Problem 81 - Calculus Single Variable

Question

Let \(p_{0}, \tau_{0}, \mu,\) and \(\Delta\) be fixed constants. The set \(\mathcal{H}\) of points \((p, \tau)\) in the first quadrant of the \(p \tau\) -plane that satisfy $$ \left(\tau-\mu^{2} \tau_{0}\right) p-\left(\tau_{0}-\mu^{2} \tau\right) p_{0}+2 \mu^{2} \Delta=0 $$ is called the Hugoniot curve. It arises in the study of gas dynamics produced by combustion. a. Find \(d \tau / d p\). b. Let \(Q=\left(p_{1}, \tau_{1}\right)\) be a point on the Hugoniot curve \(\mathcal{H}\). What is the equation of the line that is tangent to \(\mathcal{H}\) at \(Q ?\) c. Assume that \(\Delta<0 .\) (This is the mathematical condition for an exothermic reaction.) Show that the point \(\left(p_{0}, \tau_{0}\right)\) lies below the Hugoniot curve. d. Continue to suppose that \(\Delta<0 .\) Solve for those points \(Q\) on the Hugoniot curve \(\mathcal{H}\) such that the line tangent to \(\mathcal{H}\) at \(Q\) passes through the point \(P=\left(p_{0}, \tau_{0}\right) .\) The points that you find are known as the Chapman-Jouguet points. The tangent lines that you find are known as the Rayleigh lines. e. Illustrate the Hugoniot curve and its Rayleigh lines for the specifications \(p_{0}=2.1, \quad \tau_{0}=1.7, \quad \mu=1 / 2,\) and \(\Delta=-2 .\) Label the Chapman-Jouguet points.

Step-by-Step Solution

Answer

Find the derivative of the Hugoniot curve for slope, then use it to get the tangent line. Verify points below the curve via given signs of parameters. Chapman-Jouguet points lead to tangent lines passing through a specific point.

Step 1: Differentiate the Hugoniot curve

To find \(d\tau/dp\), differentiate the equation of the Hugoniot curve implicitly:\[(\tau - \mu^2 \tau_0) \cdot \frac{d}{dp}[p] + p \cdot \frac{d}{dp}[\tau - \mu^2 \tau_0] - (\tau_0 - \mu^2 \tau)p_0 \cdot \frac{d}{dp}[p] - (\tau_0 - \mu^2 \tau_0)p_0 \cdot \frac{d}{dp}[\tau] = 0\]This simplifies to:\[(\tau - \mu^2 \tau_0) + p \cdot \frac{d\tau}{dp} - \mu^2 p_0 \cdot \frac{d\tau}{dp} = 0\]Solve for \(d\tau/dp\):\[\frac{d\tau}{dp} = \frac{\mu^2 (p_0 - p)}{\tau - \mu^2 \tau_0 + \mu^2 p_0}\]

Step 2: Tangent line equation

The tangent line at \(Q = (p_1, \tau_1)\) has the slope \(d\tau/dp\). Therefore, the equation of the tangent line at \(Q\) is given by the point-slope formula:\[\tau - \tau_1 = \frac{\mu^2 (p_0 - p)}{\tau - \mu^2 \tau_0 + \mu^2 p_0} (p - p_1)\]Expand this to find the full equation. Replace \(\frac{d\tau}{dp}\) with the expression obtained in Step 1.

Step 3: Show \((p_0, \tau_0)\) is below the curve

Substitute \((p_0, \tau_0)\) into the equation of the Hugoniot curve:\[(\tau_0 - \mu^2 \tau_0) p_0 - (\tau_0 - \mu^2 \tau_0) p_0 + 2\mu^2 \Delta = 0\]This simplifies to:\[2\mu^2 \Delta = 0\]Since \(\Delta < 0\), this inequality implies that the left side is negative, confirming that \((p_0, \tau_0)\) is below the Hugoniot curve.

Step 4: Find Chapman-Jouguet points

Find points \(Q = (p_1, \tau_1)\) such that the tangent line passes through \((p_0,\tau_0)\). Set up the equation for the line passing through \((p_0,\tau_0)\) with the slope from Step 1:\[\tau_0 - \tau_1 = \frac{\mu^2 (p_0 - p_1)}{\tau_1 - \mu^2 \tau_0 + \mu^2 p_0} (p_0 - p_1)\]Solve this equation for \(p_1\) and \(\tau_1\) to get the Chapman-Jouguet points.

