Physics/matlab questions

Diffusion+Advection+Reaction (solution)

Pure diffusion in a 1D 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?

Boundary conditions:

Initial condition: 𝛿𝛿(𝑥𝑥) = � 0, 𝑥𝑥 > 0

+∞, 𝑥𝑥 = 0 0, 𝑥𝑥 < 0

Conservation of mass:

Dirac delta function

� −∞

+∞ 𝛿𝛿 𝑥𝑥 𝑑𝑑𝑥𝑥 = 1

Recall dimensional analysis:

• Buckingham π-theorem – Consider a process that can be described by 𝑚𝑚 dimensional

variables (parameters)

– This full set of variables contains 𝑛𝑛 different physical dimensions (e.g. 𝐿𝐿, 𝑇𝑇, 𝑀𝑀)

– Then there are (𝑚𝑚 − 𝑛𝑛) independent non-dimensional groups that can be formed from these variables…

– …and the variables can be related according to 𝜋𝜋1 = 𝑓𝑓(𝜋𝜋2, … , 𝜋𝜋𝑚𝑚−𝑛𝑛)

One-dimensional pure diffusion in an infinite pipe:

Number of parameters: 𝑚𝑚 = 5

Number of dimensions: 𝑛𝑛 = 3

Number of (independent) dimensionless groups: 𝑚𝑚 − 𝑛𝑛 = 2

𝜋𝜋1 = 𝑓𝑓(𝜋𝜋2) 𝐶𝐶

⁄𝑀𝑀 (𝐴𝐴 𝐷𝐷𝑡𝑡) = 𝑓𝑓(

𝑥𝑥 𝐷𝐷𝑡𝑡

)

𝐶𝐶 = ⁄𝑀𝑀 (𝐴𝐴 𝐷𝐷𝑡𝑡) 𝑓𝑓( 𝑥𝑥 𝐷𝐷𝑡𝑡

)

One-dimensional pure diffusion in an infinite pipe:

𝐶𝐶 𝑥𝑥, 𝑡𝑡 = 𝑀𝑀

𝐴𝐴 𝐷𝐷𝑡𝑡 𝑓𝑓(

𝑥𝑥 𝐷𝐷𝑡𝑡

)

𝜂𝜂 = 𝑥𝑥 𝐷𝐷𝑡𝑡

New variable:

𝜕𝜕𝜂𝜂 𝜕𝜕𝑥𝑥

= 1 𝐷𝐷𝑡𝑡

𝜕𝜕𝜂𝜂 𝜕𝜕𝑡𝑡

= 𝑥𝑥 − 1 2

𝐷𝐷𝑡𝑡 − 1 2−1𝐷𝐷 = −

𝜂𝜂 2𝑡𝑡

𝐶𝐶 𝑥𝑥, 𝑡𝑡 = 𝑀𝑀

𝐴𝐴 𝐷𝐷𝑡𝑡 𝑓𝑓(𝜂𝜂)

𝜕𝜕𝐶𝐶 𝜕𝜕𝑡𝑡

= 𝜕𝜕 𝜕𝜕𝑡𝑡

𝑀𝑀 𝐴𝐴 𝐷𝐷𝑡𝑡

𝑓𝑓(𝜂𝜂) = 𝜕𝜕 𝜕𝜕𝑡𝑡

𝑀𝑀 𝐴𝐴 𝐷𝐷𝑡𝑡

𝑓𝑓 𝜂𝜂 + 𝑀𝑀

𝐴𝐴 𝐷𝐷𝑡𝑡 𝜕𝜕𝑓𝑓 𝜕𝜕𝜂𝜂

𝜕𝜕𝜂𝜂 𝜕𝜕𝑡𝑡

= − 𝑀𝑀

2𝐴𝐴𝑡𝑡 𝐷𝐷𝑡𝑡 𝑓𝑓 + 𝜂𝜂

𝜕𝜕𝑓𝑓 𝜕𝜕𝜂𝜂

𝜕𝜕2𝐶𝐶 𝜕𝜕𝑥𝑥2

= 𝑀𝑀

𝐴𝐴𝐷𝐷𝑡𝑡 𝐷𝐷𝑡𝑡 𝜕𝜕2𝑓𝑓 𝜕𝜕𝜂𝜂2

(diffusion equation in 1D)𝐷𝐷 𝜕𝜕2𝐶𝐶 𝜕𝜕𝑥𝑥2

= 𝜕𝜕𝐶𝐶 𝜕𝜕𝑡𝑡

One-dimensional pure diffusion in an infinite pipe:

