## Benchmark F1

### Flow in Rigid Channel

##### Description

Fluid flows from the reservoir with the pressure ${p}_{S}$ through upstream restrictor with resistance ${R}_{u}$ throught rigid channel and finaly through downstream restrictor to the outer space with pressure ${p}_{\mathrm{out}}$

##### Parameters

The fluid:

$\begin{array}{ccc}\hfill \rho & \hfill =\hfill & 997.05\mathrm{kg}·{\mathrm{m}}^{-3}\hfill \\ \hfill {\mu }_{d}& \hfill =\hfill & 0.00089106\mathrm{kg}·{\mathrm{m}}^{-1}·{\mathrm{s}}^{-1}\hfill \\ \hfill c& \hfill =\hfill & 1494\mathrm{m}·{\mathrm{s}}^{-1}\hfill \end{array}$

The rigid chanel:

$\begin{array}{ccc}\hfill l& \hfill =\hfill & 0.11\mathrm{m}\hfill \\ \hfill {l}_{0}& \hfill =\hfill & 0.15\mathrm{m}\hfill \\ \hfill {D}_{0}& \hfill =\hfill & 0.025\mathrm{m}\hfill \\ \hfill {p}_{S}& \hfill =\hfill & 4000\mathrm{Pa}\hfill \\ \hfill {p}_{\mathrm{out}}& \hfill =\hfill & 0\mathrm{Pa}\hfill \\ \hfill {R}_{u}& \hfill =\hfill & {10}^{9}\mathrm{Pa}·{\mathrm{s}}^{2}·{\mathrm{m}}^{-6}\hfill \\ \hfill {R}_{d}& \hfill =\hfill & {10}^{11}\mathrm{Pa}·{\mathrm{s}}^{2}·{\mathrm{m}}^{-6}\hfill \\ \hfill N& \hfill =\hfill & 31\hfill \\ \hfill {\epsilon }_{S}& \hfill =\hfill & 0\hfill \end{array}$

##### Theory

Under assumption of the Poiseuille flow, the flow rate $Q$ is given by formula:

$Q=\Delta p\frac{{A}_{0}^{2}}{8\pi {\mu }_{d}}\frac{1}{\frac{N-1}{N}l}$

where $\frac{N-1}{N}l$ is the distance between the points ${p}_{1}$ and ${p}_{N}$.

##### Results
Theory femCalc
$\Delta p\text{\hspace{0.17em}}\left[\mathrm{Pa}\right]$ $\Delta p\text{\hspace{0.17em}}\left[\mathrm{Pa}\right]$ $\text{Ratio}\text{\hspace{0.17em}}\left[\mathrm{-}\right]$
$1.947$ $1.946$ $0.9995$
Theory femCalc
$Q\text{\hspace{0.17em}}\left[{\mathrm{m}}^{3}·{\mathrm{s}}^{-1}\right]$ $Q\text{\hspace{0.17em}}\left[{\mathrm{m}}^{3}·{\mathrm{s}}^{-1}\right]$ $\text{Ratio}\text{\hspace{0.17em}}\left[\mathrm{-}\right]$
$1.96815·{10}^{-4}$ $1.96815·{10}^{-4}$ $1.0000$