Modeling 2D Heat Flow
(10pts)
In this problem, you'll use our Two-dimensional Diffusion/Decay explorer to model the flow of heat through a circular domain that's missing a triangle. Everyone gets their own domain and parameters describing the problem but they're all similar. You can get your randomly generated problem by clicking here.
Comments
As can be observed in the screenshot above, the temperature near the midpoint of the insulated edge of the triangle (see my randomly generated problem for details of the model) is 1.04558.
Per the screenshot above, the temperature near the midpoint of the insulated side of the triangle is approximately 0.04795. See my random questions for more details.
Correction! I forgot to set the Dirichlet boundary on the circle as well. The temperature at the midpoint of the insulated edge of the triangle is now 0.31130 after 1s.
The temperature near the midpoint of the insulated edge at $t=1 s$ is $2.18727$.
The temperature near the midpoint of the insulated edge of the triangle at $t=1s$ is $0.57029$
Given my specific domain and parameters, the temperature at $t=1s$ near the midpoint of the insulated edge of the triangle is $0.57228$.
For my problem, the basic heat equation was:
$$u_t = 0.47\Delta u$$
The vertical side of the triangle is insulated
The Dirichlet condition on the circle and the 2 other sides of the triangle is
$$u(x,y,t) = 0.87^2 - (x^2+y^2)$$
The initial condition is:
$$u(x,y,0) = 0.87^2 - (x^2+y^2)$$
I found the temperature of the domain near the midpoint of the insulated edge at time, $t = 0.99$s to be 0.11568
@dmajor It looks to me like you're over-insulated!
The temperature of the midpoint at the insulated edge at t=1s is 0.16587 as shown in the image below.
The temperature of the midpoint at the insulated edge at t=1s is 1.91ish as shown in the image below.
The temperature near the midpoint of the insulated edge is around 1.83 after 1s
Given my domain and parameters, the temperature at t=1s is about 0.00
The temperature at the midpoint of the insulated side of the triangle at 1s was 0.096.
The temperature at the midpoint of the insulated side of the triangle when we reached one second is around 0.1725.
This is the heat flow model from the parameters given in my particular problem. At about $t=1s$, the temperature near the midpoint of the insulated edge is approximately 0.32308 heat units.
Given my parameters, at t=1s the temperature will be 0.00053
The temperature near the midpoint of the insulated edge of the triangle at $t=1s$ is about $0.17972$
Given my parameters, the temperature near the midpoint of the insulated side of the triangle at $t=1s$ is about $0.52492$.
Given the boundary and initial conditions, $u(t=1s) \approx 0.92$ [temperature units] at the midpoint of the insulated boundary.
At $t = 1s$, the temperature near the midpoint of the insulating edge of the triangle is $0.29975$.
At $t=1s$, the temperature near the midpoint of the insulated edge of the triangle is approximately $.15480$.