Dear Fellow Applied Physicists,
I have come across a simple problem turned complicated, where I attempt to solve a stationary 2D Laplace Equation (L.E) in order to better understand the effects of sharp edges on the solutions.
Starting from a simple setup, I solved, using a classic separation of variables technique, the L.E in 2D for a infinitely extended edge (with an opening angle of 120°) for the simple Dirichlet Boundary Condition of constant potential across the edge's infinite contour. The solution of this setup is easily found using 2D polar coordinates. However, once I add an additional geometric constraint to my problem, the Separation of Variables hypothesis no longer holds; thus I require some help here.
I have added one finite boundary to one of the edge's lengths, a finite value of length L, where this new boundary (at x = 0) behaves like a symmetry boundary condition.
The Laplace Equation (2 dimensional):
∇2 ϕ = 0
The boundary conditions in Cartesian coordinates are as follows:
1) ϕ = { V=const 0 < x < L & y = 0
L < x < +∞ & y = mx - mL for m = slope of boundary
}
2) ∇ ϕ|(x = 0 , y) = 0
I switched to Cartesian coordinates because the 2nd BC removes the separation of variables property when using polar coordinates, thus rendering it no more useful than a Cartesian formulation.
If you guys can, please either refer me to texts that tackle similar problems, or advise me on how to properly select solution methods for this problem. Separation of Variables doesn't work as far as I can tell, due to the BC.1 having y = f(x).
Please inform me if more details are needed.
Best,
Christopher
Hi! This may turn out not be a way to solve the problem, but I am speculating whether it might be possible to use a Schwarz-Christoffel transformation to convert the domain into a semi-infinite strip. My guess is that that type of transformation may well yield coefficients to the first and second order derivatives in the transformed Laplacian which are then functions of the new coordinates. The question of whether one can then use separation of variables will then depend on the complexity of those coefficients. Polar coordinates clearly has coordinate-dependent coefficients which still permit separation of variables, but whether such a transformed wedge problem does is unknown to me at present.
Andrew
Dear Andrew,
Thank you for your reply!
As I am not a trained mathematician, I cannot give my opinion on the Schwarz-Christoffel transformation (and mapping techniques), that you referred to, before reading up on the literature.
However, I am curious as to whether there are any other direct (and standard) solution methodologies that do not assume a separation of variables hypothesis. I am guessing that oceanographers come across similar boundary conditions when including bathymetry into their equations, but alas, I am still at a loss here when it comes to the solution method and the literature to look up.
Best,
Chris
I do not understand BC 2), you wrote only the gradient (that drive to a vector field) I suppose it must be projected along the normal unit to the boundary to produce homogeneous Neumann BC, isn't that?
Dear Filippo,
What I mean by BC 2 is that both the partial derivatives (w.r.t x and y) at x=0 for any y are zero.
I am basically using BC 2 as a symmetry condition for the solution, so I can use the mirror of the solution for the x
Hi Christopher Nahed,
If you can solve
1) ∇2 ϕ1 = 0
ϕ1 = { V/2 , -∞ < x < 0 & y = 0
V , 0
Dear Victor,
Thank you for your reply!
I will solve this setup soon, and get back to this thread with my results.
Best,
Chris
Dear Christopher ,
I had overlooked that for |x|>L/2, y=0 function ϕ(x,y)=ϕ1 (x-L/2,y)+ϕ2 (x+L/2,y) will be discontinuous. Please forgive me for this mistake.
Best regards.
Victor
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