Problem 2 - Advanced Engineering Mathematics

Question

In Problems 1 and 2 , find the steady-state temperature \(u(r, \theta)\) in a circular plate of radius \(c\) if the temperature on the circumference is as given. $$ u(c, \theta)=\left\\{\begin{array}{lr} 1, & 0<\theta<\pi / 2 \\ 0, & \pi / 2<\theta<3 \pi / 2 \\ 1, & 3 \pi / 2<\theta<2 \pi \end{array}\right. $$

Step-by-Step Solution

Answer

Solve the Laplace equation using separation of variables, apply boundary conditions, and utilize Fourier series for the solution.

Step 1: Understanding the Problem

We need to find the steady-state temperature distribution \( u(r, \theta) \) within a circular plate of radius \( c \). The boundary conditions are given on the circumference at \( r = c \), where the temperature is 1 in the intervals \( 0 < \theta < \pi/2 \) and \( 3\pi/2 < \theta < 2\pi \), and 0 otherwise.

Step 2: Utilize the Laplace Equation

The steady-state solution satisfies the Laplace equation \( abla^2 u = 0 \) in polar coordinates. For a circular plate, this becomes \[ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0. \] This equation helps us find the temperature distribution given the boundary conditions.

Step 3: Formulate the Boundary Conditions

The boundary conditions on \( r = c \) are specified piecewise. Convert this to a formal boundary condition expression: \[ u(c, \theta) = \begin{cases} 1, & 0 < \theta < \pi/2 \; \text{or} \; 3\pi/2 < \theta < 2\pi, \ 0, & \pi/2 < \theta < 3\pi/2. \end{cases} \] We will use these to find \( u(r, \theta) \).

Step 4: Apply Separation of Variables

Assume a solution of the form \( u(r, \theta) = R(r)\Theta(\theta) \). Substitute into the Laplace equation and separate variables: \[ r^2 \frac{R''}{R} + r \frac{R'}{R} = -\frac{\Theta''}{\Theta} = \lambda, \] where \( \lambda \) is a separation constant.

Step 5: Solve the Radial Equation

Solve the radial part: \( r^2 R'' + r R' - \lambda R = 0 \). This is a Cauchy-Euler equation with solutions \( R(r) = A r^m + B r^{-m} \) where \( m = \sqrt{\lambda} \). For the steady-state (bounded as \( r \rightarrow 0 \)), \( B \) must be zero.

Step 6: Solve the Angular Equation

Solve the angular equation: \( \Theta'' + \lambda \Theta = 0 \) with solutions \( \Theta(\theta) = C \sin(m\theta) + D \cos(m\theta) \), given the periodicity in \( \theta \).

Step 7: Construct the General Solution

Combine solutions, considering both \( R(r) \) and \( \Theta(\theta) \): \[ u(r, \theta) = \sum_{n=1}^{\infty} \left( A_n r^{n} \sin(n\theta) + B_n r^{n} \cos(n\theta) \right). \] The coefficients \( A_n \) and \( B_n \) will be determined using Fourier series from the boundary conditions.

Step 8: Determine Fourier Coefficients

To satisfy the boundary condition, express \( u(c, \theta) \) as a Fourier series. Given the piecewise nature, perform Fourier series expansion on each piece, determine \( A_n \) and \( B_n \) accordingly.

Step 9: Assemble Solution

Use the determined coefficients to express the steady-state solution \( u(r, \theta) \). The final solution will respect both the inner behavior governed by the differential equation and the outer boundary conditions.

Key Concepts

Laplace Equation

The Laplace Equation is a fundamental mathematical tool when examining steady-state systems and potential distributions like electric or temperature fields. It states that the second derivative, or Laplacian, of a function equals zero, represented mathematically as \(abla^2 u = 0\). In polar coordinates, this becomes:

• \(\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0\)

In the context of our problem, it tells us about the behavior of the temperature distribution across the plate. Since we're working with a steady-state condition, the temperature does not change over time. The equation becomes central due to its ruling function that dictates how the temperature gradients spread out evenly throughout the circular plate regulated by the boundary conditions.

Polar Coordinates

Polar Coordinates provide an alternative to Cartesian coordinates for describing points in a plane, perfect for circular domains. Instead of using x and y, it employs two parameters: radius \(r\) and angle \(\theta\). Here, the position is determined by:

• \(r\) — the distance from the origin
• \(\theta\) — the angle measured from a reference direction

In our problem of determining the steady-state temperature in a circular plate, polar coordinates simplify the mathematics since they align with the circle's natural geometry. Thus, calculations regarding temperature profiles become more intuitive because the Laplace Equation and boundary conditions can be directly applied in this coordinate system.

Boundary Conditions

Boundary Conditions are critical in solving differential equations, providing essential information on how a function behaves at the boundary of its domain. For this problem, the condition is set at the circular plate's edge:

• \(u(c, \theta) = 1\) for \(0 < \theta < \pi/2\)
• \(u(c, \theta) = 0\) for \(\pi/2 < \theta < 3\pi/2\)
• \(u(c, \theta) = 1\) for \(3\pi/2 < \theta < 2\pi\)

These conditions dictate the output of the Laplace equation. They guide how the steady-state temperature will spread across the plate, ensuring it fits these initial constraints when expressed through the entire system. Essentially, they provide the temperature 'rules' our solution must follow at the boundary.

Separation of Variables

Separation of Variables is a powerful technique for solving partial differential equations, especially helpful here with the Laplace Equation. It involves splitting a complex multivariable problem into simpler, single-variable problems. For the stationary heat equation with polar coordinates, we express \(u(r, \theta)\) as the product of two functions \(R(r)\) and \(\Theta(\theta)\). This method transforms the Laplace equation into:

• \(r^2 \frac{R''}{R} + r \frac{R'}{R} = -\frac{\Theta''}{\Theta} = \lambda\)

Where \(\lambda\) is a separation constant. By solving these individual ordinary differential equations, we find solutions for each part, which are then combined to form the complete solution. This method makes it easier to incorporate the boundary conditions and find specific aspects of the temperature distribution.