Step 5: Illustrate the Hugoniot curve

Using the given values \(p_0 = 2.1\), \(\tau_0 = 1.7\), \(\mu = 1/2\), and \(\Delta = -2\), plot the Hugoniot curve by solving:\[(\tau - \frac{1}{4} \cdot 1.7) p - (1.7 - \frac{1}{4} \tau) \cdot 2.1 + 2 \cdot \frac{1}{4} \cdot (-2) = 0\]Use the slope equation from Step 1 to plot the Rayleigh lines at Chapman-Jouguet points. Label these points on the graph.

Key Concepts

Hugoniot Curve

The Hugoniot curve is a fascinating concept in the study of gas dynamics. It represents a set of points in the pressure-volume plane, \(p\tau\) plane, which satisfy a particular equation derived from the principles of fluid mechanics. This curve arises specifically when analyzing phenomena like shock waves and combustion processes in gases. It is expressed through the equation:
\[\left(\tau-\mu^{2} \tau_{0}\right) p-\left(\tau_{0}-\mu^{2} \tau\right) p_{0}+2 \mu^{2} \Delta=0\]
In this equation, \(p_0\), \(\tau_0\), and \(\Delta\) are constants and reflect initial conditions or specific physical parameters such as initial pressure, volume, and energy level involved in the reaction.

• Initial pressure (\(p_0\)): The starting pressure for the gas dynamics process.
• Initial volume (\(\tau_0\)): The beginning volume of the gas mixture.
• Energy parameter (\(\Delta\)): This value is often related to the energy absorbed or released by the reaction.

The Hugoniot curve thus serves as a tool for studying wave propagation in gas dynamics, providing rich insights into the nature of interactions in reactive gases.

Gas Dynamics

Gas dynamics explores how gases behave under the influence of forces, resulting in motion and changes in state. It plays a critical role in understanding how shock waves and other high-velocity phenomena influence gases.
When studying gas dynamics, several principles come into play, such as:

• Conservation of Mass: The mass of a closed system must remain constant over time.
• Conservation of Momentum: This principle asserts that the momentum of a system is conserved unless acted upon by an external force.
• Energy Conservation: Energy within a system remains constant, barring the influx or loss outside the system.

Gas dynamics helps us understand the pressure, density, and velocity changes within a gas, especially in responses to shocks and expansions caused by processes like explosions or compressions. The Hugoniot curve relates directly to these principles by providing a visual and mathematical representation of such dynamics over specific conditions.

Implicit Differentiation

Implicit differentiation is a powerful mathematical tool often used to find derivatives involving functions that are not easily separable into explicit forms. In the context of the Hugoniot curve, implicit differentiation helps determine the rate of change between pressure and specific volume along the curve.
To differentiate the Hugoniot curve implicitly, we treat \(p\) and \(\tau\) as dependent variables without needing to solve one explicitly in terms of the other. Here's a simple explanation of the process:

• Identify the equation that relates the variables \(p\) and \(\tau\).
• Apply the chain rule to differentiate both sides with respect to \(p\).
• Isolate \(\frac{d\tau}{dp}\) to find the derivative.

For the Hugoniot curve, the implicit differentiation reveals the slope of the tangent line at any given point on the curve:
\[\frac{d\tau}{dp} = \frac{\mu^2 (p_0 - p)}{\tau - \mu^2 \tau_0 + \mu^2 p_0}\]
This expression is essential for solving many problems related to this concept, including calculating tangents and understanding the dynamics between pressure and volume.

Chapman-Jouguet Points

The Chapman-Jouguet (CJ) points possess crucial importance in analyzing detonation waves in reactive gas mixtures. These points are situated on the Hugoniot curve where the tangent lines, known as Rayleigh lines, intersect at a specific angle with particular significance.
The CJ points represent conditions where the speed of the detonation wave is synchronized with the speed of sound in the downstream gas, reassoning that if the detonation wave travels slower or faster, it will not be stable at these conditions.
To find the CJ points on the Hugoniot curve, the line tangent to the curve must pass through the point \(P = (p_0, \tau_0)\). This creates a condition represented by the slope equation obtained during implicit differentiation:
\[\tau_0 - \tau_1 = \frac{\mu^2 (p_0 - p_1)}{\tau_1 - \mu^2 \tau_0 + \mu^2 p_0} (p_0 - p_1)\]
The CJ points provide precise foundational understanding crucial for predicting how pressure and volumetric conditions react when the detonation wave reaches or surpasses a critical threshold, offering invaluable insights into the behavior of explosive reactions.