Momentum equation for Inviscid fluid flow (a partial case of Navier–Stokes equation):
| (1) | \frac{\partial {\bf u}}{\partial t} + ({\bf u} \cdot \nabla) {\bf u} = - \frac{1}{\rho} \, \nabla p + {\bf g} +\frac{1}{\rho} \cdot {\bf f}_{\rm cnt} |
where
\rho | fluid density |
\nu | fluid kinematic viscosity |
{\bf g} | resulting volumetric body force exerted on fluid body |
{\bf f}_{\rm cnt} | volumetric density of all contact forces exerted on fluid body |
Approximations
1D non-stationary fluid flow (for example along a pipeline trajectory) takes form:
| (2) | \rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial l} \right)= -\frac{\partial p}{\partial l} + \rho \, g \, \cos \theta + f_{\rm cnt, \, l} |
where
l | measured length along the 1D flow trajectory |
\theta | the angle between the body force and the tangent to the flow trajectory |
1D stationary fluid flow (for example along a pipeline trajectory) takes form:
| (3) | \frac{d p}{d l} = -\rho \, u \, \frac{\partial u}{\partial l} + \rho \, g \, \cos \theta + f_{\rm cnt, \, l} |
which is a guiding equation in practical pipe flow simulations.
In case of incompressible fluid \rho = \rm const in Earth's Gravity with no friction it simplifies to what is called a Bernoulli equation:
| (4) | \frac{p(l)}{\rho} + \frac{u^2}{2} - g \cdot z(l) = \rm const |
where
z(l) = l \, \cos \theta | elevation along the 1D flow trajectory |
g | standard gravity constant |
See also
Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Fluid flow / Navier–Stokes equation