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Lets say, I have a fluid in a rectangular enclosure (2D). I apply electric potential $U=U_1$ at left boundary and $U_2$ at right boundary. In the lower and upper boundaries, the potential varies linearly from $U_1$to $U_2$.

I am simulating this problem. So what initial condition should I take for potential of the fluid (inside the four boundaries)? Also how will the potential develop?

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What physical effect are you studying?

If you are investigating the properties of the steady state, I would try to start as close to it as possible. I would recommend something that only depends on the transverse coordinate: linear velocity change with the potential.

If you are investigating turbulence you should break all the symmetries and try many different initial conditions.

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