Question

Suppose we record the temperature profile along a thin, laterally insulated bar at some instant of time, say \(t=t_{0}\), and the smooth curve shown in Figure \(9.3\) is obtained. Use the concavity of the curve and the heat equation to show that $$ \frac{\partial u\left(x_{1}, t_{0}\right)}{\partial t}>0 \text { and } \frac{\partial u\left(x_{2}, t_{0}\right)}{\partial t}<0 \text {. } $$

Short answer

Answer: At point \(x_{1}\), the time derivative of the temperature is positive (\(\frac{\partial u(x_1, t_0)}{\partial t} > 0\)), and at point \(x_{2}\), the time derivative of the temperature is negative (\(\frac{\partial u(x_2, t_0)}{\partial t} < 0\)).

Step-by-step solution

Identify and recall the heat equation

The heat equation is a partial differential equation (PDE) that describes how the temperature changes in a given region over time. In this problem, we focus on a 1D region (bar). The heat equation can be written as: $$ \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}, $$ where \(u(x, t)\) is the temperature at point \(x\) and time \(t\), and \(k\) is the thermal diffusivity constant.

Relate concavity to the second derivative of the temperature

The concavity of a curve can be determined by the second derivative of the curve with respect to \(x\). If the second derivative is positive, the curve is concave up, and if the second derivative is negative, the curve is concave down. Let's consider the given temperature profile at time \(t = t_{0}\) to find the concavity at points \(x_{1}\) and \(x_{2}\).

Determine the concavity at \(x_{1}\) and \(x_{2}\)

By observing the provided temperature curve at \(t = t_{0}\), we can notice that the curve is concave up at \(x = x_{1}\) and concave down at \(x = x_{2}\). This information results in: $$ \frac{\partial^2 u(x_1, t_0)}{\partial x^2} > 0, $$ and $$ \frac{\partial^2 u(x_2, t_0)}{\partial x^2} < 0. $$

Apply the heat equation to find the time derivatives at \(x_{1}\) and \(x_{2}\)

Using the heat equation and the results from Step 3, we can find the time derivatives of the temperature at \(x = x_{1}\) and \(x = x_{2}\) For \(x = x_{1}\): $$ \frac{\partial u(x_1, t_0)}{\partial t} = k \frac{\partial^2 u(x_1, t_0)}{\partial x^2} > 0 $$ since \(\frac{\partial^2 u(x_1, t_0)}{\partial x^2} > 0\) and \(k > 0\). For \(x = x_{2}\): $$ \frac{\partial u(x_2, t_0)}{\partial t} = k \frac{\partial^2 u(x_2, t_0)}{\partial x^2} < 0 $$ since \(\frac{\partial^2 u(x_2, t_0)}{\partial x^2} < 0\) and \(k > 0\). Thus, we have shown that: $$ \frac{\partial u(x_1, t_0)}{\partial t} > 0 \text{ and } \frac{\partial u(x_2, t_0)}{\partial t} < 0. $$

Key concepts

Partial Differential Equations

Fundamental to understanding phenomena like heat transfer, wave propagation, and fluid dynamics are partial differential equations (PDEs). A PDE is a mathematical equation involving partial derivatives of a function which depends on several variables. An example of a PDE is the heat equation, which models how temperature changes within a certain material over time.

The heat equation takes the form \[ \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}, \]where \(u(x, t)\) represents the temperature at a given location \(x\) and time \(t\), and \(k\) represents the thermal diffusivity, a measure of how quickly heat spreads through a material. In essence, the heat equation relates the time rate of change of temperature to the spatial changes in temperature.

In solving a problem with the heat equation, one usually looks for a solution that satisfies certain initial conditions, such as a known temperature distribution at the start, and boundary conditions, like fixed temperatures at the ends of a bar. These conditions guide the uniqueness and behavior of the solution within the context of the physical problem.

Temperature Distribution

Temperature distribution is a specific arrangement of temperature across a medium, such as a metal rod or a room. It's a key concept in solving the heat equation problem, as it represents the initial or boundary condition necessary for the solution process. In a graph representing the temperature of a bar at a particular time (like the one discussed in the exercise), the shape of the curve provides valuable information about the bar's temperature characteristics.

For instance, recording the initial temperature profile of a thin, insulated bar can yield a curve with various concavities. In the context of the heat equation, these concavities relate directly to the heat flow within the material. A concave up point on the curve indicates a spot where the temperature is expected to increase over time, while a concave down point suggests a decreasing temperature. Understanding the temperature distribution helps in predicting how heat will evolve within the medium.

Thermal Diffusivity

Thermal diffusivity is a physical property of a material that quantifies how quickly heat diffuses through it. Represented by the symbol \(k\) in the heat equation, this property is crucial for predicting how rapidly temperature changes within the material. The higher the thermal diffusivity, the faster the heat spreads.

In the heat equation \[ \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}, \]the coefficient \(k\) influences the rate of temperature change over time (\(\frac{\partial u}{\partial t}\)). A proper understanding of thermal diffusivity can, therefore, provide insights into the behavior of a material subjected to thermal processes. This aspect is relevant in various practical applications, such as in designing materials for thermal insulation or in metallurgical processes.