(1)	$\begin{array}{l}\displaystyle \rho \left( \frac{\partial {\bf u}}{\partial t} + ({\bf u} \cdot \nabla) {\bf u} \right)= -\nabla p +\mu \nabla^2 {\bf u} + \frac{1}{3} \mu \nabla (\nabla {\bf u}) + \rho \, {\bf g} + {\bf F}_{\rm ext}\end{array}$

For incompressible fluids this will simplify to (see

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):

(2)	$\begin{array}{l}\displaystyle \rho \left( \frac{\partial {\bf u}}{\partial t} + ({\bf u} \cdot \nabla) {\bf u} \right)= -\nabla p +\mu \nabla^2 {\bf u} + \rho \, {\bf g} + {\bf F}_{\rm ext}\end{array}$

For a 1D flow along the pipe the equation (1) can be projected to the axis line fo the pipeline trajectory:

(3)	$\begin{array}{l}\displaystyle \rho \left( \frac{\partial {\bf u}}{\partial t} + u \frac{\partial u}{\partial l} \right)= -\frac{\partial p}{\partial l} +\mu \nabla^2 {\bf u} + \frac{1}{3} \mu \nabla (\nabla {\bf u}) + \rho \, g \, \cos \theta + F_{\rm ext, \, l}\end{array}$

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See also