

EQUATIONS OF MOTION 



61 



where p = -fe+^/v+J^ and ^ _ Coefficient of Viscosity. 



But, considering the elementary volume B x . B y . B z, we see that the 

 total force acting on this volume, due to any variation of stress on opposite 

 faces of the volume in the direction X, is 



x 3 y 3 z ) 

 So that if an external force, having a component X per unit mass in 

 this direction, also act on the element, we have 



Total force on element) _ |"/3 Pxx 3 p yx 8 p zx \ , n v ~| 

 in direction OX ~ LV^T " " ~^J ~ ' Tz) ~* p J 



B x . B y . B z. 



This equals mass X acceleration) 5, 5. ^ ,du 



PI . ! > s\ ^ r = P o x. o u . o z X 



of element in direction O X \ d f 



.'. pX-\ 



(9) 



<l *' 



Similarly, considering the accelerations in directions 

 Y and Z, we get 



^ ^ 3 ^ 



+ 



d 



1895, p. 508. In this discussion, p xx , etc., are reckoned positives when tensions, while p, as is 

 * common in hydrodynamical problems, is reckoned positive when a compression. This 

 I accounts for the negative sign before jw in (7). 



1 This may be proved as follows. The difference of normal stress on the faces O B and 



$D (Fig. 30A) = ^2 8 *. 



3 x 



.*. Difference of normal force on these faces = , xx S x . 



S y . 5 2. 



Also the difference of tangential stress in the direction 



X, on the faces O C and B D = -^r S y, while the 



8 y 



resultant force due to these tangential stresses in this direc- 

 tion 



Similarly the tangential stresses on the faces D and 

 B C in the direction X give rise to a resultant force 



/>=* 



. 8 x . 5 y . 8 z. 



FIG. 30A. 



These include all the forces due to variation of stress across the element, which have a 

 component in the direction X. Summing these, we get the resultant force in the direction 

 of increasing x. Similarly for the forces in the directions Z and Y. 



