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The base driving equations are:

Euler equationSteady-state 1D inviscid fluid flow


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\frac{d p}{d l} =
-\rho \, u \, \frac{d u}{d l}  + \rho \, g \, \cos \theta + f_{\rm cnt, \, l}




where

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bodyl

distance along the fluid flow streamline

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body\theta(l)

inclinational deviation,  

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body\displaystyle \cos \theta = dz/dl

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bodyz(l)

elevation along the 1D flow trajectory 

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bodyp

fluid pressure

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body\rho

fluid density

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body--uriencoded--%7B\bf u%7D

fluid velocity vector 

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body--uriencoded--u = %7C %7B\bf u%7D %7C

fluid velocity module

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bodyg

standard gravity constant

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body{\bf f}_{\rm cnt}

volumetric density of all contact forces exerted on fluid body

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body--uriencoded--%7Bf%7D_%7B\rm cnt, l%7D = %7B\bf e%7D_u \cdot %7B\bf f%7D_%7B\rm cnt%7D

projection of 

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body--uriencoded--%7B\bf f%7D_%7B\rm cnt%7D
 onto the unit fluid velocity vector:  
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body--uriencoded-- %7B\bf e%7D_u = %7B %7C %7B\bf u%7D %7C%7D %5e%7B-1%7D \, %7B\bf u%7D




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\left( 1 -  \frac{c(p) \, \rho_s^2 \, q_s^2}{\rho \, A^2}   \right )  \frac{dp}{dl} = \rho \, g \, \frac{dz}{dl}  - \frac{\rho_s^2 \, q_s^2 }{2 A^2 d} \frac{f(p)}{\rho}



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q(l) = \frac{\rho_s \cdot q_s}{\rho}



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u(l) = \frac{\rho_s \cdot q_s}{\rho(p) \cdot A}


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