𝐷𝐷 𝑀𝑀

𝐴𝐴𝐷𝐷𝑡𝑡 𝐷𝐷𝑡𝑡 𝜕𝜕2𝑓𝑓 𝜕𝜕𝜂𝜂2

= − 𝑀𝑀

2𝐴𝐴𝑡𝑡 𝐷𝐷𝑡𝑡 𝑓𝑓 + 𝜂𝜂

𝜕𝜕𝑓𝑓 𝜕𝜕𝜂𝜂

𝜕𝜕2𝑓𝑓 𝜕𝜕𝜂𝜂2

+ 1 2

𝑓𝑓 + 𝜂𝜂 𝜕𝜕𝑓𝑓 𝜕𝜕𝜂𝜂

= 0

𝑑𝑑2𝑓𝑓 𝑑𝑑𝜂𝜂2

+ 1 2

𝑓𝑓 + 𝜂𝜂 𝑑𝑑𝑓𝑓 𝑑𝑑𝜂𝜂

= 𝑑𝑑 𝑑𝑑𝜂𝜂

𝑑𝑑𝑓𝑓 𝑑𝑑𝜂𝜂

+ 1 2 𝑓𝑓𝜂𝜂 = 0 �𝑑𝑑

𝑑𝑑𝑓𝑓 𝑑𝑑𝜂𝜂

+ 1 2 𝑓𝑓𝜂𝜂 = �𝑑𝑑𝜂𝜂

𝑑𝑑𝑓𝑓 𝑑𝑑𝜂𝜂

+ 1 2 𝑓𝑓𝜂𝜂 = 𝐶𝐶0 = 0

1 𝑓𝑓 𝑑𝑑𝑓𝑓 = −

1 2 𝜂𝜂𝑑𝑑𝜂𝜂 𝑓𝑓 = 𝐶𝐶1𝑒𝑒𝑥𝑥𝑒𝑒 −

𝜂𝜂2

4 (it can be shown that 𝐶𝐶0 = 0 satisfies initial and boundary

conditions)

One-dimensional pure diffusion in an infinite pipe:

𝐶𝐶 𝑥𝑥, 𝑡𝑡 = 𝑀𝑀

𝐴𝐴 𝐷𝐷𝑡𝑡 𝑓𝑓(𝜂𝜂)

𝑀𝑀 = � 𝑉𝑉

𝐶𝐶 𝑥𝑥, 𝑡𝑡 𝑑𝑑𝑑𝑑 = � −∞

∞ � 0

𝑎𝑎 𝑀𝑀 𝐴𝐴 𝐷𝐷𝑡𝑡

𝑓𝑓(𝜂𝜂) 2𝜋𝜋𝜋𝜋𝑑𝑑𝜋𝜋𝑑𝑑𝑥𝑥

𝑑𝑑𝑥𝑥 = 𝑑𝑑𝜂𝜂 𝐷𝐷𝑡𝑡

(balance of mass)

1 = � −∞

∞ 𝑓𝑓 𝜂𝜂 𝑑𝑑𝜂𝜂

𝑓𝑓(𝜂𝜂) = 𝐶𝐶1𝑒𝑒𝑥𝑥𝑒𝑒 − 𝜂𝜂2

4

� −∞

∞ 𝐶𝐶1𝑒𝑒𝑥𝑥𝑒𝑒 −

𝜂𝜂2

4 𝑑𝑑𝜂𝜂 = 1 𝐶𝐶1 =

1

∫−∞ ∞ 𝑒𝑒𝑥𝑥𝑒𝑒 −𝜂𝜂

2

4 𝑑𝑑𝜂𝜂 =

1 2 𝜋𝜋

(solution from the integral tables)

