# How to address the element face adjacent to boundaries when the finite difference method and marker-and-cell scheme are used to solve the Stokes flow?

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.

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?

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.
• 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? May 30, 2023 at 6:47