Question

Find the steady-state temperature distribution in a rectangular plate covering the area \(0

Short answer

The steady-state temperature distribution is \ T(x, y) = \sum_{n=1}^{\infty} \frac{2(1-\cos(n\pi))}{n\pi \sinh(n\pi(2))} \sin(n\pi x) \sinh(n\pi(2-y)).

Step-by-step solution

Define the Problem

We need to find the steady-state temperature distribution in a rectangular plate with given boundary conditions. The plate covers the area defined by \(0<x<1,0<y<2\). The boundary conditions are: \(T=0\) for \(x=0, x=1, y=2\), and \(T=1-x\) for \(y=0\).

Formulate the Laplace Equation

Since we need the steady-state temperature, we use the Laplace equation for temperature distribution: \[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \]

Apply Separation of Variables

Assume a solution of the form \(T(x, y) = X(x)Y(y)\). Substituting into the Laplace equation, we have: \[ X''(x)Y(y) + X(x)Y''(y) = 0 \]

Separate the Variables

Divide both sides by \(X(x)Y(y)\) to separate the variables: \[ \frac{X''(x)}{X(x)} = -\frac{Y''(y)}{Y(y)} = -\lambda \]

Solve the X Equation

We get two ordinary differential equations (ODEs). First, solve for \(X(x)\): \[ X''(x) + \lambda X(x) = 0 \]. The boundary conditions are \(X(0) = X(1) = 0\). This is a standard Sturm-Liouville problem, and it yields solutions of the form \( X_n(x) = \sin(n\pi x) \) where \(\lambda_n = (n\pi)^2 \).

Solve the Y Equation

Next, solve for \(Y(y)\): \[ Y''(y) - (n\pi)^2 Y(y) = 0 \]. The boundary conditions are \(Y(2) = 0\) and an expression for \(Y(0)\). Solving this yields \(Y(y) = A\sinh(n\pi(2-y)) \).

Combine the Solutions

The general solution for \(T(x, y)\) is: \[ T(x, y) = \sum_{n=1}^{\infty} B_n \sin(n\pi x) \sinh(n\pi(2-y)) \]

Apply the Boundary Condition at y=0

Using \(T(x, 0) = 1-x\), we solve for the coefficients \(B_n\). This gives a Fourier sine series for \(1-x\): \[ 1-x = \sum_{n=1}^{\infty} B_n \sin(n\pi x) \sinh(n\pi(2)) \]

Solve for Coefficients

Using orthogonality, integrate to find \(B_n\): \[ B_n = \frac{2}{\sinh(n\pi(2))} \int_0^1 (1-x) \sin(n\pi x) dx \]. Evaluating this integral gives \[ B_n = \frac{2}{n\pi \sinh(n\pi(2))} (1-\cos(n\pi)) \]

Write the Final Solution

Substitute \(B_n\) back into the expression for \(T(x, y)\). The final solution is: \[ T(x, y) = \sum_{n=1}^{\infty} \frac{2(1-\cos(n\pi))}{n\pi \sinh(n\pi(2))} \sin(n\pi x) \sinh(n\pi(2-y)) \]

Key concepts

Laplace equation

For steady-state temperature distribution, we use a fundamental concept called the Laplace equation. The equation is written as:
\[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \]
This partial differential equation (PDE) describes how the temperature varies across a region when there is no change in temperature over time. It's significant in physics and engineering for problems involving heat conduction, electrostatics, and fluid flow.
The Laplace equation in two variables indicates that the sum of the second partial derivatives in each spatial direction is zero. This describes a harmonic function, crucial in describing steady-state problems.

boundary conditions

Boundary conditions specify the temperature values along the edges of the plate. These conditions are essential to find a unique solution to the Laplace equation.
In our example, the boundary conditions are:

• For x=0 and x=1: The temperature is 0, i.e., \( T(0,y) = 0 \) and \( T(1,y) = 0 \).
• For y=2: The temperature is also 0, i.e., \( T(x,2)=0 \).
• For y=0: The temperature varies linearly with x, given by \( T(x,0) = 1-x \).

Boundary conditions turn the general solution of a PDE into a specific solution matching the physical problem.

separation of variables

To solve the Laplace equation, we use a powerful method called separation of variables. This technique assumes that the solution can be written as a product of functions, each depending on only one variable:
\( T(x, y) = X(x)Y(y) \).
Substituting this into the Laplace equation allows us to separate the functions depending on x and y. We obtain:
\[ X''(x)Y(y) + X(x)Y''(y) = 0 \]
By dividing both sides by \( X(x)Y(y) \), we separate the variables into:

\[ \frac{X''(x)}{X(x)} = - \frac{Y''(y)}{Y(y)} = - \lambda \].
This approach transforms a PDE into two ordinary differential equations (ODEs), simplifying the problem significantly.

Fourier series

After separation of variables, we use Fourier series to solve for specific values satisfying boundary conditions.
The general solution for the temperature distribution can be written using a Fourier sine series:
\( T(x, y) = \sum_{n=1}^{\infty} B_n \sin(n\pi x) \sinh(n\pi(2-y)) \).
Fourier series allow us to express complex periodic functions as sums of simpler sinusoidal functions.
Understanding and computing the coefficients \( B_n \) is crucial since they determine the exact form of the solution with given boundary conditions.
This requires evaluating the integrals and using the orthogonality of sine functions.

ordinary differential equations (ODE)

The process of separation of variables in the Laplace equation results in two ordinary differential equations (ODEs).
Solving these ODEs is a standard part of analytical methods in mathematical physics. The equations are:

• \( X''(x) + \lambda X(x) = 0 \) for the x-direction,
• \( Y''(y) - \lambda Y(y) = 0 \) for the y-direction.

The boundary conditions provide specific solutions to these ODEs.
For instance, solutions of the x-direction ODE commonly involve sine functions, while the y-direction may involve hyperbolic sine.
By solving these ODEs and combining the solutions, we describe the temperature distribution within the rectangular plate.