1ρKyoka HyodoDevelopment Department, Particle Counter Division. Joined Rion in 2023. Currently, she’s mainly responsible for making design modifications and engaging in underlying technology research and development for liq-uid-borne particle counters. Her current dream is to revisit Tanegashima, which she hasn’t visited for quite a while, to breathe in the island’s serene atmo-sphere and gaze up at the star-filled night sky.For a rigid body, the equation of motion is expressed as ma = F. For a fluid, these are known as the Navier-Stokes equations. The equations are named after the two scientists who formulated them: French mathematician and physicist Claude-Louis Navier and British mathematician George Gabriel Stokes. The equations consist of terms for time, convection, pressure, and viscosity. No general solution for the Navier-Stokes equations is currently known. It’s one of the mathematical problems selected for the Millennium Prize.EPILOGUE SCIENCE, SCIENCE! 19A Column by Rion’s Staff on Their Obsession with ScienceThe first time I was introduced to the Navier-Stokes equations was in a fluid mechanics class at university. When I saw the equations on the blackboard, I just thought, “OK, so here’s this set of equations.” I had no idea they would come to play such a significant role in my life.The Navier-Stokes equations describe how fluids like water and air move over time. The equations have four terms: a time term, a convection term, a pressure term, and a viscosity term. These terms describe how variations in pressure and viscosity affect fluid flow. They’re used in various real-world applications that affect our daily lives, including the calculations used to predict atmospheric flow in weather forecasting, in analyses of airflow around airplane wings, and in analyses of water flow through pipes when designing plumbing layouts. * *In my junior year at college, I joined the fluid dynamics research lab. That’s when my relationship with the equations changed completely. In the lab, the students held weekly study sessions in which we took turns leading discussions based on a book titled A First Course in Turbulence. It wasn’t just about taking turns reading sections of the book, but also about understanding how the equations in the book were derived. My second encounter with the Navier-Stokes equations happened during one of these sessions. Attempting the derivation, I still remember thinking, “Wow, these equations consist of long and complex variables.” And they were dicult indeed. When I tried to listen and understand explanations of how the equations were derived, I couldn’t understand anything, and gradually grew to dislike them. But over the next four years, including my years in graduate school, I had to deal with the derivation of the Navier-Stokes equations every year. Maybe for that reason, my original aversion gradually faded. I started to think, “Hmm, maybe I don’t hate them as much as I thought.” I think the reason the dislike faded was that I was taught we could take the equations apart into their four terms and examine each term individually. When viewed as a set of equations, the Navier-Stokes equations look incredibly complicated. The Navier-Stokes EquationsNo. 011The Navier–Stokes EquationsRion is supported by many science-loving and math-loving staff mem-bers.In Part 11, we present an engineer who’s maintained a curious rela-tionship with the set of complex equations that govern fluid dynamics! * *My partner for life? But when broken down and examined term by term, the equations become much easier to understand. In short, this experience completely changed how I approach complex mathematical formulas.Nearly 180 years have passed since the Navier-Stokes equations were first derived. Remarkably, no general solution has yet to be found. It’s one of the mathematical problems selected as Millennium Prize Problems: anyone who finds a solution will get a prize. The core of the mystery lies in the nonlinearity of the convection term contained within the equation. Because this term is structured so that the flow velocity at the moment determines the flow velocity at the next moment, even a small disturbance can be amplified over time, gradually making it more dicult to predict the behavior of turbulent flow. It’s this turbulence that makes finding a general solution so challenging. And since these equations deal with quantities that change continuously in both time and space, their behavior has to be considered at ever finer scales. So it becomes very dicult to find a single general solution, and we don’t know whether values change smoothly or there exists an abrupt divergence (resulting in infinity). Even today, more than 180 years later, no general solution has been found. I believe it is precisely these difficulties that make the Navier–Stokes equations so captivating.Looking at it from another perspective, I remember being a member of a rocket research club as a college student. Our club participated in the annual rocket contest held on Tanegashima. No one in our club attempted to use the Navier-Stokes equations to analyze the fluid dynamics of our rockets, but the equations are relevant in fields like rocket science. I’m sure the analytical software we used incorporated them. Come to think of it, I’ve been interested in rockets, satellites, space, and stars—all things that are related to the Navier-Stokes equations—ever since I was in elementary school. And now, I’m working on particle counters that deal with fluids. I may have been connected to these equations in one way or another ever since childhood! I can’t help wondering if I’m destined to stay in the permanent company of these equations. Maybe we’re life partners. I look forward to seeing what roles the Navier-Stokes equations will continue to play in my life in the years to come.Article by Kyoka Hyodo∂2ui∂xj∂xj∂p∂xi+ν∂uiuj+∂t∂ui∂xj=−Because We’re Science and Math Lovers
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