Fluid, Fluid Everywhere: The Fascinating Tale Of Fluid Mechanics
“Fluid mechanics is a part of applied mathematics, of physics, of many branches of engineering, certainly civil, mechanical, chemical, and aeronautical engineering, and of naval architecture and geophysics, with astrophysics and biological and physiological fluid dynamics to be added.” — Sydney Goldstein, renowned mathematician and fluid-dynamicist
Fluids… We are surrounded by them. Fluids are present in the atmosphere around us. They constitute all the water bodies on our planet and are found inside the solid earth. More importantly, fluids are also a major constituent of the human body. We utilize them for activities like cooking, washing, and so on. This list of ‘uses of fluids’ does not end here!
This article is a small attempt to explore the initiation and development of the subject of fluid dynamics since ancient times. Giving an overview of this subject across different scales, it talks about its active areas of research in the present-day world.
Traveling Back In Time
Because fluids are everywhere, the laws of fluid dynamics control a diverse range of crucial phenomena that happen all around us. They also have crucial applications in various areas of science, engineering, and technology. For example, the laws of fluid motion control how airplanes fly, how ships sail, and how our bodies function. They also govern the settling of dust particles as well as critical processes like the shining of the Sun (how heat is transferred from the sun’s core to the outside) and the formation of magnetic fields around planets. A branch of study with so many practical applications is bound to have an interesting history — don’t you think so?
Well, this is exactly the case. Fluid mechanics can be thought of as a sub-branch of classical physics and its history is quite sporadic! This history is best understood by dividing the timeline into so-called eras, although there are no clear cut boundaries separating them. We shall start with the earliest developments, which started from 500 BC and lasted till 1400AD, covering a huge span of 2000 years. Evidence suggests that the ancient Romans, Greeks and other civilizations possessed some basic knowledge of fluid flow. They knew activities like sailing and irrigation. Closer to home, people of ancient Indian civilizations (like the Indus Valley) could build excellent water supply and drainage systems. Nevertheless, a large part of this understanding remained undocumented for a long time.
The Eureka Moment!
The first major insight into “Fluid Mechanics” came from Archimedes, a renowned Greek mathematician, physicist, engineer, inventor, and astronomer. He quantified the buoyancy force exerted by a liquid on an object immersed in it.
We shall loosely call the second era the medieval times, which spread from 1400 AD to 1600 AD (about 200 years). During this time, Leonardo Da Vinci, a renowned scientist, made significant contributions to fluid mechanics by carrying out extensive studies on how water flows past objects and how (gravity) waves propagate and behave on the surface of any free fluid. Following Da Vinci, a second notable discovery was done by a physician named William Harvey, who observed the mechanics of blood flow within the human body. This was at the time when people believed that lungs are, somehow, responsible for circulating blood. Harvey was amongst the first to give the world an accurate model of blood flow!
The medieval era was followed by the post-medieval times, starting from 1600 AD and ending around 1687 AD. In these 80 odd years, several prominent developments took place in fluid mechanics. The famous Galileo and his students, Torricelli and Castelli, worked on the flow of fluids in rivers and canals. Toricelli also quantified the behavior of liquid jets. Around the same time, another renowned physicist and inventor, Blaise Pascal, showed how pressure is adjusted in liquids under hydrostatic equilibrium. His laws are still widely used today in areas like hydraulics design.
Tracing The Beginning — The First Revolution
It can also be referred to as the Newtonian era which started in 1687 AD and lasted approximately till 1800 AD. This was the time when the field of fluid mechanics really started to take off! Sir Isaac Newton was the first one to propose how viscous forces act in fluids. Today, the fluids (such as water) which obey his law of viscosity are known as the Newtonian fluids. A large chunk of fluid dynamics still revolves around this family of fluids.
Later, Daniel Bernoulli explained how fluid motion (pressure and velocity) obeys energy conservation. He formulated the elegant and simple Bernoulli’s equation that is still relevant today. Inspired by the latest developments, French physicist Jean d’Alembert proposed a puzzling paradox that directly contradicted the existing theories at the time. Those theories predicted that any object should not experience any drag due to fluid moving past it and yet, the experiments (and also everyday experience) clearly suggested otherwise. Surprisingly, it took about 150 years for this paradox to be fully resolved! In 1757, one of the most prominent scientists of this era, Leonhard Euler wrote down one of the fundamental sets of equations, derived using Newton’s laws of motion, that govern the motion of inviscid (ideal: viscosity is zero) fluids.
