If you cannot get any food, you’ll starve and you don’t want that! This is not only a weekly motivation for us to go grocery shopping, but a central principle of biology: Animals, plants, bacteria, and in fact every cell in our body relies on a steady supply of nutrients. Yet, different from bacteria, cells in our tissue cannot just start roaming around to find new food sources. Instead, all our tissue is interwoven by a highly adaptive network that transports nutrients by fluid flow to all parts of our body: our blood vasculature. One of my research interests is to understand how the physics of fluid flow shapes our life. What are the principles that control the spread of solutes through complex reticulate media? We believe this demands the close interconnection between theory and experiment as well as the study of biological systems in vivo and inanimate matter to challenge our hypotheses in both vascular systems and porous media. For us, porous media are a paradigmatic model system to explore transport dynamics, as porous media are experimentally accessible and designable, observable with a brightfield microscope, and their transport dynamics are analytically tractable.
This project here started as a students’ project. Previously, we had already found that flow fields show fundamentally different statistics if obstacles in the sample porous media are either randomly distributed or arranged in a perfect lattice structure. By tuning the “disorder” of obstacles we can morph from one flow field to the other. This should have effects on the transport as well, right? So my former colleague at the Max Planck Institute for Dynamics and Self-Organization, Lucas Goehring, and his student Thomas Darwent set out to study this effect experimentally, while I set my student Felix Meigel to explore the system numerically. Felix then analyzed how the transport dynamics varied as we gradually increased the disorder in the porous media. Indeed, the dynamics changed, so things ran as we expected until Felix presented to me how the variance of a dispersive front changes as a function of disorder in one of our regular meet-ups. As I expected, the variance increased from the perfect lattice configuration to media with high disorder, but it did so in a non-monotonic fashion!? Felix’s simulations showed that minimal variance in front width was achieved not in the case of perfect obstacle arrangement, but at intermediate disorder values. I was puzzled because this non-monotonicity was not predicted by the current theories that linked flow rate statistics in the pore space to macroscopic transport properties. We knew that as we increase the disorder, the variance of pore-based flow rates increases monotonically. Based on these flow statistics, the seminal work of Saffmann [1] predicts a monotonic increase in dispersion – contrary to our observations. So we thought of two options: Either there is a mistake in the simulations or we missed something in our theories that we need to explain. So Felix set off to carefully check his simulations again – no bug was found. Instead, Felix showed that the effect could be reproduced in a minimal network of an H-structure: Two parallel pores with an inflow and outflow each, connected by an orthogonal pore in the middle. Measuring the dispersion variance, we again found a non-monotonicity as we increased disorder in the H-structure. Now, we decided to further investigate this non-monotonicity in experiments and explain it analytically. What did we miss? Here, the H-structure hinted to us the direction to go, as we could fully track the transport in this simple structure analytically. At a junction between pores, neither fluid nor flow can be created or destroyed. Hence, solute fluxes are locally coupled! And indeed, as we take the local correlations between pores into account, we can intuitively understand the non-monotonicity we observe. Moreover, we can now precisely predict in which parameter regimes we expect this non-monotonicity. So we discussed with Lucas and Thomas, whether we could also explore these parameter regimes experimentally. We were on a good road. But then Felix moved on to do his Ph.D. at the Max Planck Institute for the Physics of Complex Systems in Dresden and I moved with my lab to Munich. We already had the major insights in this project, but still a tad was missing on the theory side, the numerics were not polished and we had no paper drafted yet. How to proceed with this project? Sometimes it's important to have an honest excitement if you work in science. There are these moments when you realize that you understood something about nature. Suddenly things that puzzled you before totally make sense. And you come to the realization that actually the things you’re working on are kind of cool. And I think it's part of my job as a supervisor to bring this spark to my students. And for Felix, this spark sprung over! He agreed to finish this project with me and work on this project on his weekends. There were a few more difficulties we had to struggle with, such as small extensions on the theory side or inhomogeneous illumination in the experimental setup (Thanks to Leonie Bastin for helping us out!). Sure, in the end, we had some delays in our story and I want to thank Felix for sticking in and Lucas for all his patience, but I think it’s fair to say that it was a really fun project for all of us! To have this combination of analytical tractability, confirmation in experiments, and new conceptual insight into the physics of transport through complex media – Keep track of your local correlations! – is always something beautiful. So, what’s next? We know that biological networks locally adapt, so we want to continue on this theme of investigating the local correlation. Can we apply our new insights to study adapting porous media and what can we learn about the physics of our vasculature? I’ll stay excited about the work of my students. There is still so much we can learn.
References:
[1] Saffman, P. G. A theory of dispersion in a porous medium. J. Fluid Mech. 6, 321–349 (1959)
[2] Meigel et al., Dispersive transport dynamics in porous media emerge from local correlations, Nat. Commun., 13, 5885 (2022)