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There is no finite volume formulation in this problem
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# Pressure boundary conditions in Stokes Equation in 2D

I am solving the steady-state incompressible Stokes equations in 2D:

$$$$\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} = 0,$$$$

$$$$\mu\left[\frac{\partial^2 u_x}{\partial x^2} + \frac{\partial^2 u_x}{\partial y^2}\right] - \frac{\partial p}{\partial x} = 0,$$$$ and $$$$\mu\left[\frac{\partial^2 u_y}{\partial x^2} + \frac{\partial^2 u_y}{\partial y^2}\right] - \frac{\partial p}{\partial y} = 0.$$$$

I'm using finite differences with a staggered grid approach (velocities defined at the faces and pressure at the cell centers).

I'm struggling to find how to implement pressure boundary conditions in the left and bottom faces. My pressure stencil is an "L" stencil: I use $$p_{i,j+1}$$, $$p_{ij}$$, and $$p_{i+1,j}$$. Hence, at the left and bottom-left volumes, I don't use the pressure value in the "ghost cell". Any clue on how to implement pressure BCs in this setting? I would like, for instance, to validate my code with a parallel plate simulation (prescribed pressure at left and right faces, and no-slip at top and bottom faces).