Electrohydrodynamics: How will electric potential develop in a fluid when potential is applied from ends

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?