514 



BAGNOLD 



[chap. 21 



direction of flow), is small, the term tarijS can also be neglected. But when j3 

 is increased until P = (f> the stationary grain bed will move as an avalanche 

 without any fluid stress being applied. 



It should be noted that relation (2) expresses a state of shear equilibrium 

 between an a^^pliod and a resisting thrust, the thrusts being equal in value but 

 different in kind. For this reason the externally applied fluid thrust is dis- 

 tinguished for clarity by a special symbol ^ — a capital script T, The whole 

 applied thrust, including the tangential gravity component, if any, on the solid 

 phase, would be ^ F + [{ps —p)lps\gnib sin jS. 



The distinctions between the four different stresses essentially involved in 

 two-phase equihbrium are shown schematically in Fig. 3. 



FLUID STRESSES 



FLUID FLOW 



APPLIED STRESSES ^- 

 RESISTING STRESSES 



iiiiiiiiiiniiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii 



MOVING 



SEDIMENT 



LOAD 



IIIIIIIIIIPIIIIIIIIII 



GRAIN STRESSES 

 P.-P 



P 



gm sin /3 



P'P 

 T = -^-^ gm cos /3 tan <^ 



llilllllllllllllllllllllllllllll 



P- P r\ 



= ^^-^ am cosyS 



Fig. 3. Sclieniatic diagram showing conditions of tw'O -phase stress eqviihbrium at base of 

 a bed load. A part of the applied fivxid stress, ^p, is transferred to the bed -load grains 

 above the bed boundary, and is transmitted to the bed as a grain stress. The overall 

 equilibrium condition is expressed by 



ar_ . Ps-P 



Cy p 



Ps 



gmsin^ = tq + 



Ps-p 

 ps 



gm cos jS tan (f>. 



The implications of the general equilibrium relation (2) as applied to trans- 

 jjort over a homogeneous grain bed have been discussed in a further paper 

 (Bagnold, 1956). It was there shown that the residual fluid element to cannot 

 increase above its value rt = ^ Pt at the threshold of grain motion over the bed 

 when m\) is zero ; and as ^ p is increased to must, in fact, decrease progressively 

 to a negligible value. 



Any increase in ^ p must cause erosion of the bed grains, with a resulting 

 increase in the bed-load weight \{f>s —p)lps]gnib cos /3 imposed on the new bed 

 siu'face and holding it down, together with a corresponding increase in the solid- 

 phase resisting thrust as given by the first term on the right of relation (2). 

 Moreover, as the bed load increases, so does the concentration of grains in its 

 bottom-most sheared layer, with the effect that the intensity of turbulence in 

 the fluid between the grains is appreciably reduced. It is reasonable to conclude 

 that when a certain transitional stage of bed movement is exceeded the fluid 

 flow may be considered as unaffected by the existence of an underlying 



