I thank all of you for being here. Wherever “here” is for you during these days of sickness and distance. I think I speak for everyone when I say that this semester has felt “remote” – distant – as if each of our interactions had a pane of glass between us and the rest of the world. I have heard similar descriptions given for depression. To get here was not easy, to stay here was not easy, to make it to the end was not easy. But it is done. And so are we. So where do we go from here?
The first and simplest answer to that is “wherever you want to go”. I believe in personal freedom and autonomy – call it “liberty” – and as such I think the only thing that should limiting how far a person goes is how willing they are to put one foot in front of the other. We find now that such an answer cannot be wholly satisfying. The world will not allow us to go wherever we want. We can’t just do whatever we want. There are constraints. Impositions. Mandates. Systems in place. There are rules.
The rules of biofluid mechanics are simple: three factors accelerate a particle of fluid – a lonely atom – a gravitational field, a pressure gradient, and stress. The rest is details. The conservations of mass and momentum define all fluid motion. A series of equations helpfully laying those out – considered a jewel in the crown of fluid mechanics research – bears the names of a couple of men – and yes, in fluids, it’s just about always men – you never met, taught at a distance by two by two men you may have never met. We trust those equations – and the men who present them to us – due to myriad factors of history, circumstance, using the tools for ourselves. Typically, it is enough to get us through.
But this year has posed its issues, none of which you need me outlining up here. I know some of us have suffered during all of this. Others know others who have. Our world is depressed. Under glass. It’ll get better I hope.
And but so we must consider then a second, subtler answer to the question, where do we go from here? Recall, the advice of Bruce Lee from the beginning of the semester. “Be “like water making its way through cracks. Do not be assertive, but adjust to the object, and you shall find a way around or through it. If nothing within you stays rigid, outward things will disclose themselves. Empty your mind, be formless, shapeless, like water. If you put water into a cup it becomes the cup. You put water into a bottle and it becomes the bottle. You put it in a teapot, it becomes the teapot. Now, water can flow or it can crash. Be water, my friend.” Which is to say, we can go where we are needed, where the field takes us, along our own particular particle’s path. It is my hope that you will learn what you have learned here to better our biomedical landscape, wherever you may roam along it.
In this class, we roamed far and wide, establishing both the fundamentals and creatively “peaking outside the frame” of typical content to include things like scientific literature, decades old fluid mechanics demonstration, and our own peers’ understandings of shared material. If I may briefly summarize. We began by introducing physical parameters necessary to understand fluid mechanics including mass, density, and viscosity. Given that fluids can twirl, swirl, and go everywhere we established systems of dimensions. We saw how pressure was measured at a point, how it develops along a field, and how a gradient of pressure can be established by the presence of gravity. Going from static fluids to dynamic fluids, we solved Newton’s Second Law along and across streamlines, expanded our view of motion by considering velocity and acceleration fields in the kinematic behavior of fluids, and constrained ourselves to control volumes and control surfaces by which means we could box in answers via the Reynolds Transport Theorem. We then honed in on the differential presentation of fluids as a system of ODEs that can be simplified and solved. Simplification was expanded into a science once we started to dimensionally analyze situations. Finally we studied the specifics of flow through pipes as blood pumped through our very vessels, as air filled our lungs.
As it turns out, mine and Dr. G.’s weren’t the only lungs to be filled with air to help talk this class along as we walked along that biomedical landscape.
Many students were integral to us establishing the foundations. These included each and every single one of you for participating in a weekly Homework Walkthrough sessions. From the praises I heard sung and the general vibe I got from around the virtual rooms, I think for many of us, this may have been our favorite part of the class. And I get that: learning from your peers is what the University experience is all about. I said the semester before the pandemic that “this University as a place on earth is something special. The halls are hallowed by lofty conversations, the rooms are charged with hope, expectation, ambition (thwarted or otherwise). But that is only be- cause you all attend this fine institution. “The only thing special about this place is the people. And you are the people. Emptied of you, this place has the mere residue of wonder.” That is because you are all smart, talented, capable, driven, even if you don’t think you are, even if you don’t believe you are, even if you’d fight me on the subject! You all are a special lot, and our times here together are special by consequence. “ And, well, I still mean it. The problem is, where has our University gone? Wherever you are, I thank you for bringing yourself here, now, to learn and share with us all. I thank you all for having shared consistently with us all, what you know and how you know it and helping us understand the homework as we could. Together, in parallel, decentralizedly, we solve 100 problems in biofluid mechanics and shared it with one another. At the very least, I am impressed.
