6.021/Notes/2006-09-12

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Review

  • Fick's first law: [math]\phi(x,t)=-D\frac{\partial c(x,t)}{\partial x}[/math]
    • Flux is proportional to the concentration gradient
  • Continuity equation: [math]-\frac{\partial\phi(x,t)}{\partial x} = \frac{\partial c(x,t)}{\partial t}[/math]
    • A flux gradient leads to change in concentration over time
  • Diffusion equation: [math]\frac{\partial c(x,t)}{\partial t} = D\frac{\partial^2 c(x,t)}{\partial x^2}[/math]
  • Steady state solution: time-invariant ([math]\frac{\partial c}{\partial t}=0, \frac{\partial\phi}{\partial t}=0[/math])
    • [math]\phi(x,t)=\phi_0[/math]
    • [math]c(x,t) = -\frac{\phi_0}{D}X+\alpha[/math]
  • Equilibrium: Zero flux + time invariant
    • [math]\phi(x,t)=0, C(x,t)=[/math]constant (uniform distribution)
  • Impulse response: Gaussian [math]c(x,t)=\frac{n_0}{\sqrt{4\pi Dt}}e^{-x^2/(4Dt)}[/math]
    • [math]x^2 \approx Dt[/math]