Training Multiphysics Differential Equations Reaction–Diffusion Systems
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Reaction–Diffusion Systems

30 min Multiphysics Differential Equations

Reaction–Diffusion Systems

Reaction–diffusion equations describe how chemical species spread in space while reacting with each other. They are the foundation of many multiphysics models: flame propagation, tumor growth, pattern formation in morphogenesis, battery electrochemistry, and combustion.

The simplest form for a single species $u(x,t)$ is $u_t = D\,u_{xx} + R(u)$. When $R(u)$ is nonlinear, the system can exhibit traveling waves, bistability, or chaotic patterns. Coupled two-species systems (e.g., activator–inhibitor) produce the Turing patterns that explain zebra stripes and animal coat markings.

We derive the canonical 1-D equations, find traveling-wave solutions for the Fisher–KPP model, and discuss the Damköhler number that balances reaction against diffusion.

Single-Species Reaction–Diffusion

$$\frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2} + R(u)$$

Fisher–KPP: $R(u) = r u(1 - u/K)$ gives logistic reaction.

Damköhler Number

$$\text{Da} = \frac{\text{reaction rate}}{\text{diffusion rate}} = \frac{k_r L^2}{D}$$

$\text{Da} \gg 1$: reaction-limited, mass transfer fast.  $\text{Da} \ll 1$: diffusion-limited.

Example 1 — Fisher wave speed

For $D = 10^{-5}\,\text{m}^2/\text{s}$ and reaction rate $r = 0.1\,\text{s}^{-1}$, find the minimum wave speed.

Fisher's classical result: $c_{\min} = 2\sqrt{r D} = 2\sqrt{0.1 \cdot 10^{-5}} = 2\sqrt{10^{-6}} = 2\times 10^{-3}\,\text{m/s} = 2\,\text{mm/s}$.

Example 2 — Damköhler number

For a catalyst pellet $L = 1\,\text{mm}$, $k_r = 5\,\text{s}^{-1}$, $D = 10^{-9}\,\text{m}^2/\text{s}$, find Da.

$\text{Da} = 5 \cdot (10^{-3})^2 / 10^{-9} = 5 \times 10^{-6}/10^{-9} = 5000$. Diffusion-limited — the reactant cannot reach the pellet interior fast enough.

Example 3 — Steady pellet concentration

For the same pellet, solve $D c'' = k_r c$ on $(0, L)$ with $c(L) = c_s$ and $c'(0) = 0$.

Let $\phi = L\sqrt{k_r/D}$ (Thiele modulus). The solution is $c(x) = c_s \cosh(\phi x/L)/\cosh(\phi)$.

With $\phi = 10^{-3}\sqrt{5\cdot 10^{9}} \approx 70.7$, the interior is essentially depleted: $c(0)/c_s = 1/\cosh(70.7) \approx 0$.

Interactive Demo: Fisher Wave Speed
c_min =2.000mm/s
time to cross 1 m =8.33min

Practice Problems

1. Give the PDE for Fisher–KPP.
2. If $D=2\times10^{-5}$ m²/s and $r=0.5$ /s, find $c_{min}$.
3. State the Thiele modulus.
4. Explain why a large Thiele modulus implies diffusion limitation.
5. Name one biological pattern-formation system modelled by reaction–diffusion.
6. What physical quantity distinguishes reaction-limited from diffusion-limited regimes?
Show Answer Key

1. $u_t = D u_{xx} + r u (1 - u/K)$.

2. $c_{min} = 2\sqrt{0.5 \cdot 2\times 10^{-5}} = 2\sqrt{10^{-5}} \approx 6.32\times 10^{-3}$ m/s = 6.32 mm/s.

3. $\phi = L\sqrt{k_r/D}$.

4. $\phi \gg 1$ ⇒ reactant cannot diffuse into the catalyst before being consumed; internal concentration drops sharply.

5. Turing patterns in animal coats, zebrafish pigmentation, Belousov–Zhabotinsky reaction.

6. Damköhler number Da (or equivalently the Thiele modulus).

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