Why you don’t need turbulent flow

Molecular diffusion helps homogenise the distribution of molecules in laminar flow

This post is about the 1999 paper Monitoring Reaction Kinetics in Solution by Continuous-Flow Methods: The Effects of Convection and Molecular Diffusion under Laminar Flow Conditions by Lars Konermann (Konermann 1999).

Here this refers to fluids in narrow tubes (75 μm tube diameter) hence microfluidics. A tube is full of fluid, so flow is under pressure.

Why I’m reading it?

If you’ve ever play Pooh sticks, you’ll know that two sticks dropped at the same time move at different speeds under a bridge. In the project I have just joined, we want to “keep the sticks together”. Or in our case the proteins, such that for any group of proteins at any time, they have on average had the same experience.

What’s problem?

If you flow slowly (laminar flow) the (sticks at the) centre of the flow moves fastest as there is less friction than close to the walls of the tube, creating a parabolic velocity profile (flow speed of sticks or protein in the direction of travel).

This means that under laminar flow molecules have a different “age” in the sense that the disperse and after time some molecules will have travelled further (centre) than others (near the wall.)

Historically it was believed that you needed turbulent flow to homogenise the mixing of molecules so they all move (approximately) together. This uses larger diameter tube with fast flows and therefore required lots of sample, which can be expensive or difficult to make.

What I learnt from the paper

What Lars Konermann shows is that under laminar flow over a long enough distance homogenisation of the occurs due to diffusion of molecules.

Diffusion means molecules move perpendicular to the direction of flow due to random Brownian motion and in doing so they sample different velocities, therefore on average they experience the same “age” . The rate of diffusion is independent of the flow rate.

Obviously, read the paper for the full story, but I’ve put some of my Claude chat output at the bottom (Section 5).

Visualising diffusion and laminar flow

I got Claude to read the paper and had a chat with it and asked it to create some animations to demonstrate these ideas.

• The first animation visualises diffusion: as a molecule moves closer to the centre would speed up if it was coming towards you, and slow down if it moves to wards the edge.

• The second animation visualises laminar flow with or without diffusion to illustrate how the flow parabola is affected by diffusion. Mixing here means how well mixed are the molecules across different velocity regions in the tube. This mixing is what reduces the dispersive effects of the parabolic flow profile.
  
  

Claude Chat Output

Experiments and simulations

The author compared simulations with experimental conditions of:

• 75 μm tube diameter
• Flow rate of 30 μL/min
• Average flow velocity of 0.113 m/s

Summary of key findings

Here are the key points from the paper:

1. Under laminar flow, without diffusion:

• The velocity profile is parabolic (from equation 2 in the paper)
• Maximum velocity at center of tube
• Zero velocity at walls
• This would cause significant dispersion/blurring of the time axis

2. The role of diffusion:

• Molecules undergo random 2D motion perpendicular to the flow direction (radial diffusion)
• This causes molecules to continuously change their radial positions
• As molecules change position, they experience different flow velocities
• This averaging effect helps counteract the dispersion caused by the parabolic velocity profile

3. Key equations related to diffusion:

• Mean square displacement: ⟨s²⟩ = 4Dt (equation 14)
• Root-mean-square deviation: √⟨s²⟩ = 2√Dt
• For significant diffusion effects: τ > R²/(36D) (equation 16)

4. Practical implications:

• At short distances/times: diffusion effects are minimal, dispersion dominates
• At longer distances/times: diffusion helps average out velocity differences
• Smaller tube diameters increase the importance of diffusion effects
• For D = 1×10⁻¹⁰ m²/s in a 10 μm tube, significant diffusion effects occur around τ = 30 ms

The paper emphasizes that this interplay between laminar flow and molecular diffusion is what makes continuous-flow experiments feasible under laminar conditions, contrary to the previous belief that turbulent flow was essential.

References

Konermann, Lars. 1999. “Monitoring Reaction Kinetics in Solution by Continuous-Flow Methods: The Effects of Convection and Molecular Diffusion Under Laminar Flow Conditions.” The Journal of Physical Chemistry A 103 (36): 7210–16. https://doi.org/10.1021/jp9908350.