503 



g T -"g = g-'x g-" 



rik 



v.g = V g. 



(8) 



and where x^'^ are the contravariant components of 

 the symmetric stress tensor. In terms of quantities 

 defined in {5)-(8), the local field equations which 

 follow from the integral forms of the three- 

 dimensional conservation laws for mass, linear 

 momentum and moment of momentum, respectively, are 



t. 



* h 

 p g = 



* * k * Is * 



+ pfg =pgv 



X T 



(9) 



where p* is the three-dimensional mass density, f* 

 is the body force field per unit mass, and a comma 

 denotes partial differentiation with respect to S""". 



A material line (not necessarily a straight line) 

 in S can be defined by the equations, 9*^ = 6"(5); 

 the equation resulting from (1) with 9" = 9"(C) rep- 

 resents the parametric form of this material line in 

 the current configuration and defines a 1-parameter 

 family of curves in space, each of which we assume 

 to be smooth and nonintersecting. We refer to the 



space curve , 



= , in the current configuration 



by c. Any point of this curve is specified by the 

 position vector, r, relative to the same fixed ori- 

 gin to which r* is referred, where 



r = r(S,t) = r (0,0,5,t) 



(10) 



Let a3 denote the tangent vector along the 5-curve. 

 By (10) and (3)i, 



3r 

 as = a3(C,t) = JF = g3(0'0.5-t) 



(11) 



where k and x denote, respectively, the curvature 

 and the torsion of c. In the special case that c 

 is a plane curve, we may choose aj as the unit 

 normal to the curve and then a2 will be perpendicu- 

 lar to the plane of ai and 337 If c is a straight 

 curve, then there is no unique Serret-Frenet triad 

 and a. may be chosen as any orthogonal triad with 

 ai,a2 as unit vectors. Equations (13) are not 

 xdentical to the formulas of Prenet because the pa- 

 rameter, £, , is not necessarily the arc length of c. 

 It may be noted here that the convected coordinate, 

 £, may be chosen to coincide with the arc length in 

 any one configuration of the material curve, e.g., 

 in the present configuration. However, in a general 

 motion (involving different configurations) the arc 

 length between any pair of particles changes while 

 the convected coordinates of each particle must re- 

 main the same. Therefore, arc length would not 

 qualify as a convected coordinate. 



A material surface in 6 can be defined by the 

 equation, E, = E,(B'^) ; the equation resulting from 

 (1) with 5 = 5(9") represents the parametric form 

 of this material surface in the current configura- 

 tion and defines a 1-parameter family of surfaces 

 in space, each of which we assume to be smooth and 

 nonintersecting. We refer to the surface, 5=0. 

 in the current configuration by s. Any point of 

 the surface, s, is specified by the position vector, 

 r, relative to the same fixed origin to which r* is 

 referred, where 



r = r(9'^,t) = r*(90',0,t) 



(14) 



Let a,^ denote the base vectors along the 9 -curves 

 on the surface, s. By (14) and (3)i, 



3r 

 a = a (9^,t) = ^^ = g (e'^,0,t) , (15) 

 39" 



and the unit principal normal, aj , and the unit bi- 

 normal vector, 52, to c may be introduced as 



and the unit normal, a3 = a3 (9 ,t) , to s may be 

 defined by** 



§1 = ai (C,t) = 



32 = a2(5,t) 



i§3l = (333) 



|3a3/3C| 



I §3 I 



X a-i 



(12) 



a • a, = , a, • a, = 1 , 



~a -3 ~ 6 ~i 



a^ = a^ [a-^&^a^] > 



(16) 



In the next four paragraphs we provide appropri- 

 ate definitions for jet-like and sheet-like bodies 

 in fairly precise terms. 



333 = ?3 • ?3 ' 

 [aia2a3] > , 



(12) 



Definition of a Jet-like Body. A Representation 

 for the Motion of a Slender Jet. 



where the notation |a3| stands for the magnitude of 

 §3. The system of base vectors, aj , are oriented 

 along the Serret-Frenet triad and satisfy the dif- 

 ferential equations 



3ai 



3a2 

 ~3S 



333 



x(a33) a2 - Ka3 , 



- x(a33) a 



33' ai , 



3a 



33 



-TZ = a33Kai + ^ 

 3C ~^' -'■ 2a 



33 



8? 



§3 ' 



(13) 



Consider a space curve c defined by the parametric 

 equations, 9" = 0, over a finite interval, S 1^5^52- 

 Let r be the position vector of any point of c and 

 let ai,a2 and 33 denote its unit principal normal, 

 unit~binormal, ~and the tangent vector, respectively. 

 At each point of c, imagine material filaments ly- 

 ing in the normal plane, i.e., the plane perpendicu- 



The use of the same symbols for base vectors of a surface 

 in (15) -(16) and for the triad of a space curve in (11) -(12) 

 should not give rise to confusion. The main developments 

 for jets and sheets are dealt with separately in the rest 

 of the paper; this permits the use of the same symbol for 

 different quantities in the case of jets and sheets without 

 confusion. 



