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Benchmark F1

Flow in Rigid Channel

Description

Fluid flows from the reservoir with the pressure $p_{\mathrm{S}}$ through upstream restrictor with resistance $R_{\mathrm{u}}$ 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:

$\begin{array}{ccc}\hfill l& \hfill =\hfill & 0.11\mathrm{m}\hfill \\ \hfill l_0& \hfill =\hfill & 0.15\mathrm{m}\hfill \\ \hfill D_0& \hfill =\hfill & 0.025\mathrm{m}\hfill \\ \hfill p_{\mathrm{S}}& \hfill =\hfill & 4000\mathrm{Pa}\hfill \\ \hfill p_{\mathrm{out}}& \hfill =\hfill & 0\mathrm{Pa}\hfill \\ \hfill R_u& \hfill =\hfill & {10}^9\mathrm{Pa}·{\mathrm{s}}^2·{\mathrm{m}}^{-6}\hfill \\ \hfill R_d& \hfill =\hfill & {10}^{11}\mathrm{Pa}·{\mathrm{s}}^2·{\mathrm{m}}^{-6}\hfill \\ \hfill N& \hfill =\hfill & 31\hfill \\ \hfill {\epsilon }_{\mathrm{S}}& \hfill =\hfill & 0\hfill \end{array}$

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 $p_{\mathrm{N}}$.

Results