I am similarly impressed by the efforts of
- E. C., N. G., N. H., E. K., E. L., L. L., R. S., C. S., F. Y., and A., Z. to present our first of six “commentary” discussions on the world renowned NSF Fluid Mechanics video series, this on the subject of Eulerian and Lagrangian Descriptions in Fluid Mechanics. I am always heartened to see intrepid spirits start us off on good paths. Thank you setting the pace.
- J. A., E. A., M. B., A. G., S. K., B. M., K. S., A. S., Z. S., and T. T. Each commented on Pressure Fields and Fluid Acceleration. This at a time when we were all beginning to feel the pressure and things start to accelerate. For helping keep the pace, thank you.
- Z. F., L. F., B. L., A. P., A. R., A. R., S. S., A. S., and J. W. facilitated discussions on the Deformation of Continuous Media to 90 other people. The their efforts were diffuse, the message was unified and helped to undergird our core content. Thank you all for your help.
- M. A., M. B., Y. H., M. J., M. L., F. L., A. M., S. R., B. R., and C. S. demonstrated the Rheological Behavior of Fluids emphasizing the effects of viscous forces in relation to inertial body forces. When stressed, fluids go with the flow. Thank you all for showing us how.
- R. B., C. H., X. L., J. L., M. P., K. R., J. S., H. T., and Z. Z. slowed things down with some Low-Reynolds-Number Flows. As biological flows are generally in the lower-numbered flow regions, understanding this behavior is critical to our understanding of biofluid flows. Thank you all for providing that critical element.
- And D. C., E. D., E. D., P. F., S. K., and S. N. taught us the Fundamentals of Boundary Layers, which demarcate the bounds between those spatial regions in flows where viscous effects dominate, and those regions where the inertial thrust of a fluid keeps it going. Thank you for showing us the bounds.
And while these video commentaries were designed to reinforce our fundamentals, about half the class elected to present a commentary on a piece of our scientific literature to showcase how what we learn can be applied and how it can be applied creatively!
- A. R., D. C., D. D., S. J., J. J., A. K., L. B., P. S., J. T., and P. P. each taught us something inspiring. Whether that was the biofluid mechanics of an artificial lung, an application of computational fluid mechanics to air flow in an infant incubator, a fluid mechanics explanation of the effectiveness of common materials for respiratory masks, or any of the other half dozen explorations of respiratory physiology, on behalf of the class, thank you.
- E. B., M. B., J. C., E. D., M. D., K. F., P. R., M. W., and D. W., each presented a commentary on cardiovascular flows including but not limited to improving hemodynamics of distal graft anastomoses, computational models of the aortic arch, and a general demystification of computational and clinical methods associated with cardiovascular dynamics. Thank you for that.
- D. W., L. W., L. W., M. P., H. L., A. K., S. J., E. F., and L. B. Each help us learn a little more about tissue-centric “flows” such as blood flow during bone repair and maintenance, fluid mechanical signaling in organogenesis, and our lymph system. We even explored more far out concepts like brain organoids on a chip, the effects of acupuncture, and bioprinting functional contractual cardiac tissues! For helping us all learn, thank you.
- M. G., E. C., J. C., S. C., I. H., J. O., S. Z., B. R., and J. L. showed us how the sausage gets made in the disposal of our wastes including showing us the “Magenstrasse” “stomach road” for gastric emptying, a human urodynamic simulator, globular filtration rates for people and lampreys, and even how nano carrier drug delivery systems absorb, distribute, metabolize, and excrete biomaterials. If you’ll pardon me the terrible pun, it turns out, learning about waste, was not a waste of our time.