One-dimensional pure diffusion in an infinite pipe:

𝐶𝐶1 = 1

2 𝜋𝜋

𝑓𝑓(𝜂𝜂) = 𝐶𝐶1𝑒𝑒𝑥𝑥𝑒𝑒 − 𝜂𝜂2

4

𝐶𝐶 𝑥𝑥, 𝑡𝑡 = 𝑀𝑀

𝐴𝐴 𝐷𝐷𝑡𝑡 𝑓𝑓 𝜂𝜂

𝜂𝜂 = 𝑥𝑥 𝐷𝐷𝑡𝑡

𝐶𝐶 𝑥𝑥, 𝑡𝑡 = 𝑀𝑀

𝐴𝐴 4𝜋𝜋𝐷𝐷𝑡𝑡 𝑒𝑒𝑥𝑥𝑒𝑒 −

𝑥𝑥2

4𝐷𝐷𝑡𝑡

Gaussian distribution, with a standard deviation, 𝜎𝜎2 = 2𝐷𝐷𝑡𝑡

Solution for Diffusion+Advection in 1D:

(translating with u)

Solution for Diffusion+Advection in 1D:

• Stronger advection, 𝑢𝑢 ↑, less time to spread out, narrower cloud for each 𝑡𝑡𝑖𝑖

• Faster diffusion, 𝐷𝐷 ↑, cloud spreads out more between 𝑡𝑡𝑖𝑖, profiles overlap

𝐶𝐶 𝑥𝑥, 𝑡𝑡 = 𝑀𝑀

𝐴𝐴 4𝜋𝜋𝐷𝐷𝑡𝑡 𝑒𝑒𝑥𝑥𝑒𝑒 −

𝑥𝑥2

4𝐷𝐷𝑡𝑡

Advection-Diffusion-Reaction:

𝜕𝜕𝐶𝐶 𝜕𝜕𝑡𝑡

= − 𝜕𝜕 𝑢𝑢𝑖𝑖𝐶𝐶 𝜕𝜕𝑥𝑥𝑖𝑖

+ 𝐷𝐷 𝜕𝜕2𝐶𝐶 𝜕𝜕𝑥𝑥𝑖𝑖

2 ± 𝑅𝑅

𝐶𝐶 𝑥𝑥, 𝑡𝑡 = 𝑀𝑀

𝐴𝐴 4𝜋𝜋𝐷𝐷𝑡𝑡 𝑒𝑒𝑥𝑥𝑒𝑒 −

𝑥𝑥 − 𝑥𝑥0 + 𝑢𝑢𝑡𝑡 2

4𝐷𝐷𝑡𝑡 𝑒𝑒𝑥𝑥𝑒𝑒 ±𝑘𝑘𝑡𝑡

• First-order transformation (reaction), in one dimension:

Reaction

Advection Diffusion

𝑑𝑑𝐶𝐶 𝑑𝑑𝑡𝑡

= ±𝑘𝑘𝐶𝐶

𝐶𝐶 𝑡𝑡 = 0 = 𝐶𝐶0 𝐶𝐶 𝑡𝑡 = 𝐶𝐶0𝑒𝑒±𝑘𝑘𝑘𝑘

Reaction-diffusion for endothelial cell networks:

2000 - Helmlinger - Formation of endothelial cell networks

(across the glass slide)

Reaction-diffusion for endothelial cell networks:

On periphery

In the center

2000 - Helmlinger - Formation of endothelial cell networks

Reaction-diffusion for endothelial cell networks:

2000 - Helmlinger - Formation of endothelial cell networks

Homework assignment:

• Homework #5 – due Wednesday, 10/31, 11:59pm