# Intuition about Stokes flow beween solid walls

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1. Dec 8, 2014

### Hemmer

Hi there,

I have a question about incompressible Stokes flow in a channel between solid walls (with no-slip boundary conditions at $y = 0, L_y$). It is my intuition that, if the flow direction is $x$ (periodic), and the direction normal to the walls is $y$, then there cannot be a net velocity in that direction, i.e. $\langle v_y \rangle = 0$, as this somehow implies fluid must be passing through the walls? Is this correct and if so how should I go about showing this? I guess it might require some integration of the incompressibility condition but I've not got anywhere yet. Full equations:

$$\eta\nabla^2 \textbf{v} - \nabla p = \textbf{f}, \qquad\nabla . \textbf{v}=0$$

Please let me know if you require any additional information, and any replies greatly appreciated!

2. Dec 13, 2014