The user wants a real-time fluid simulation in a single `index.html` file.
Key requirements:
- Full-screen canvas.
- Mouse dragging pushes swirling colored dye.
- Self-contained (CSS in `<style>`, JS in `<script>`).
- No external resources/CDNs.
- High performance (real-time).

I should implement a Stable Fluids simulation based on Jos Stam's method (or a simplified version of it).
The core components of a fluid simulation:
1.  **Advection**: Moving the fluid (and dye) along the velocity field.
2.  **Diffusion**: Spreading out the fluid/dye.
3.  **Projection**: Ensuring the velocity field is divergence-free (making it look like an incompressible fluid).
4.  **External Forces**: Adding velocity and dye based on mouse input.

Since it's for a web app, using a grid-based approach (Eulerian) is common. To make it performant, I might use a relatively small grid and upscale it to the full canvas, or use a more efficient approach.

Plan:
1.  Create a basic HTML structure with a full-screen `<canvas>`.
2.  Implement a `Fluid` class that manages the velocity field (`u`, `v`) and the dye density (`rho`).
3.  Implement the solver steps: `diffuse`, `advect`, `project`.
4.  Implement interaction: tracking mouse position and velocity to inject force and dye.
5.  Implement rendering: mapping the density values to colors and drawing them to the canvas.

Wait, for "swirling colored dye", I can use multiple color channels or a single density with a color mapping. Let's go with a simple RGB approach or a single density with a hue shift based on velocity or position. Actually, a better way for "colored dye" is to have several density arrays (R, G, B) or just one density and vary the color over time or based on movement.

Actually, a simpler and visually appealing way is to use a single density field but map its value to a gradient, or maintain three density fields for RGB.

Let's refine the simulation:
- Grid size: maybe 64x64 or 128x128 for performance on a single thread.
- Time step: fixed.
- Visualization: Linear interpolation to fill the full screen.

Let's write the code.