Circuler grid need to be solved by Finite difference method pls help me

[nafiz27me]

Circuler grid need to be solved by Finite difference method! pls help me...

hi this is the picture of the problem.. i have studied the rectangular grid but not the circular grid... now pls someone help me to find out the way to solve a heat conduction problem for circle using finite difference method.

1/r δ/δr (r δT/δr)+ 1/r^2 ((δ^2 T)/(δΦ^2 ))=0
this is the two dimensional equation and the discritise values are below...

1) 1/r δ/δr (r δT/δr)= 1/r (δT/δr)+(δ^2 T)/(δr^2 )=1/r (T_((i,j,k)-T_((i+1,j,k) ) )/Δr)+((T_((i+1.j.k) )+T_((i-1,j,k) )-2T_((i,j,k) ))/(Δr^2 ))

2) 1/r^2 ((δ^2 T)/(δΦ^2 ))= 1/r^2 ((T_((i.j+1.k) )+T_((i,j-1,k) )-2T_((i,j,k) ))/(ΔΦ^2 ))

 

[Navin A S]

so the final discretized form is: (T_((i,j,k)-T_((i+1,j,k) ) )/Δr)+((T_((i+1.j.k) )+T_((i-1,j,k) )-2T_((i,j,k) ))/(Δr^2 )) + (T_((i.j+1.k) )+T_((i,j-1,k) )-2T_((i,j,k) ))/(ΔΦ^2 )) =0or rearranging the terms, you get:T_((i,j,k)) = (T_((i+1,j,k) ) + (T_((i-1,j,k) )/(Δr^2 )) + (T_((i.j+1.k) )+T_((i,j-1,k) )/(ΔΦ^2 )) / (1/Δr + 1/Δr^2 + 1/ΔΦ^2 )Hope this helps!