A Ws, W 



Us,U 



Figure A-2. Definition sketch. 



To derive the horizontal momentum equation we consider the 

 elementary control volume indicated in Figure A-2. The momentum 

 equation states that the sum of the forces acting on the fluid within 

 this element must equal the rate of increase in momentum plus the net 

 momentum flux out of this control volume. Of forces acting on the 

 fluid within the element we have the pressure force. 



6F = 

 P 



3p 

 8x 



6x6z 



(A-61) 



and a force resisting the fluid motion within the porous medium. In 

 an unsteady motion this force will consist of a drag force, 5F^, and 

 an inertia force, 6Fj. For steady flow the drag force component is 

 expressed as a volume force which may be taken as 



6F 



-P (ex + 6 



/7 



W^)U 



6x5z 



CA-62) 



where the hydraulic properties of the porous medium are given by the 

 coefficients a and B. The coefficient a expresses the laminar flow 

 resistance and S is associated with the turbulent flow resistance. 



The inertial force, 6Fj, is associated with the fluid acceleration. 

 The fluid velocity, as seen by a solid particle in the porous medium, 

 is the seepage velocity and by analogy with inertia forces acting on a 

 single particle we may take 



116 



