Cartesian Coordinates
A two-dimensional heat equation describes how heat distributes on any surface in two dimensions. We will start in cartesian coordinates using a sheet metal rectangle plate as an example.
The Equation:
.png){kind=small}
This is the thermal diffusivity, this is material specific. It measures the rate of transfer of heat of a material from the hot end to the cold end
Where k is the thermal conductivity, p is the density, and cp is the specific heat capacity.
The curvature of the heat in the plate, this takes the curvature at all the coordinates in the plate.
This is the rate of change of temperature of the plate over time
Just U(x,y,t) would mean the actual temperature of the plate over time
Why the Second Derivative?
The heat equation is based on curvature not the linear component meaning that it must be based on the second derivative, which is curvature of the values.
If we were to visualize a linear temperature profile in theory then this would mean equilibrium, which means that there would be not heat transfer or diffusion. This explains why we must take the second derivative - the curvature - in order to see how heat will transfer until it is in equilibrium.
Initial Conditions
The initial conditions are what the temperature of the plate looks like at t = 0
.png){kind=small}
This will mean that the surface initially has this temperature profile - showing which parts are hot and cold
This gives the plate's initial temperature change - showing which parts are being heated and cooled
Boundary Conditions
The x and y in the boundary condition will give where the boundary is enacted. For an example, if it is a 1x1 plate and the boundary condition has x=1 then the condition is true for the rightmost side of the plate.
When the function is a constant this means that it is being held at a constant temperature. When the function is positive it means that the plate is being heated at that location and when the function is negative it means that the plate is being cooled at this location
When the function is any combination of positive and negative values (often sine or cosine) the plate is being cooled or heated the value of the function along that line.
When the boundary conditions contains either Ut or U the temperature change is dependent on the temperature of the plate itself.
Having Ut is altering with the slope of the temperature, therefore if there is a lot of differences on the plate there will be more heat transfer at the boundary.
​
Having U depends on the actual temperature meaning that the more extreme the temperature values the greater the exchange at the boundary.
​
These transfer into having variable input and output.
​
​
​
This boundary condition means that its temperature gradient is 0 at x=1. As heat flows down its temperature gradient, this means that the rod is insulated at x=1. No heat can leave the rod at x=1.
Polar Coordinates
Now we can use this information and knowledge to understand the 2D heat equation in polar coordinates, using a sheet metal disc as an example.
The Equation:
Same as in Cartesian
Same as in Cartesian
Similar to the previous section. This takes the curvature of the surface along all of its radius and angles which will give the curvature of the full disc
Why 1/r?
Before when using cartesian coordinates all of the grid pieces were the same size. When we take equal increments of angles and equal increments of the radius the grid is not uniform. These 1r term makes it so that the grid scales to be bigger as the radius increases, matching what we see below:
