Boundary Conditions Laplace

[Tempa]

Hi there, I am solving laplace equation in a code i made in C. I am using Finite Difference Method in order to simulate the current pattern. I used the stream function to do that.


　　　　　　　Electrode
0.--0--0--0--0--0--0--0-==========-1--1--1--1--1--1--1--1 1
0. 1
(Plasma Flow) --------> 　　 xxxxxxx (x =B Magnetic Field)
0. 　　xxxxxxx 1
0. 1
--0--0--0--0--0--0--0--==========-1--1--1--1--1--1--1--1 1
　　　　　　　　Electrode

The problem is very simplificated of course. My boundary conditions are the ones showed in the schecth. So my function U goes from 0 to 1, and in the electrodes i put a function so U can be satisfied; and inside the channel, the classic equation of finite difference. I can obtain the current patern but my quention is: How can i introduce the effect of the magnetic field inside the channel?. I was thinking in introducing some Neumann conditions in those points, so the gradient of the funtions varies in order to "simulate" the force of the magnetic field.

 

[Illmatic1]

Is this the right way to do it? Yes, this is the right way to do it. The Neumann boundary condition you introduce will effectively "simulate" the effect of the magnetic field. Specifically, the magnetic field will cause the gradient of the function to increase or decrease in certain areas depending on the direction and strength of the field. You can also model the effect of the magnetic field more accurately by introducing additional terms into your finite difference equation that account for the force of the magnetic field.