DU DU 



6Fj = -p(i + k) ^ 5¥-^ = -P(l + K) ^ (1-n) 6x6z , (A-63) 



where SVjjj is the volume of solids within the control volume and k is 

 the added mass coefficient. Little information is available on the 

 magnitude of k for a closely packed ensemble of irregular grains. For 

 isolated spheres k = 0.5 may serve as an indication of the order of 

 magnitude. 



The rate of change of momentum within the fixed control volume is 

 given by 



3M 9 ^^c 



and the momentum flux out of the control volume is given by 



3U 3U 



where the continuity equation has been invoked. 

 Now formulating the momentum equation, 



6F + 6F^ + 6F^ = 1^ + M^ , CA-66) 



p d 1 3t F 



and introducing equations (A-61 through A-65) the horizontal momentum 

 equation is obtained as 



DU y-^ J 



p(l + K (1 - n)) jrr^ = - ^ - p(a + e/u + W ) U , (A-67) 



Ut oX 



which corresponds to the equation given by Polubarinova-Kochina (1962) 

 for unsteady flow in a porous medium with the resistance term expressed 

 by a Dupuit-Forchheimer type of formula. The coefficient attached to 

 the acceleration term is somewhat different from the expression given 

 by Sollitt and Cross (1972), but is believed to be correct in the 

 present derivation. 



117 



