511 



dt 



[aia2d] 



(62) 



and can alternatively be expressed in the form 



[(d • a3)a" - (d • a")a3] • v + 33 • w = .(63) 



~->~ "'"'*' ~',0t*' 



For an incompressible inviscid fluid sheet, which 

 models the properties of the three-dimensional in- 

 viscid fluid at constant temperature, we introduce 

 the constitutive assumption that N",m,M'^ do not de- 

 pend explicitly on the kinematical quantities, v ^, 

 w,w „, and are furthermore workless, i.e.. 



, V = y 



\ = <ii 



(68) 



and we note that the velocity components, u,v,A,w, 

 may be regarded as functions of either 6 ,6 ,t or 

 of x,y,t. From (67) follow the expressions 



V = uei + veo + ^e3 , w = we3 (69) 



and 



u = u + uu + vu , V = V + uv vv , 

 t X y t X y 



A = A^ + uA + vA , w - w + uw + vw , (70) 

 t X y t X y 



V + mw + M w =0 



(64) 



provided y (^ and w satisfy the constraint condition 

 (63) . With the use of (51) , it can then be shovm 

 that [see Green and Naghdi (1976, 1977)] 



N« = - p'{(d • a3)a°' = (d • a°')a3} 



P; e *ag X b , 



m = - p 'a 



M = 



(65) 



where p^^^ is an arbitrary scalar function of 6^,t 

 and e^S is the alternating tensor in 2-space. With 

 the help of the energy equation (60) and the fact 

 that the mechanical power vanishes identically for 

 an incompressible inviscid fluid at constant tempera- 

 ture, it can be shown that [see the appendix of Green 

 and Naghdi (1976)] 



q = 



7. WATER WAVES OF VARIABLE DEPTH 



Within the scope of the restricted theory of the 

 previous section, we include here an outline of a 

 derivation of a system of nonlinear differential 

 equations governing the two-dimensional motion of 

 incompressible fluids for propagation of fairly long 

 waves in a stream of water of variable initial depth. 

 Our developments include the effects of gravity and 

 surface tension but we assume that the mass density 

 of the fluid does not vary with depth. However, a 

 more general derivation for a nonhomogeneous inviscid 

 fluid in which the mass density is allowed to vary 

 with depth is given by Green and Naghdi (1977) . Let 

 ei,e2,e3 be a set of right-handed constant orthonormal 

 base vectors associated with rectangular Cartesian 

 axes and choose the unit vector, b, to coincide with 

 63. Then, the position vector, r, in (45) j and the 

 director, d, in (51) 1 can be represented as 



r = xej + ye2 + ipe^ 



pes 



(66) 



where x,y,i|j,(j) are functions of 6 ,6 ,t. The velocity, 

 V, and the director velocity now take the forms 



V = uej + ve2 + Ae3 , w = we3 , (67) 



where 



where the subscripts, x,y,t, designate partial dif- 

 ferentiation with respect to x,y,t, when u,v,A,w are 

 regarded as functions of x,y,t. With the use of (57) 

 and (70) , the incompressibility condition (64) as- 

 sumes the simpler form 



1 (u + V ) + w = 

 X y 



(71) 



In order to complete our development, we need to 

 specify values for the assigned force, f, and the 

 assigned director force, Z, and to identify the co- 

 efficients ,y <k and k, which, in general, require 

 constitutive equations. For this purpose we consider 

 the corresponding fluid sheet in the three-dimensional 

 theory in which an incompressible homogeneous fluid 

 under gravity || ,-g*e3, flows over a bed specified by 

 the position vector 



xej + ye2 + a(x,y)e3 



and we specify the surface of the fluid by 



xej + ye2 + B(x,y,t)e3 



(72) 



(73) 



In (72), a is a given function of x,y but g in (73) 

 depends on x,y,t. At the surface (73) of the stream 

 there is constant pressure, p^, a constant normal 

 surface tension, T. At the bed the (unknown) pres- 

 sure, p, depends on x,y and t. Thus, the normal 

 pressure, p* , at the top surface (73) is 



p* = p - q , 



T{ (1 + 



2B 3 6 + (1 + 

 x y xy 



yy 



(74) 



(1 + 



32)3/2 



y 



At the bed (72) the normal velocity of the fluid is 

 zero and the pressure, p* takes the value 



p* = p(x,y,t) 



(75) 



where p is to be determined. 



To proceed further, we recall the notation in (3) , 

 let the surface, C = 0, defined by (15) coincide with 

 the surface, J , and consider the three-dimensional 

 region of space between the surfaces (72) and (73) 

 occupied by the fluid. Any point in this three- 

 dimensional region is then specified by 



" We use g* (instead of g) for gravity, since the letter, g, 

 is used for a different quantity in (3), (5) and elsewhere in 

 the paper. 



