The simplest such parameter having the right dimensions is the time re- 

 quired for a particle to settle through a distance equal to its diameter 

 in the fluid at rest. If the settling velocity is denoted by Vs then we 

 can write 



*i = c i x (4_18) 



c being a constant of proportionality 



A basic characteristic of our model is that the criterion of equili- 

 brium is governed mainly by the lift force exerted by the flow on the 

 particle. Since the coefficient of this lift force assumes a constant 

 value even for very small values of the Reynolds number the settling 

 velocity at equilibrium may be written as 



/ g (P s - Pf ) D 



V = = S V i ~^ ~ (4-19) 



s 2 pf 



Combining now (4-18) and (4-19) and making the proper substitutions for & 

 and t equation (4-16) becomes 



J- £ P ^ ° 2 /ISIS (4-20) 



13 D Of 



A? A T 



■ PY S D^ J g( K s 



" Pf 

 which is the "bed-load" equation; after being rearranged (4-20) becomes 



/gCPs - Pf) 



Y S A 2 A L D^ A 3 A lD - V.Dp 



This is identical with Einstein's (1950) equation (38) provided that the 

 material forming the bed is uniform. The significance of equation (4-21) 

 is that it describes the equality between the rate of deposition (left 

 hand side term) and the rate of erosion (right hand side). The remarkable 

 characteristic of either equation (4-20) or (4-21) is that they indicate 

 that the bed-load rate is only indirectly related to the flow intensity 

 through the probability p. Eliminating this probability between equations 

 (4-13a) and (4-20) we obtain the fundamental relationship between flow 

 intensity and bed-load rate. 



Before we conclude this section, it would be necessary to define a 

 little better the average distance of travel I - AlD. A rather small 

 value of p implies that only on a small fraction of the bed surface the 

 lift force is strong enough to remove a grain of a given size at any 

 instance. Therefore a particle that has been lifted by the flow will 

 probably come to rest immediately upon completion of the first step since 

 it is very probable that the conditions locally will favor deposition. 

 In this case I = A'lD where A'l is the true constant of proportionality 

 between distance of travel in a single step and particle diameter. If, on 



17 



