The basic concept regarding the pattern of motion was that the loose bed 

 is sliding in layers under the action of the flow above. The top layer 

 of the bed is set into motion by the "tractive force" or shear, which in 

 flows where energy is dissipated mainly to overcome friction, is equal per 

 unit area of the bed to the product of the unit weight of the water, of the 

 depth of flow and of the energy gradient. When this force becomes larger 

 than the force resisting motion per unit area, which is proportional to 

 the submerged weight of the particles forming the bed, the latter begin 

 to move. The rate of transport determined experimentally was found to be 

 a function of the difference between the two forces. No effort has been 

 made in developing this theory to explain the actual mechanism of inter- 

 action between the solid particles and the flow field. General information 

 therefore cannot be deduced from this theory and its application is neces- 

 sarily confined to the narrow range of conditions used in its derivation. 

 A major weakness is that it deals only with an average value of the shear 

 which is assumed in each case to be constant in time and space. The 

 implication is that all particles of a certain size will start moving 

 simultaneously over the entire bed, whenever the average shear is larger 

 than critical. This, of course, is contrary to the well-established fact 

 that motion near the bed takes place in the form of sudden jumps by indi- 

 vidual particles alternating with rather long periods of rest. 



Jeffreys (1929) perhaps, was the first investigator to base a theory 

 on the stability of the individual solid particle. He suggested the appli- 

 cation of the solution from classical hydrodynamics regarding the stability 

 of a long circular cylinder of radius r resting on the flat bed of a deep 

 stream with its axis perpendicular to the flow. The expression of the 

 complex potential is then of the form 



W = -nr U coth TTr/z (4-1) 



and the upward thrust transmitted to the solid can be shown to be equal fo 



L - v ♦ * - 2 ) uV 



It follows that the condition of motion is given by the inequality 



where U c is the critical value of the stream velocity. He postulated that 

 for three-dimensional elements the values would be slightly larger. The 

 numerical values given by Jeffreys for water and sand were U c = 4.32 cm/sec 

 for r = 0.01 cm and U c = 13.6 cm/sec for r = 0.1 cm. This model, of course 

 gives the correct answer as long as there is sufficient justification for 

 describing dynamic effects in flows of real fluids by means of the irrota- 

 tional theory. Jeffreys answer to this question is affirmative; he claims 

 that during the initial stage, when a particle is just dropped on the bed 



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