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Steady-state 1D inviscid fluid flow
Equation of State (EOS)


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




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\rho = \rho(p, T)


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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bodyT(l)

fluid temperature

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bodyp(l)

fluid pressure

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body\rho(l)

fluid density

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body--uriencoded--%7B\bf u%7D(l)

fluid velocity vector 

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

fluid velocity module

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body

gstandard gravity constant LaTeX Math Inlinebody{\bf f}_{\rm cnt}

--uriencoded--%7B\bf f%7D_%7B\rm cnt%7D(l)

volumetric density of all contact forces exerted on fluid body

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body--uriencoded--%7Bf%7D_%7B\rm cnt, l%7D(l) = %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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bodyj_m

fluid mass flux

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body\dot m

mass flowrate

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bodyg

standard gravity constant




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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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