Boundary conditions for a stream function in a hydrodynamics problem

[GilSE]

TL;DR

The water is pouring out from the bath with a hole, I want to find streamlines. Needed boundary conditions for Laplace equation.

The situation is like this: we have a bath of a square cross-section. The cross-section is modeled by an area $\{x, y: 0\leq x\leq 1, 0\leq y\leq 1\}$ in 2d Cartesian coordinates (the gravity fied direction is opposite to the y-axis). There is a hole in the floor of the bath, it starts at $x=a$ and ends at $x=1-a$. The bath is filled with ideal incompressible liquid which is pouring out from the hole. Assuming the liquid is pouring out slowly, I think, we can suppose the flow is steady when the considered time interval is small enough. Also, I assume the flow is vortexless because why not. I'd like to find the streamline equation for this flow.

In this case, the stream function can be found from Laplace equation $\Delta\psi (x,y)=0$.
The question is the boundary conditions for the equation. Of course, on the walls of the bath they should be $\psi(x,y)=const$, but for the $y=1$ boundary and for the hole it's more complicated question for me.

My reasoning:
$\psi(x_0,y_0)$ could be found by integrating partial derivative $\frac {\partial \psi (x,y_0)}{\partial x}$ with respect to $x$ from $0$ to $x_0$, if set $\psi(0,y)=0$. As the derivative $\frac {\partial \psi}{\partial x} = -v_y$, all we need to obtain values of $\psi$ at these boundaries it is the vertical component of velocity $v_y$ at them. But only I have no idea how to figure out $v_y(x,0)$ and $v_y(x,1)$. I thought the Bernoulli theorem could be the key, but it deals with the magnitudes of velocities, not with components.

 

[Baluncore]

Also, I assume the flow is vortexless because why not.

The elimination of Coriolis force is a very wise move.
You may need to consider the hole, as having a vena contracta, below the cut edge.

Don't put a round hole in a square bath. It messes with your coordinates.

If the bottom of the bath, in plan, is square, with vertical walls, then use a symmetrical square hole to partly simplify the maths.

Ideally, for math simplicity, the bottom of the bath would be round, the hole could then also be round, and in the centre of a radial model.

 

[GilSE]

Don't put a round hole in a square bath. It messes with your coordinates.

Of course! The bath is infinite in the direction normal to the Oxy plane, and the hole is an infinitely elongated rectangle. I think now it fits to my coordinates perfectly. (I am not in the mood to mess with solving an equation in cylindrical coordinates, let it be simple Cartesian)

 

[GilSE]

You may need to consider the hole, as having a vena contracta, below the cut edge.

What I've (I hope) understood about "vena contracta" is that there is a level slightly below the hole where $v_x$ component is equal to 0. Maybe it can be helpful for my problem, but I didn't figure out how exactly, for now.

 

[Baluncore]

Fluid, converging towards the hole, continues to converge after passing through the hole. That results in a flow after the hole, that is narrower than the actual hole.
https://en.wikipedia.org/wiki/Vena_contracta

 

[GilSE]

Fluid, converging towards the hole, continues to converge after passing through the hole. That results in a flow after the hole, that is narrower than the actual hole.
https://en.wikipedia.org/wiki/Vena_contracta

Actually, I did an attempt to solve the equation assuming $v_y$ is constant at $y=0$ and at the hole. The picture was like this:

Spoiler

Configuration of the streamlines at the hole looks like they really will form the picture from the Wikipedia page if would be extrapolated, but I don't think that my assumption about constant $v_y$ is physically justified.