- And L. N., A. O., E. B., A. B., E. R., T. E., J. S., L. G., C. C., and R. J. gave us a sense of what lies “beyond” in the the farther reaches of biofluidic research. This included things like bloodstain pattern analysis, altered knee joint mechanics, dynamics of tear films, biofluid flow models incorporating magnetic drug targeting, and modeling disease as a fluid flowing. Thank you all for taking us beyond the bounds.
You all have helped make this class what it is. We might forget the influence of others around us, just as the fish might not know it is help up by water. Whether it is better to see or not see the water around us, I leave to others. My role in this course – my philosophy of this content – is that I have to teach you to see water in a new way. If after this class you do not conceive of a glass of water differently, we haven’t yet done our job. The fluids of life, air, water, blood, bile, bio fluids, surround us, infuse us, are us. And they can be modeled rather useful by conserving mass, conserving momentum, and adhering to the kinematics of the field, all more or less in a handful of equations. The rest, as I said, is details. And it can be hard to see and understand and work with this fluids if simply told. Especially if simply told by a quote “authority”. The old fish doesn’t get the two younger fish to see water simply by asking them about it. But that old fish does get them inquiring. And if the pair of them have learned anything from life, it’s that they can learn from each other. You have done so time and again in this class. You walked through our homeworks together, you shared your thoughts on scientific literature with us, and we all even had a biweekly reading discussion in which examples were gone through and the chapter summarized by a peer of ours. A fellow fish.
Fish do better in school. Maybe I say this as a schoolteacher. Maybe I say this as a lonely atom adrift in empty hallowed halls. All the same, I think we can all agree that this semester has been rough. And the next one is liable to be tough too. You all have done incredible work, regardless of the circumstance. Each week I continued to be impressed by your tenacity and perseverance. You all learned fluid mechanics way better than I did my first time around. And you did it at a distance. Sometimes alone. I view this as both a testament to what we were all able to achieve here together and as a reminder to ourselves and others, just because you don’t get something now doesn’t mean you won’t get it eventually. As another fellow fish reminds us, Just keep swimming.
Sometimes it can be hard to just keep swimming, especially when the waters as viciously viscous and shear against us. Those who model persistence are consistent models of excellence. I want to take a moment to thank those particularly persistent fish who swam to class every day: A. R.; C. C.; D. W.; E. K.; H. L; L. L.; L. W; P. R.; S. K.; and S. J.
Their continued efforts, along with the dozens of others who consistently participated, kept this class going.
But more than any others, our team of IAs kept us going. Whether it was answering Piazza questions, grading homework, hosting dozens of office hours a week, or any of the other critical roles they filled each week, this class simply could not have functioned as well as it did without the dedication of these six individuals. Please join me in an especially thunderous round of applause for C. F., J. G., J. L., G. T., R. W., and A. W.
Thank you thank you thank you for all your efforts here.
Our time is coming to its end. To everything there is a season we are told. What can be said of this season which has not already been lamented? What can be hoped of the next which has not be bewailed? A dark winter. A school of fish, not sharing in the same sea. Our water is similar, but like the lone fish in its tank, we might not really know what surrounds us. Just as critically, it is harder to ask the fish in the next tank over how their water is. And when you cannot ask, you cannot know.
And while there is plenty more for us to know, for now let us content ourselves with better answering that parabled question: what the hell is water? A fluid whose motion can be characterized by three factors – acceleration due gravity, due to pressure differentials, and viscous shearing forces – critical to sustaining multiple biological systems. Ours included. And while indeed it is great for us each to have our own cup of that knowledge, we have seen time and again that pooling together with others betters all.
With that, I thank you for the opportunity to dismiss this class one final time. I wish I could do so under better circumstances. But happenstance has brought us to this point. Where shall we go from here? Wherever you’re headed, I wish you the best of luck on your upcoming exams and your ever-to-the-horizon journeys. I have done what I can for you here and I trust you dan do more.
Good luck.