The Second Revolution
Beginning from the early 1800s, the second revolution spanned almost the entire 19th century and witnessed enormous advancements in mankind’s understanding of the laws that govern fluid motion. Claude-Louis Navier (French civil engineer) and George Gabriel Stokes (British mathematician) are two of the most important figures of these times. While on the one hand, Navier gave equations that are applied to study deformations in elastic solids, Stokes applied these equations to fluids and gave what is called the “infamous” Navier-Stokes equation, which is diabolically difficult to solve. This equation governs the motion of real viscous fluids that obey Newton’s law of viscosity (more precisely, Stokes Law of viscosity). Stokes further revolutionized our understanding of fluid motion by introducing several key concepts such as Stoke’s Hypothesis and Stoke’s Stream function which are utilized even in the present times. A few years down the line, Augustin-Louis Cauchy wrote another fundamental equation, which was able to quantify motions of complex (non-Newtonian) fluids as well, a paradigm that so far had remained elusive.
In 1856, French physicist Henry Darcy proposed a pioneering law for the flow of fluids through porous substances; this law, today is known as Darcy’s Law, in his honor. In the first half of the 19th century, another French physicist Jean Leonard Marie Poiseuille demonstrated his results for viscous fluid flow through tubes, by correlating the pressure drop, volume flow rate, and tube length. In the latter half of the 19th century, a new realm of fluid mechanics opened up, as German physicist Ernst Mach made one of the most significant breakthroughs in fluid dynamics, by experimenting on high speed supersonic and compressible flows and demonstrated, for the first time in his lab, the formation of shock waves.
Another Milestone — The Third Revolution
Our understanding of fluid dynamics improved by leaps and bounds during this era of rapid advancement, which started from around 1883 and continued till about the 1930s. To begin with, we must talk about the work of Osborne Reynolds first, who transformed our understanding of fluid motion through his experiments on flow through pipes; he showed how turbulence (defined as a chaotic irregular flow, as opposed to orderly laminar flows) sets in, as certain geometrical or flow parameters (such as velocity) are changed. He showed that a particular non-dimensional number, now called the Reynolds number, decides whether the flow of any fluid remains laminar or turbulent. However, interestingly, this number was first derived by Stokes, although Reynolds was the one who informed the world about the physical significance of this number! Reynolds’s experiments revolutionized fluid dynamics and unleashed a novel area of research on turbulence and instabilities.
In 1904, a German engineer Ludwig Prandtl developed the Boundary Layer theory that finally resolved the d’Alembert’s paradox once and for all. This theory once again fundamentally transformed our understanding of fluid motion around solid objects and explained how drag forces are generated in high speed flows. It was so influential that it also led to the creation of a new branch of mathematics called the Singular Perturbation or Matched Asymptotic Expansion. Apart from Prandtl, many other scientists and researchers also made significant contributions towards expanding our understanding of fluid dynamics; the list includes people like Theodore von Karman, Paul Henrich Blasius, Lord Rayleigh, Lord Kelvin, William M. Orr, Arnold Sommerfeld, Werner Heisenberg, and GI Taylor, in the long list of many others. GI Taylor also gave the Leaky Dielectric Model that talks about how electric fields and poorly conducting fluids interact with each other.
The Final Frontier!
This brings us to the final era, which started in around the 1930s and what a beginning it had! Starting in 1941, Andrey Kolmogorov, a Russian Mathematician came up with a series of unmatched insights on the structure of turbulence in various fluids. He proposed the Kolmogorov scales — a combination of velocity, length, and time scales that is at the heart of every turbulent motion in nature. He also gave a series of hypotheses on how energy is distributed across the eddies (read, vaguely defined turbulent flow structures) of various scales when the Reynolds number is sufficiently large. I would also like to take this opportunity to talk about another little known fact about a huge contribution to non-linear dynamics that came from fluid mechanics. In 1963, Edward Lorenz, renowned mathematician and meteorologist, was trying to study the thermal convection in the atmosphere using the Navier-Stokes equations, and in the process became one of the first people to come across deterministic chaos. He ended up with a set of equations (now called the Lorenz equations) and demonstrated how a slight change in the initial conditions leads to a drastic modification in the final solutions. This is the reason why weather prediction is so difficult!
