Find the steady-state temperature u(r,θ) in a semicirular plate of radius r=2 if: u(2,θ) = {1, 0<θ<π/2 and 0, π/2<θ<π} and the edges θ = 0 and 0 = π are insuated.
Find the steady-state temperature u(r,θ) in a semicirular plate of radius r=2 if: u(2,θ) = {1, 0<θ<π/2 and 0, π/2<θ<π} and the edges θ = 0 and 0 = π are insuated.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Find the steady-state temperature u(r,θ) in a semicirular plate of radius r=2 if:
u(2,θ) = {1, 0<θ<π/2 and 0, π/2<θ<π} and the edges θ = 0 and 0 = π are insuated.
![**Problem Statement:**
Find the steady-state temperature \( u(r, \theta) \) in a semicircular plate of radius \( r = 2 \).
**Boundary Conditions:**
\[
u(2, \theta) =
\begin{cases}
1 & \text{for } 0 < \theta < \pi/2 \\
0 & \text{for } \pi/2 < \theta < \pi
\end{cases}
\]
Additionally, the edges \( \theta = 0 \) and \( \theta = \pi \) are insulated.
---
**Explanation:**
The problem involves determining the temperature distribution, \( u(r, \theta) \), over a semicircular plate. The boundary conditions specify that along the curved edge of the semicircular plate (at \( r = 2 \)), the temperature is set to 1 in the first quadrant (\( 0 < \theta < \pi/2 \)) and 0 in the second quadrant (\( \pi/2 < \theta < \pi \)). The straight edges of the semicircle, corresponding to \( \theta = 0 \) and \( \theta = \pi \), are insulated, meaning no heat flows across these boundaries.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6987128-ed27-4e9c-9c0e-830e4852a138%2Ffa1750ed-9474-4ea2-9b38-e79a597273f5%2Fita1fz9_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the steady-state temperature \( u(r, \theta) \) in a semicircular plate of radius \( r = 2 \).
**Boundary Conditions:**
\[
u(2, \theta) =
\begin{cases}
1 & \text{for } 0 < \theta < \pi/2 \\
0 & \text{for } \pi/2 < \theta < \pi
\end{cases}
\]
Additionally, the edges \( \theta = 0 \) and \( \theta = \pi \) are insulated.
---
**Explanation:**
The problem involves determining the temperature distribution, \( u(r, \theta) \), over a semicircular plate. The boundary conditions specify that along the curved edge of the semicircular plate (at \( r = 2 \)), the temperature is set to 1 in the first quadrant (\( 0 < \theta < \pi/2 \)) and 0 in the second quadrant (\( \pi/2 < \theta < \pi \)). The straight edges of the semicircle, corresponding to \( \theta = 0 \) and \( \theta = \pi \), are insulated, meaning no heat flows across these boundaries.
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