https://file.aiquickdraw.com/mathbot/file/1730928491916s5jokxne.jpg

Here's the plot showing the initial concentration profile \( n(x, 0) \) along the length \( L \) of the test tube. This visualization helps us see how the concentration varies with position at \( t = 0 \), following the given oscillatory function.

### Summary of the Solution Steps
1. **Diffusion Equation**: \(\frac{\partial n}{\partial t} = D \frac{\partial^2 n}{\partial x^2}\).
2. **Time-dependent Concentration \( n(x, t) \)**: Using separation of variables, we derived:
   \[
   n(x, t) = n_0 + n_1 \left( \cos \frac{\pi x}{L} e^{-\left( \frac{\pi}{L} \right)^2 D t} + \frac{1}{9} \cos \frac{3 \pi x}{L} e^{-\left( \frac{3 \pi}{L} \right)^2 D t} + \frac{1}{25} \cos \frac{5 \pi x}{L} e^{-\left( \frac{5 \pi}{L} \right)^2 D t} \right)
   \]
3. **Graph**: The graph for \( n(x, 0) \) illustrates the initial condition with concentration oscillations.

Would you like further details on any part, or have any questions?

Here are five related questions to expand your understanding:
1. How does the diffusion coefficient \( D \) affect the rate at which concentration levels stabilize?
2. How would the concentration profile change if \( n_1 \) had a different value?
3. What would the solution look like for a different initial concentration distribution?
4. How does the addition of higher-frequency terms (like \(\cos(7 \pi x / L)\)) affect the initial concentration profile?
5. How would the solution differ if the diffusion were anisotropic (direction-dependent)?

**Tip**: In diffusion problems, oscillatory initial conditions tend to decay over time, with higher-frequency components decaying faster due to the exponential term in their solution.

A test tube of length L contains a sugar solution. The concentration of sugar at time t=0 is given by the formula n(x, 0) = n0 + n1 (cos(πx/L) + 1/9 * cos(3πx/L) + 1/25 * cos(5πx/L)). Assume that the concentration variations n(x, t) are described by the diffusion equation with constant D. Consider n0, n1, and L as known constants.

1) Write the diffusion equation for n(x, t).
2) Calculate n(x, t) for t > 0.
3) Graphically represent n(x, 0).

Learn to solve the diffusion equation for a sugar solution in a test tube with an oscillatory initial concentration profile. This problem covers key mathematical concepts like partial differential equations, Fourier series, and the diffusion equation solution using separation of variables. Suitable for undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation