V²T = a²T a²T + = 0 მx2' მy2 Don't worry: we will not actually be solving this equation as it would require knowledge of partial differential equations Instead, consider the following temperature field that is a proposed solution: T(x, y) = 6y²x - 2x3 Verify that the above temperature field satisfies the heat equation. c. Make a rough sketch of the above temperature field on the following 5x5 rectangular domain: Write down the numerical value for the temperature at each of the 25 (x,y) location above. The location in the bottom left has coordinates (1,1), and the coordinate of the top right is (5,5). • Ans: T(1,1) = 4, T(5,5) = 500, T(1,5) = 148, T(5,1) = -220 Problem 2 (The Laplacian and the Heat Diffusion Equation) One differential operator that we did not get a chance to go over in class is known as the Laplacian (often denoted by V2T), which is a combination of the gradient and the divergence. Even though it shares a similar name, it has absolutely nothing to do with the Laplace transform. V²T = V (VT) The Laplacian shows up very often in subjects like heat transfer and fluid mechanics. It represents diffusion of a quantity over a region of space. You can physically think of diffusion as a passive spreading of an agent in space. If you spray air freshener into the air, then it will slowly spread in all directions. This is diffusion! Answer the following questions: a. Prove that the Laplacian of a scalar field T(x, y, z) is given by the following expression by first taking the gradient of T(x, y, z) then taking the divergence of that: √²T = a²T a²T ²T + + Əx² Əy² Əz² b. One common application of the Laplacian is the heat equation. Specifically, if you assume that your conditions are steady-state and there is no generation/destruction of heat, the heat equation in 2D is given by the following:
V²T = a²T a²T + = 0 მx2' მy2 Don't worry: we will not actually be solving this equation as it would require knowledge of partial differential equations Instead, consider the following temperature field that is a proposed solution: T(x, y) = 6y²x - 2x3 Verify that the above temperature field satisfies the heat equation. c. Make a rough sketch of the above temperature field on the following 5x5 rectangular domain: Write down the numerical value for the temperature at each of the 25 (x,y) location above. The location in the bottom left has coordinates (1,1), and the coordinate of the top right is (5,5). • Ans: T(1,1) = 4, T(5,5) = 500, T(1,5) = 148, T(5,1) = -220 Problem 2 (The Laplacian and the Heat Diffusion Equation) One differential operator that we did not get a chance to go over in class is known as the Laplacian (often denoted by V2T), which is a combination of the gradient and the divergence. Even though it shares a similar name, it has absolutely nothing to do with the Laplace transform. V²T = V (VT) The Laplacian shows up very often in subjects like heat transfer and fluid mechanics. It represents diffusion of a quantity over a region of space. You can physically think of diffusion as a passive spreading of an agent in space. If you spray air freshener into the air, then it will slowly spread in all directions. This is diffusion! Answer the following questions: a. Prove that the Laplacian of a scalar field T(x, y, z) is given by the following expression by first taking the gradient of T(x, y, z) then taking the divergence of that: √²T = a²T a²T ²T + + Əx² Əy² Əz² b. One common application of the Laplacian is the heat equation. Specifically, if you assume that your conditions are steady-state and there is no generation/destruction of heat, the heat equation in 2D is given by the following:
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Transcribed Image Text:V²T
=
a²T a²T
+
= 0
მx2' მy2
Don't worry: we will not actually be solving this equation as it would require
knowledge of partial differential equations Instead, consider the following
temperature field that is a proposed solution:
T(x, y) = 6y²x - 2x3
Verify that the above temperature field satisfies the heat equation.
c. Make a rough sketch of the above temperature field on the following 5x5
rectangular domain:
Write down the numerical value for the temperature at each of the 25 (x,y)
location above. The location in the bottom left has coordinates (1,1), and the
coordinate of the top right is (5,5).
•
Ans: T(1,1) = 4, T(5,5) = 500, T(1,5) = 148, T(5,1) = -220
Transcribed Image Text:Problem 2 (The Laplacian and the Heat Diffusion Equation)
One differential operator that we did not get a chance to go over in class is known as the
Laplacian (often denoted by V2T), which is a combination of the gradient and the
divergence. Even though it shares a similar name, it has absolutely nothing to do with
the Laplace transform.
V²T = V (VT)
The Laplacian shows up very often in subjects like heat transfer and fluid mechanics. It
represents diffusion of a quantity over a region of space. You can physically think of
diffusion as a passive spreading of an agent in space. If you spray air freshener into the
air, then it will slowly spread in all directions. This is diffusion!
Answer the following questions:
a. Prove that the Laplacian of a scalar field T(x, y, z) is given by the following
expression by first taking the gradient of T(x, y, z) then taking the divergence of
that:
√²T =
a²T a²T ²T
+ +
Əx² Əy²
Əz²
b. One common application of the Laplacian is the heat equation. Specifically, if you
assume that your conditions are steady-state and there is no
generation/destruction of heat, the heat equation in 2D is given by the following:
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