# Tutorial 4 : The Lattice Boltzmann Method in ESPResSo - Part 4¶

### 6 Poiseuille flow in ESPResSo¶

Poiseuille flow is the flow through a pipe or (in our case) a slit under a homogeneous force density, e.g. gravity. In the limit of small Reynolds numbers, the flow can be described with the Stokes equation. We assume the slit being infinitely extended in $y$ and $z$ direction and a force density $f_y$ on the fluid in $y$ direction. No slip-boundary conditions (i.e. $\vec{u}=0$) are located at $x = \pm h/2$. Assuming invariance in $y$ and $z$ direction and a steady state, the Stokes equation is simplified to:

\begin{equation} \mu \partial_x^2 u_y = f_y \end{equation}

where $f_y$ denotes the force density and $\mu$ the dynamic viscosity. This can be integrated twice and the integration constants are chosen so that $u_y=0$ at $x = \pm h/2$ to obtain the solution to the planar Poiseuille flow :

\begin{equation} u_y(x) = \frac{f_y}{2\mu} \left(h^2/4-x^2\right) \end{equation}

We will simulate a planar Poiseuille flow using a square box, two walls with normal vectors $\left(\pm 1, 0, 0 \right)$, and an external force density applied to every node.

Use the data to fit a parabolic function. Can you confirm the analytic solution?

The solution is available at /doc/tutorials/04-lattice_boltzmann/scripts/04-lattice_boltzmann_part4_solution.py

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