Engineering physics/ matlab

Diffusion through an Interface

Diffusion-Advection-Reaction:

𝜕𝜕𝑀𝑀 𝜕𝜕𝑡𝑡

= �𝐽𝐽𝑖𝑖𝑖𝑖 − �𝐽𝐽𝑜𝑜𝑜𝑜𝑜𝑜 ± 𝑆𝑆

𝛿𝛿�̇�𝑚 = − 𝜕𝜕 𝑢𝑢𝑖𝑖𝐶𝐶 𝜕𝜕𝑥𝑥𝑖𝑖

+ 𝐷𝐷 𝜕𝜕2𝐶𝐶 𝜕𝜕𝑥𝑥𝑖𝑖

2 𝛿𝛿𝑥𝑥𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿

𝑆𝑆 = ±𝑅𝑅𝛿𝛿𝑥𝑥𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝑀𝑀 = 𝐶𝐶𝛿𝛿𝑥𝑥𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿

𝜕𝜕𝐶𝐶 𝜕𝜕𝑡𝑡

= − 𝜕𝜕 𝑢𝑢𝑖𝑖𝐶𝐶 𝜕𝜕𝑥𝑥𝑖𝑖

+ 𝐷𝐷 𝜕𝜕2𝐶𝐶 𝜕𝜕𝑥𝑥𝑖𝑖

2 ± 𝑅𝑅 Reaction

Advection Diffusion

One-dimensional pure diffusion in an infinite pipe:

𝑎𝑎 – radius of the tube

• A mass of tracer 𝑀𝑀 is injected at the point 𝑥𝑥 = 0 at time 𝑡𝑡 = 0, uniformly across the cross-section area 𝐴𝐴 = 𝜋𝜋𝑎𝑎2

• How does the tracer spreads in time, due to molecular diffusion alone?

𝐶𝐶 𝑥𝑥, 𝑡𝑡 = 𝑀𝑀

𝐴𝐴 4𝜋𝜋𝐷𝐷𝑡𝑡 𝑒𝑒𝑥𝑥𝑒𝑒 −

𝑥𝑥2

4𝐷𝐷𝑡𝑡

𝐶𝐶 𝑥𝑥, 𝑡𝑡 = 𝑀𝑀

𝐴𝐴 4𝜋𝜋𝐷𝐷𝑡𝑡 𝑒𝑒𝑥𝑥𝑒𝑒 −

𝑥𝑥 − 𝑥𝑥0 + 𝑢𝑢𝑡𝑡 2

4𝐷𝐷𝑡𝑡 𝑒𝑒𝑥𝑥𝑒𝑒 ±𝑘𝑘𝑡𝑡

• Solution for Diffusion-Advection-(first-order)Reaction in 1D:

(finite amount of substance)

Step-function initial distribution:

(initial distribution)

• Consider diffusion of infinitesimal mass:

(use solution for instantaneous point source)

• Using superposition for all bits of mass:

−∞ 0

Step-function initial distribution: • Using superposition for all bits of mass:

−∞ 0

Step-function initial distribution:

MATLAB: erf(x)

𝐶𝐶 0, 𝑡𝑡 = 𝐶𝐶0 2

−∞ 0

What if concentration is fixed at x = 0?

𝐶𝐶0 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑡𝑡

(infinite supply of substance)

(valid only for x > 0)

(solution invalid for x < 0)

Diffusion in hypoxic cell culture:

(Am J Pathol 2010, 176:710–720)

Diffusion of O2 into stagnant water volume:

𝐶𝐶0

𝐶𝐶𝑎𝑎𝑖𝑖𝑎𝑎

𝐶𝐶0 < 𝐶𝐶𝑠𝑠𝑎𝑎𝑜𝑜

(infinite supply of O2) (initial condition)

• A semi-infinite volume of water in contact with air (infinite supply of O2), initial concentration of O2: 𝐶𝐶 𝛿𝛿, 0 = 𝐶𝐶0 < 𝐶𝐶𝑠𝑠𝑎𝑎𝑜𝑜

• Dissolution of O2 is very fast, concentration in the surface layer of water quickly becomes saturated: 𝐶𝐶 𝛿𝛿𝛿𝛿, 𝛿𝛿𝑡𝑡 = 𝐶𝐶𝑠𝑠𝑎𝑎𝑜𝑜

• How does the concentration of O2 changes in time?

Diffusion of O2 into stagnant water volume:

𝜕𝜕𝐶𝐶 𝜕𝜕𝑡𝑡

= − 𝜕𝜕 𝑢𝑢𝑖𝑖𝐶𝐶 𝜕𝜕𝑥𝑥𝑖𝑖

+ 𝐷𝐷 𝜕𝜕2𝐶𝐶 𝜕𝜕𝑥𝑥𝑖𝑖

2 ± 𝑅𝑅

0 0

𝜕𝜕𝐶𝐶 𝜕𝜕𝑥𝑥

= 𝜕𝜕𝐶𝐶 𝜕𝜕𝛿𝛿

= 0

(assume no advection or reaction) (semi-infinite -> no change in X and Y)

𝜕𝜕𝐶𝐶(𝛿𝛿, 𝑡𝑡) 𝜕𝜕𝑡𝑡

= 𝐷𝐷 𝜕𝜕2𝐶𝐶(𝛿𝛿, 𝑡𝑡) 𝜕𝜕𝛿𝛿2

(boundary and initial conditions)

𝐶𝐶 𝛿𝛿, 0 = 𝐶𝐶0

𝐶𝐶 −∞, 𝑡𝑡 = 𝐶𝐶0

𝐶𝐶 0, 𝑡𝑡 = 𝐶𝐶𝑠𝑠𝑎𝑎𝑜𝑜 𝐶𝐶0

𝐶𝐶𝑎𝑎𝑖𝑖𝑎𝑎 z

(concentration far away from the interface)

(concentration at the interface)

(initial concentration)

−∞

Diffusion of O2 into stagnant water volume:

𝐶𝐶0

𝐶𝐶𝑎𝑎𝑖𝑖𝑎𝑎 z

−∞

𝐶𝐶𝑠𝑠𝑎𝑎𝑜𝑜

𝐶𝐶 𝛿𝛿, 𝑡𝑡 − 𝐶𝐶0 𝐶𝐶𝑠𝑠𝑎𝑎𝑜𝑜 − 𝐶𝐶0

= 1 − 𝑒𝑒𝑒𝑒𝑒𝑒 −𝛿𝛿 4𝐷𝐷𝑡𝑡

𝛿𝛿 < 0