# 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:

$$\mu \partial_x^2 u_y = f_y$$

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 [8]:

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

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

## References¶

[1] S. Succi. The lattice Boltzmann equation for fluid dynamics and beyond. Clarendon Press, Oxford, 2001.
[2] B. Dünweg and A. J. C. Ladd. Advanced Computer Simulation Approaches for Soft Matter Sciences III, chapter II, pages 89–166. Springer, 2009.
[3] B. Dünweg, U. Schiller, and A.J.C. Ladd. Statistical mechanics of the fluctuating lattice-Boltzmann equation. Phys. Rev. E, 76:36704, 2007.
[4] P. G. de Gennes. Scaling Concepts in Polymer Physics. Cornell University Press, Ithaca, NY, 1979.
[5] M. Doi. Introduction to Polymer Physics. Clarendon Press, Oxford, 1996.
[6] Michael Rubinstein and Ralph H. Colby. Polymer Physics. Oxford University Press, Oxford, UK, 2003.
[7] Daan Frenkel and Berend Smit. Understanding Molecular Simulation. Academic Press, San Diego, second edition, 2002.
[8] W. E. Langlois and M. O. Deville. Exact Solutions to the Equations of Viscous Flow. In: Slow Viscous Flow, Springer, Cham, 2014.