Fluid flow with fluid pressure is not changing in time:

p(t, {\bf r}) = p({\bf r})

This immediately leads to stationary fluid velocity :

{\bf u}(t, {\bf r}) = {\bf u}({\bf r})

In the most general case (both reservoir and pipelines) the fluid velocity is a function of pressure and pressure gradient and can be written as:

{\bf u}(t, {\bf r})= F({\bf r}, p, \nabla p)

with right side not dependent on time in stationary flow:

\frac{\partial {\bf u}(t, {\bf r})}{\partial t}= 0

which leads to .

The fluid temperature is supposed to vary slowly enough to provide quasistatic equilibrium.

This flow regime is often observed in pipeline fluid flow and reservoir fluid flows.

See also