Question

Find the steady-state temperature distribution \(T(\rho, \varphi, z)\) in a semiinfinite solid cylinder of radius \(a\) if the temperature distribution of the base is \(f(\rho, \varphi)\) and the lateral surface is held at \(T=0\).

Short answer

The steady-state temperature distribution in the semi-infinite solid cylinder is \(T(\rho, \varphi, z) = \sum_{n=-\infty}^{\infty} \int_0^a B_n (r) J_0 (\lambda_n r) e^{in\varphi} e^{-\lambda_n z}dr\), where \(B_n(r)\) are the Fourier-Bessel coefficients, \(J_0\) is the Bessel function of the first kind and \(\lambda_n\) are the roots of the equation \( J_0(\lambda a) = 0\).

Step-by-step solution

Heat Equation and Boundary Conditions

The heat equation in cylindrical coordinates is \(\frac{1}{\rho} \, \frac{\partial}{\partial \rho}(\rho \, \frac{\partial T}{\partial \rho}) + \frac{1}{\rho^2} \, \frac{\partial^2 T}{\partial \varphi^2} + \frac{\partial^2 T}{\partial z^2} = 0\). Apply the boundary conditions: \(T(\rho, \varphi, 0) = f(\rho, \varphi)\) and \(T(a, \varphi, z) = 0\) for \(0 \leq \rho \leq a\), \(0 \leq z < \infty\) and \(0 \leq \varphi < 2 \pi\).

Fourier-Bessel Series Expansion

Expand the given initial temperature distribution \(f(\rho, \varphi)\) in a Fourier-Bessel series. We have \(f(\rho, \varphi) = \sum_{n=-\infty}^{\infty} \int_0^a B_n (r) J_0 (\lambda_n r) \, e^{in\varphi} dr\), where \(B_n(r)\) are the Fourier-Bessel coefficients and \(J_0\) is the Bessel function of the first kind.

Finding the Steady-State Temperature Distribution

Using method of separation of variables, we solve \( \frac{1}{\rho} \, \frac{\partial}{\partial \rho}(\rho \, \frac{\partial T}{\partial \rho}) + \frac{1}{\rho^2} \, \frac{\partial^2 T}{\partial \varphi^2} + \frac{\partial^2 T}{\partial z^2} = 0\) to find \(T(\rho, \varphi, z) = \sum_{n=-\infty}^{\infty} \int_0^a B_n (r) J_0 (\lambda_n r) e^{in\varphi} e^{-\lambda_n z}dr\). Where \(\lambda_n\) are the roots of the equation \( J_0(\lambda a) = 0\).

Key concepts

Heat Equation in Cylindrical Coordinates

The heat equation is a fundamental partial differential equation (PDE) that describes how heat diffuses through a given medium. In cylindrical coordinates, this equation accounts for the symmetry and shape inherent to cylinders, making it particularly relevant for problems involving cylindrical geometries.

Starting with the general heat conduction equation, which in stable, or steady-state conditions without a heat source simplifies to the Laplace equation, it is adapted to cylindrical coordinates due to the geometry of the problem. Specifically, the equation becomes \[\begin{equation}\frac{1}{\rho} \, \frac{\partial}{\partial \rho}(\rho \, \frac{\partial T}{\partial \rho}) + \frac{1}{\rho^2} \, \frac{\partial^2 T}{\partial \varphi^2} + \frac{\partial^2 T}{\partial z^2} = 0\end{equation}\]
In this form, the variables \( \rho \), \( \varphi \), and \( z \) represent the radial distance from the center axis of the cylinder, the angular coordinate around the axis, and the axial distance along the axis, respectively. The term \( T \) is the temperature, which is the function we aim to solve for. The heat equation in cylindrical coordinates provides a way to study the temperature distribution within cylindrical bodies, being a critical tool for engineers and scientists dealing with heat transfer in cylindrical structures.

Fourier-Bessel Series Expansion

When dealing with circular or cylindrical domains, the Fourier-Bessel series provides a powerful method to represent functions that adhere to the geometry of the problem. The development of a Fourier-Bessel series for the initial temperature distribution is a crucial step in solving the heat equation for cylindrical coordinates.

The series involves Bessel functions, which are solutions to Bessel's differential equation and occur in many physical applications. In the context of our problem, we expand the temperature function \( f(\rho, \varphi) \) as \[\begin{equation}\sum_{n=-\infty}^{\infty} \int_0^a B_n (r) J_0 (\lambda_n r) \, e^{in\varphi} dr\end{equation}\]
where \( B_n(r) \) are the Fourier-Bessel coefficients, which capture the contribution of each radial mode to the overall temperature distribution. The \( J_0 \) represents the Bessel function of the first kind of order zero, which naturally emerges from problems with cylindrical symmetry. The variable \( \lambda_n \) indicates the roots of the Bessel function, which are essential to solve the boundary value problem. This expansion is essential, as it allows for the decomposition of the temperature distribution into terms that can be solved analytically.

Method of Separation of Variables

The method of separation of variables is an essential technique for solving partial differential equations, especially when dealing with problems that exhibit symmetry across dimensions. It allows the complex PDE to be broken down into simpler, solvable ordinary differential equations (ODEs).

In our exercise, we separated variables assuming the temperature function \( T(\rho, \varphi, z) \) can be represented by a product of functions, each depending only on a single coordinate. This technique streamlined the process for solving the heat equation tailored to the cylindrical domain. Through separation of variables, the original PDE transforms into a set of ODEs with respect to \( \rho \), \( \varphi \), and \( z \), which can be handled individually. These ODEs are solved using boundary conditions, and the solutions are then recombined to provide the full temperature distribution formula:\[\begin{equation}T(\rho, \varphi, z) = \sum_{n=-\infty}^{\infty} \int_0^a B_n (r) J_0 (\lambda_n r) e^{in\varphi} e^{-\lambda_n z}dr\end{equation}\]
This result exemplifies the elegance and power of the method of separation of variables when applied to problems with appropriate symmetries and boundaries.

Bessel Functions

Bessel functions play a central role in solving PDEs within circular or cylindrical domains, such as the steady-state heat distribution in our cylinder problem. Named after the mathematician Friedrich Bessel, these functions are a family of solutions to Bessel's differential equation and are ubiquitously applicable in problems involving radial symmetry.

Specifically, in our context, we encounter the Bessel function of the first kind, \( J_0 \), which arises when solving the radial part of the PDE after applying the separation of variables. Bessel functions exhibit oscillatory behavior similar to sine and cosine functions, but with respect to a radial coordinate. The zeros of Bessel functions, denoted \( \lambda_n \), are of particular interest as they are used to satisfy the boundary conditions of the problem. To solve the heat equation for the cylinder, these zeros determine the eigenvalues that are critical in constructing the solution as a Fourier-Bessel series. Understanding Bessel functions, their properties, and their applications is essential for those who delve into wave phenomena, electromagnetic theory, and, as illustrated, heat transfer in cylindrical geometries.