- #1
GilSE
- 8
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- TL;DR Summary
- 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.
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.
