Fluid flow with fluid pressure $\begin{array}{l}p(t, {\bf r})\end{array}$ is not changing in time:

(1)	$\begin{array}{l}\displaystyle p(t, {\bf r}) = p({\bf r})\end{array}$

This immediately leads to stationary fluid velocity $\begin{array}{l}{\bf u}(t, {\bf r})\end{array}$ :

(2)	$\begin{array}{l}\displaystyle {\bf u}(t, {\bf r}) = {\bf u}({\bf r})\end{array}$

Derivation

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:

(3)	$\begin{array}{l}\displaystyle {\bf u}(t, {\bf r})= F({\bf r}, p, \nabla p)\end{array}$

with right side not dependent on time in stationary flow:

(4)	$\begin{array}{l}\displaystyle \frac{\partial {\bf u}(t, {\bf r})}{\partial t}= 0\end{array}$

which leads to (2).

The fluid temperature $\begin{array}{l}T(t, {\bf r})\end{array}$ is supposed to vary slowly enough to provide quasistatic equilibrium.

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

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See also