheet-like bodies. To this end, 

 consider a finite three-dimensional body, B, in a 

 Euclidean 3-space, and let convected coordinates, 

 e-"- (i = 1, 2, 3) , be assigned to each particle (or 

 material point) of 6. Further, letTr* be the posi- 

 tion vector, from a fixed origin, of a typical parti- 

 cle of g in the present configuration at time, t. 

 Then, a motion of the (three-dimensional) body is 

 defined by a vector-valued function, r* , which as- 



'The use of an asterisk attached to various symbols is for 

 later convenience. The corresponding symbols without the 

 asterisks are reserved for different definitions or designa- 

 tions to be introduced later. 



where 



Recall that when the particles of a continuum are referred 

 to a convected coordinate system, the numerical values of 

 the coordinates associated with each particle remain the 

 same for all time. Although the use of a convected coordi- 

 nate system is by no means essential, it is particularly 

 suited to studies of special bodies (such as sheets, jets, 

 shells, and rods) and often results in simplification of 

 intermediate steps in the development of the subject. 



" The choice of positive sign in (5) is for definiteness. 

 Alternatively, for physically possible motions we only need 

 to assume that g'5 5* with the understanding that in any 

 given motion [919293! Is either > or < . The condition 

 (5) also requires that 6^ be a right-handed coordinate 

 system. 



