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The Stokes equations are $$-\Delta \mathbf u + \nabla p = f \text{, in }\Omega,$$ and $$ -\nabla \cdot \mathbf u = g, \text{ in } \Omega$$ where $\mathbf u = \left( u, v \right)$ is the flow velocity, and $p$ is the pressure.

I tried to solve the equations by the finite difference method and marker-and-cell scheme, and I found this paper. It helps.

Simply put, I generate a 2D structured mesh. For vertical element faces, the unknown is $u$. For horizontal element faces, the unknown is $v$.

But I have a question about dealing with the case when the element face is near the boundary of the domain.

The paper said that a ghost value should be introduced, as the figure below shows.

enter image description here

The paper presented a formula to interpolate the ghost value: $$\left(u_{0, j} + u_{1,j}\right)/2=u_D\left(x_j, 1\right)$$, here I want the (red) boundary to have a Dirichlet boundary condition $v = 0$. As I understood, $u_D$ here is $v = 0$.

The thing I do not understand is: the boundary is horizontal, and its variable is $v$, but either $u_{1,j}$ or $u_{0,j}$ is the vertical component of the flow velocity. Why can we use a known horizontal component value (at the red boundary) to interpolate the ghost value which is a vertical component of the flow velocity?

Can anyone help me?

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The paper is not specifying a Dirichlet $v=0$ at $y=1$, but $u=u_D$ at $y=1$. This could be used as a wall slip condition: for $u_D=0$, you have a no slip boundary wall; that is, the flow tangential to the wall is zero. For $u_D \neq 0$, you could also interpret this as a no slip boundary wall, however the wall is moving with some fixed horizontal velocity $u_D$.

There are of course other interpretations of what setting a Dirichlet boundary condition on $u$ means, these are just two possible examples.

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  • $\begingroup$ So, that means: apart from $v = 0$ (non-penetration) imposed at the boundary ($y = 0$), a $u_D$ is also needed at that boundary (non-slip or slip), in order to introduce the ghost value. Am I correct? $\endgroup$ May 30, 2023 at 6:47
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    $\begingroup$ yes, you specify both v and u at the wall. $\endgroup$ May 31, 2023 at 4:24

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