512 



)3^ej = xiei+ ye2 + (li) + 0'<j))e3 



(76)^ 



where the surfaces, a and 6, in (26) or (72) and (73) 



correspond to 6^ = 5l» 



i3 = 



C2» respectively. Also, 



x,y,i|' and (fi in (76) are functions of ,8 and t and 



^ + Cl<l 



e 



* + C2<t 



(77) 



Next, in order to obtain explicit values of Y/k,k,f 

 and a in relation to the top and bottom surfaces of 

 the fluid, we choose the surface, 9 = 0, so that the 

 center of mass of the three-dimensional fluid region 

 under consideration always lies on this surface and 

 we then identify this surface with the surface, J , 

 in the theory of Cosserat surface. Without loss in 

 generality, we may choose 5l = ~h, $2 - +^ (see 

 Figure 2) . This leads to the identification: 



The questions of continuous dependence upon the 

 initial data and uniqueness for solutions of initial 

 boundary-value problems for a class of symmetric 

 flows characterized by a special case of the system 

 of nonlinear partial differential equations given 

 by Green et al . (1974c) has been discussed by Green 

 and Naghdi (1975) . A similar procedure may be used 

 to establish uniqueness for the more general system 

 of equations (80) . 



For later reference, we consider here the reduc- 

 tion of the system of nonlinear differential equa- 

 tions (71) and (80) for unidirectional flow in the 

 absence of surface tension, T. Without loss in 



generality, we set the ambient pressure, p. 



0, 



and consider flows in the x-direction only. Then, 

 with q = 0, from (71) and (80) we obtain 



((>. + ((fu) = , 



pa 



p g^de- 



P <1> 



3(x,y) 

 3(6^, 02) 



p - pa 



X X 



p g'^e^de^ = , 



yk = f p*g'^(e3)2de2 



P A 3 (X'Y) 



TT 'f 7. — 



12 3(ei,e2) 



(78) 



where p is the three-dimensional mass density in 

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

 culated from the approximation (21) so that 



p (f.X = p - p*g*(( 



1 *^- 1 - P 



— -p <pw = - h p + f 



x2 ffi 



(83) 



We may solve (83)3,1, for p and p and obtain the ex- 

 pressions 



_ * * . 



p = p <j) (g + X) , 



, * 2 • 1 • 



P = %P<)> (g + A+— w) 



(84) 



g = 



3(x,y) 

 3(6l,e2) 



(79) 



Substitution of (78) and the appropriate expressions 

 for f and I into (57) to (59) results in the dif- 

 erential equations of motion 



Introduction of (84) j 2 into (83) i 2 yields a system 

 of two partial differential equations in u and w but 

 we do not record these here. A further simplifica- 

 tion of these equations results for a horizontal bed. 

 For a horizontal bottom a may be taken to be zero and 

 (77) 1 2 and {68)3 i, reduce to 



P <l>u 



-Px + <Po - ^'^x = P"x 



A = 



(85) 



p (})V = -p + (p - q) e - pa 

 y o y y 



where 



p(t>A=q-p^ + p-pg 



— p <i>V! 



• (q 



*5 p + 



p'4, 



(80) 



(81) 



Moreover, since the bed of the stream is stationary, 

 from (77) and (70)3 i, we have 



ua + va = i) 

 X y '^ 



'i w 



(82) 



The above system of equations is independent of the 

 remaining equations (58) which involve S",s. The 

 fields, S'^,s, correspond to appropriate constraint 

 responses for the restricted motion (51) . 



In (76) to (78) , we have returned to the notation 

 instead of f, introduced in (2) . 



8. FURTHER REMARKS 



The system of nonlinear differential equations (71) 

 and (80) J ^2 , 3 , 1* ' which include the effects of gravity 

 and surface tension, govern the two-dimensional mo- 

 tion of incompressible inviscid fluids for the propa- 

 gation of fairly long waves in a stream of variable 

 initial depth. They are derived here by a direct 

 approach as consequences of the conservation laws 

 (54) subject to the incompressibility condition (64) . 

 Upon specialization to unidirectional flow, the non- 

 linear differential equations (71) and (80) reduce 

 to those for inviscid fluids over a bottom of vari- 

 able initial depth given by Green and Naghdi (1976a, 

 Sections 5-6) , while the equations for two- 

 dimensional flow over a horizontal bottom were de- 

 rived earlier [Green et al. (1974c) ] . 



The differential equations governing the motion 

 of a viscous fluid sheet are discussed briefly by 

 Green and Naghdi (1976a, Section 11) and a similar 

 development can be given within the framework of the 

 restricted theory of Section 6, but we do not con- 

 sider this aspect of the subject here. The system 

 of differential equations obtained in Section 6 is 



