ScalarDiffusionAnalytic

Python scripts to compute the analytical solutions of the 2-phase diffusion problem in 1D, 2D, and 3D.

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Table of Contents

• Introduction
• Features
• Theoretical Background
  • 1D Diffusion
  • 2D Diffusion
  • 3D Diffusion
• Installation
• Usage
  • Running the Scripts
  • Example Outputs
• Dependencies
• Contributing
• License
• Acknowledgments

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Introduction

ScalarDiffusionAnalytic is a collection of Python scripts that compute analytical solutions for two-phase diffusion problems in one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) geometries. These scripts are designed to visualize the diffusion processes between two different media with varying diffusivities and interface conditions.

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Features

• Analytical Solutions: Provides exact solutions derived using methods such as Laplace transforms, separation of variables, and integral transforms.
• Dimensional Flexibility: Includes scripts for 1D, 2D (cylindrical coordinates), and 3D (spherical coordinates) diffusion problems.
• Customizable Parameters: Allows users to input physical parameters such as diffusivities, interface conditions, initial concentrations, and geometrical dimensions.
• Visualization: Generates plots of concentration profiles over time and space to aid in the visualization of diffusion processes.
• Henry's Law Interface Conditions: Incorporates different interface conditions, including Henry's Law with adjustable constants.

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Theoretical Background

1D Diffusion

Problem Statement:

We consider a one-dimensional diffusion problem where two semi-infinite media meet at an interface located at $$x = x0$$. The diffusivities in the two media are $$D_1$$ and $$D_2$$, respectively.

Governing Equations:

• For x < x0 : $$\frac{\partial c_1}{\partial t} = D_1 \frac{\partial^2 c_1}{\partial x^2}$$
• For ( x > x0 ): $$\frac{\partial c_2}{\partial t} = D_2 \frac{\partial^2 c_2}{\partial x^2}$$

Interface Conditions:

• Concentration Discontinuity (Henry's Law): $$c_1(x_0, t) = \alpha c_2(x_0, t)$$
• Flux Continuity: $$D_g\frac{\partial c_1}{\partial r}_{x_0}=D_l\frac{\partial c_2}{\partial r}_{x_0}$$

Solution Approach:

• Apply Laplace transforms to eliminate time dependence.
• Solve the resulting ordinary differential equations (ODEs).
• Apply boundary and interface conditions to determine constants.
• Perform inverse Laplace transform to obtain the time-dependent solution.

2D Diffusion

Problem Statement:

We analyze diffusion from a circular bubble (in 2D) of radius $$R_0$$ into the surrounding medium.

Governing Equations:

• Inside the bubble (gas phase): $$\frac{\partial c_g}{\partial t} = D_g \left( \frac{\partial^2 c_g}{\partial r^2} + \frac{1}{r} \frac{\partial c_g}{\partial r} \right)$$
• Outside the bubble (liquid phase): $$\frac{\partial c_l}{\partial t} = D_l \left( \frac{\partial^2 c_l}{\partial r^2} + \frac{1}{r} \frac{\partial c_l}{\partial r} \right)$$

Interface Conditions:

• Henry's Law: $$c_g(R_0, t) = \alpha c_l(R_0, t)$$
• Flux Continuity: $$D_g\frac{\partial c_g}{\partial r}_{R_0}=D_l\frac{\partial c_l}{\partial r}_{R_0}$$

Solution Approach:

• Use Laplace transforms and Bessel functions to solve the radial diffusion equations.
• Apply boundary conditions to find constants.
• Invert transforms to obtain solutions in the time domain.

3D Diffusion

Problem Statement:

We consider diffusion from a spherical bubble (in 3D) of radius $$R_0$$ into the surrounding liquid.

Governing Equations:

• Inside the bubble: $$\frac{\partial c_g}{\partial t} = D_g \left( \frac{\partial^2 c_g}{\partial r^2} + \frac{2}{r} \frac{\partial c_g}{\partial r} \right)$$
• Outside the bubble: $$\frac{\partial c_l}{\partial t} = D_l \left( \frac{\partial^2 c_l}{\partial r^2} + \frac{2}{r} \frac{\partial c_l}{\partial r} \right)$$

Interface Conditions:

• Henry's Law: $$c_g(R_0, t) = \alpha c_l(R_0, t)$$
• Flux Continuity: $$D_g\frac{\partial c_g}{\partial r}_{R_0}=D_l\frac{\partial c_l}{\partial r}_{R_0}$$

Solution Approach:

• Similar to the 2D case but involves spherical Bessel functions.
• The solutions are derived using separation of variables and series expansions.

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Installation

1. Clone the Repository:

   git clone /~https://github.com/yourusername/ScalarDiffusionAnalytic.git

2. Navigate to the Directory:

   cd ScalarDiffusionAnalytic

3. Install Dependencies:

   Ensure you have Python 3 installed along with the necessary libraries.

   pip install numpy scipy matplotlib

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Usage

Running the Scripts

Each dimension has its own script:

• 1D Diffusion: diffusion_1d.py
• 2D Diffusion: diffusion_2d.py
• 3D Diffusion: diffusion_3d.py

To run a script, use the following command:

python diffusion_1d.py

Configuring Parameters

At the beginning of each script, you can set the physical and numerical parameters:

# Physical parameters
D1 = 1.0     # Diffusivity in medium 1
D2 = 0.5     # Diffusivity in medium 2
alpha = 0.8  # Henry's law constant
c0 = 1.0     # Initial concentration

# Simulation parameters
x0 = 0.0     # Interface position
t_values = [0.1, 0.5, 1.0, 2.0, 5.0]  # Time points
x_values = np.linspace(-5, 5, 200)    # Spatial domain

Adjust these parameters according to your problem.

Example Outputs

After running a script, it will display plots of concentration profiles:

• Concentration vs. Position: Shows how concentration varies in space at different times.
• Concentration vs. Time: Shows how concentration at a fixed position evolves over time.

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Dependencies

The scripts require the following Python libraries:

• NumPy: For numerical computations.
• SciPy: For special functions and integration.
• Matplotlib: For plotting results.

Install them using:

pip install numpy scipy matplotlib

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Contributing

Contributions are welcome! If you have suggestions or improvements, please submit a pull request or open an issue.

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License

This project is licensed under the MIT License - see the LICENSE file for details.

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Acknowledgments

• Scientific Computing Resources: The scripts utilize computational methods and special functions available in scientific libraries.
• Mathematical References:
  • Crank, J. The Mathematics of Diffusion. Oxford University Press.
  • Palas Kumar Farsoiya, Stéphane Popinet, and Luc Deike. Bubble-mediated transfer of dilute gas in turbulence. Journal of Fluid Mechanics, 920(A34), June 2021.