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LaTeX Math Inline
bodyl

distance along the fluid flow streamline

LaTeX Math Inline
body\theta(l)

inclinational deviation,  

LaTeX Math Inline
body\displaystyle \cos \theta = dz/dl

LaTeX Math Inline
bodyz(l)

elevation along the 1D flow trajectory 

LaTeX Math Inline
bodyT(l)

fluid temperature

LaTeX Math Inline
bodyp(l)

fluid pressure

LaTeX Math Inline
body\rho(l)

fluid density

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

fluid velocity vector 

LaTeX Math Inline
bodyu(l)

superficial velocity of the pipe flow

LaTeX Math Inline
body--uriencoded--%7B\bf f%7D_%7B\rm cnt%7D(l)

volumetric density of all contact forces exerted on fluid body

LaTeX Math Inline
body--uriencoded--f_%7B\rm cnt, l%7D(l) = %7B\bf e%7D_u \cdot %7B\bf f%7D_%7B\rm cnt%7D

projection of 

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

LaTeX Math Inline
bodyj_m

fluid mass flux

LaTeX Math Inline
body\dot m

mass flowrate

LaTeX Math Inline
bodyg

standard gravity constant




LaTeX Math Block
anchor9QRCZ
alignmentleft
\left( 1 - j_m^2 \cdot \frac{c}{\rho}   \right )  \frac{dp}{dl} = \rho \, g \, \frac{dz}{dl}  - \frac{j_m^2 }{2  d} \cdot \frac{f(p)}{\rho}



LaTeX Math Block
anchor1
alignmentleft
j_m = \frac{\rho_0 \cdot q_0}{ A}



Alternative form:


LaTeX Math Block
anchorrho
alignmentleft
\left[ \frac{1}{c} - \frac{j_m^2}{\rho} \right] \cdot \frac{d \rho}{dl} =
\rho^2 \, g \, \cos \theta -  \frac{j_m^2 }{2d} \cdot f(\rho)



See Also

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Petroleum Industry / Upstream / Pipe Flow Simulation / Water Pipe Flow @model / Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model

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