Integral Equation for Momentum Conservation

By proceeding in a manner similar to that employed in deriving the von Karman equation for a developing boundary layer, the system of partial differential equations (Equations 4.21 and 4.22) can be reduced to the ordinary differential equation 

It can be readily shown that Equation (4.25) can also be obtained by substituting Equation (4.24) into the continuity and momentum equations directly and integrating by parts. It should be emphasized that, provided no statement is made about the stresses or the net 

The second term on the left of Equation (4.30) would, in fluid flow, be equal to the hydrostatic pressure according to Bernoulli s equation. For granular flow on an incline this term is equivalent to the driving force parallel to the inclined plane and hence may be equated to the overburden pressure in the x-direction, that is. 

Assembling the resuits obtained thus far yields the expression 

The final task before proceeding with its solution is to derive an appropriate expression for the shear stress acting over the bottom surface of the control volume, t . 

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