Question

In solving differential equations the computations can almost always be simplified by the use of dimensionless variables. Show that if the dimensionless variable \(\xi=x / L\) is introduced, the heat conduction equation becomes $$ \frac{\partial^{2} u}{\partial \xi^{2}}=\frac{L^{2}}{\alpha^{2}} \frac{\partial u}{\partial t}, \quad 0<\xi<1, \quad t>0 $$ since \(L^{2} / \alpha^{2}\) has the units of time, it is convenient to use this quantity to define a dimensionless time variable \(\tau=\left(\alpha^{2} / L^{2}\right) t .\) Then show that the heat conduction equation reduces to $$ \frac{\partial^{2} u}{\partial \xi^{2}}=\frac{\partial u}{\partial \tau}, \quad 0<\xi<1, \quad \tau>0 $$

Short answer

Short Answer: Introducing dimensionless variables \(\xi = x / L\) and \(\tau = (\alpha^2 / L^2) t\) to the heat conduction equation allows us to simplify the equation to the form \(\frac{\partial^2 u}{\partial \xi^2} = \frac{\partial u}{\partial \tau}\). This is achieved through the use of the chain rule and substituting expressions for the second-order derivatives in terms of these dimensionless variables.

Step-by-step solution

Introduce the dimensionless variable \(\xi\)

We start by introducing the dimensionless variable \(\xi = x / L\) where \(L\) is a constant with the dimension of length. To do this, we will need to rewrite the heat conduction equation in terms of the derivatives with respect to \(\xi\). Our goal is to express \(x\) derivatives in terms of \(\xi\) derivatives. First, from the definition of \(\xi\) we can write, \(x = L \xi\). Taking the first derivative with respect to both sides, we have: \(\frac{dx}{d\xi}=L\).

Apply the chain rule to the second derivative of \(u\)

Now we apply the chain rule to find the second derivative of \(u\) with respect to \(\xi\). We can write the second partial derivative as follows: $$ \frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x} \left( \frac{\partial u}{\partial x} \right) $$ Applying the chain rule, we get: $$ \frac{\partial}{\partial x} \left( \frac{\partial u}{\partial x} \right) = \frac{\partial}{\partial \xi} \left( \frac{\partial u}{\partial \xi} \frac{d\xi}{dx}\right) \frac{dx}{d\xi} $$ We know \(\frac{dx}{d\xi} = L\), so the expression simplifies to: $$ \frac{\partial^2 u}{\partial x^2} = \frac{1}{L^2} \frac{\partial^2 u}{\partial \xi^2} $$

Substitute the expression for the second derivative back into the heat conduction equation

Now we can substitute this expression for the second derivative back into the heat conduction equation: $$ \frac{1}{L^2} \frac{\partial^2 u}{\partial \xi^2} = \frac{L^2}{\alpha^2} \frac{\partial u}{\partial t} $$

Introduce the dimensionless time variable \(\tau\) and rewrite the equation

Now we introduce the dimensionless time variable \(\tau = (\alpha^2 / L^2) t\), and its corresponding derivative is \(\frac{d\tau}{dt} = \frac{\alpha^2}{L^2}\). We can then substitute this expression for the time derivative in the equation: $$ \frac{\partial^2 u}{\partial \xi^2} = \frac{L^2}{\alpha^2} \frac{\partial u}{\partial(\frac{L^2}{\alpha^2}\tau)} $$

Simplify the equation to get the desired result

Now simplifying the equation, we get: $$ \frac{\partial^2 u}{\partial \xi^2} = \frac{\partial u}{\partial \tau} $$ This is the final form of the heat conduction equation using dimensionless variables \(\xi\) and \(\tau\).

Key concepts

Heat Conduction Equation

The heat conduction equation is a fundamental concept in thermal physics. It describes how heat energy is distributed within a material over time. In its basic form, it is a partial differential equation:

• \( \frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2} \)

Here, \(u\) represents the temperature, and \( \alpha \) is the thermal diffusivity of the material. The equation states that the rate of change of temperature with respect to time is proportional to the second spatial derivative of temperature. This relates to the flow of heat within the material. In simpler terms, it's like how a drop of ink spreads out in water.
When you're using this equation, it is often useful to consider variables that don't have units – called dimensionless variables. They allow us to look at the problem in a simplified way.

Partial Derivatives

Partial derivatives are a way of analyzing how a function changes as each variable is changed, one at a time. In the context of the heat conduction equation, partial derivatives are used to see how the temperature \(u\) changes with respect to either time \(t\) or position \(x\).

• The term \( \frac{\partial u}{\partial t} \) shows how temperature changes over time.
• The term \( \frac{\partial^2 u}{\partial x^2} \) shows the change with respect to position and is a measure of how the temperature is likely to spread or "smooth out" over time.

Partial derivatives are crucial in understanding the behavior of systems that are influenced by more than one variable, like heat distribution. It's like tracking how a path changes when you adjust the curve at certain points, rather than seeing the entire picture.

Chain Rule

The chain rule is a formula for computing the derivative of a composition of two or more functions. In the context of the dimensionless heat conduction problem, the chain rule helps convert derivatives with respect to \(x\) to derivatives with respect to the dimensionless variable \(\xi = x/L\).

Understanding the Application

As part of simplifying the heat conduction equation, you are scaling the spatial dimension. By using the identity \(x = L\xi\), you change the derivative as follows:

• \( \frac{\partial}{\partial x} = \frac{1}{L} \frac{\partial}{\partial \xi} \)
• \( \frac{\partial^2 u}{\partial x^2} \) becomes \( \frac{1}{L^2} \frac{\partial^2 u}{\partial \xi^2} \)

This change uses the chain rule to account for the transformation from a dimensional to a dimensionless form. Think of it as turning ingredients quantities from cups to teaspoons in a recipe to simplify things.

Dimensionless Time Variable

In simplifying complex equations, applying a dimensionless time variable often proves very useful. The dimensionless time variable in this context, \(\tau\), is defined by:

• \( \tau = \left( \frac{\alpha^2}{L^2} \right) t \)

The introduction of \(\tau\) aids in non-dimensionalizing the time element of the heat conduction equation. The choice of \(\alpha^2 / L^2\) is not random; it naturally scales the time to match the spatial scaling with \(L\), yielding a unified equation.

Benefits of a Dimensionless Time Variable

• Leads to a simplified form of equations, making them easier to solve or analyze.
• Reduces the number of actual parameters in the equation, highlighting the underlying physical processes better.
• Allows generalization across different systems, using the same equation for different materials or scales.

Imagine a world where all clocks tick to the same rhythm, regardless of where you are; that's what a dimensionless time variable does for your physics equations.