George K Batchelor, Howard Brenner, Chandrasekhar S, JG Oldroyd, Nan-Phan Thien, and Andreas Acrivos were some other great 20th-century fluid dynamicists who carried out breakthrough research in various aspects of fluid motion. With the invention of computers, this period also saw the start of Computational Fluid Dynamics (CFD), and Suhas Patankar developed the SIMPLE algorithm that could systematically solve the Navier-Stokes equations in a computer. Around the same time, Steven Orszag created Spectral Methods for numeric computation to solve simple fluid-flow problems.
The Latest — Where This Field Stands Now?
To know this, we should first know the answer to the question — how do we quantify fluid motion? Well, a fluid is a collection of particles and is a continuous medium. We use the Eulerian description to quantify fluid motion which is essentially a field description where all the variables that we study are treated as functions of both space and time. Now, the next question that should come to your mind is — what are we trying to find out? One of the things to understand here is the difference between solids and fluids, one of the most fundamental of which is that fluids can deform continuously and indefinitely. It hints at the fact that it probably does not make much sense to talk about absolute deformations in fluids; rather, the rate of this deformation is more crucial. Based on this information, the fluid-mechanics researchers mainly focus on studying three quantities — velocity, pressure, and temperature. All these parameters are highly related to one another. Everything else (like the deformation rate) can be calculated by utilizing these quantities.
Diverse Geography — The Fluid Scales
The physical dimension of flow is an extremely vital quantity in fluid mechanics. Depending on what scale one is looking at, one might get drastically different behaviors. It can be better understood with the example of water — its behavior when flowing in a nanopore as compared to in oceans is very different! There is a huge spectrum of fluid scales, starting right from the nanometer scale and going all the way up to the astrophysical flows (10⁹ meters). An amazing feature of fluid dynamics — it doesn’t matter which scale one is looking at, the same principles, laws (conservation of mass, momentum, and energy), and even equations are applicable at all levels!
Below nanometer scales, we encounter the atomic and molecular regime where the discrete nature of matter becomes prominent and the continuum hypothesis breaks down. Essentially, the fluid no longer remains fluid and the laws of fluid dynamics cannot be applied at such small scales. Moving up, we arrive at the micro-level (1 to 100 microns) and are concerned with flows in small chips, around bacteria, porous media, dust particles, etc. The fluids in this range are termed as micro-fluids. Next in line are the millimeter-scale dimensions, ranging from several millimeters to a maximum of a few meters. These consist of flows in engineering devices viz., small pipes, arteries, and veins, to name a few. Progressing further, the scale expands to a few tens of meters and this regime talks about flows around big objects like buildings, cars, airplanes, and so on. Higher up, the scale reaches thousands of kilometers and it comprises flows in or around the atmosphere, oceans, landmarks, and so on. The highest scale is that of the astrophysical flows and these include, for example, flows in stars and gas clouds.
Some Current Areas Of Research
One of them is the turbulence. It is characterized by chaotic and rapidly fluctuating flow fields. Instability is another active area of research, which explores whether a particular flow is stable or not. High-speed compressible flows come next. This branch deals with shockwaves, supersonic and hypersonic flows (pretty useful in fields like rocket science). The field of geophysical fluid dynamics revolves around atmospheric flows, weather predictions, etc. Hydrology focuses on water resources, environmental, and subsurface flows. Rheology is concerned with constitutive relations for complex fluids (suppose, if we impose a strain on a particular fluid, how does it respond?).
A closely related field is the study of the flow of complex (non-Newtonian) fluids. Another one is the multiphase flows and it encompasses studies on a mixture of fluids, drops, and bubbles. Flows around porous media, thin films, particle motion, biological and microscale flows fall in the category of creeping (viscous) flows. Fluid-structure interaction discusses how deformable bodies behave in a flow. Moreover, computational fluid dynamics remains an active area of research and involves new algorithms for solving flow equations. A number of fluid dynamicists are also interested in magnetohydrodynamics which explains how magnetic fields and fluids interact with each other. Other than these, there are many more sub-branches of fluid dynamics that also involve other disciplines of engineering.
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This article is based on one of the sessions (delivered by Uddipta Ghosh, faculty in the Mechanical Engineering discipline) of the Virtual Seminar Series by IIT Gandhinagar. It is an online program started by the Institute in the wake of the current pandemic as a means to engage the people so that they can learn about a diversity of topics from the comfort of their homes, in an interesting manner. (The 9th article of this series can be found here and the 1st article is available here.)
