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## Super-fluids, Non-Newtonian fluids, and a Continuation of Fluid Dynamics

Intro:

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So a while back I made an article on hydrodynamics and liquid crystals. I recommend checking out that article first before reading this if you'd like to look into fluid dynamics more. You can find that article: HERE.

However, I also thought this topic might be fun to talk about some more as I recently gave a talk about the hydrodynamics of Jupiter. Superfluids are common on that planet, due the density of the atmosphere. So I think VFX wise, when it comes to simulating the atmosphere of a planet, it might be worthwhile to talk about. Here we go.

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What are Superfluids?

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A superfluid is a liquid that has zero resistance when it flows. So it is not affected by friction in any way. Superfluid's can flow without losing any energy over time. Superfluidity is more commonly described as a trait of a fluid, rather than one type of fluid or selective ones.  Fluids with this property usually have zero viscosity, and form vortexes when disturbed. They also don't usually form under normal environments here on Earth. They are commonly discovered or created in labs, on other planets, or in other extreme environments. Superfluidity was first discovered in liquid Helium by Pyotr Kapitsa and John F. Allen in 1937. It also plays a huge part in quantum hydrodynamics. But that might be talked about in another article. :)

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• To my best knowledge, you can mimic Superfluids in Houdini. However, you'll need to not only adjust and account for the viscosity and friction of the fluid, but also the velocity fields for the turbulence as well. You can create these velocity fields by either using smoke sims to displace and advect the liquid, or custom velocity trails. Persevering the rate of which the liquid flows will be difficult, but Houdini has multiple tools for reducing friction on collision objects and particles. Just remember: @Velocity is instant, @force adds over time.

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One other thing to note about Superfluids, is that they happen and exist around something called the Bose–Einstein condensate. This state of matter is used in condensed matter physics to describe the cooling of gases containing bosons to low temperatures. Superfluids formed at high temperatures do not follow this state of matter, but ones formed at low temperatures do.

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Some of the most common forms of Superfluids are in liquid Helium. 4He (Helium-4) and 3He(Helium-3) are powerful fluids that have a lot of interesting properties. Helium was first discovered by Pierre Janssen and Norman Lockyer in1868. But since then, more variations of the element have been discovered.

- 4He is an isotope of Helium, and its superfluid properties are harder to simulate as it needs to be heated to higher temperatures than its counterpart 3He. All of its atoms are made up of bosons, and because of this its particles have zero spin. However, it is one of the most abundant isotopes of Helium in the Universe. This isotope is commonly used in Spectroscopy. Which is the study of the interaction between matter and electromagnetic radiation. Superfluids are also used in gyroscopes and satellites that study infrared data and need to operate at low temperatures.

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- 3He is a sister isotope to 4He. It is more stable and lighter than 4He, and is considered a primordial isotope. This means it has existed longer than the Earth, or was created when Earth was. It can be found in other places in our Solar System, such as the lunar surface and nebulas.  It is thought that it was first appeared by cosmic rays colliding with Earth based atoms and materials.  It can also be released into the atmosphere by nuclear weapons testing. In 1996, the scientists who discovered 3He won the Nobel Prize in Physics.

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Besides Helium, Superfluidity can be observed in fermionic gases. This was first proven by a team in 2005 led by Wolfgang Ketterle. They were studying quantum vortices, and discovered that the gas can turn into a superfluid form after being affected by external magnetic fields.

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Besides Helium and fermionic gases, superfluidity can exist in other forms of exotic matter. Which makes it a trait to look for in astrophysical systems. It is thought that Superfluids exist in neutron stars, as they are so condensed and operate at extreme temperatures. This means the nucleons inside them would start to operate differently than in other celestial bodies.

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Now that we've talked about the basics of Superfluids, there is one important theory that these liquids fit into. Which might be of some interest when it comes to the overall theory of the Universe.  Superfluid Vacuum Theory (SVT) is a theory in quantum mechanics where any vacuum, including the vacuum of space, is considered a superfluid. There is an overall goal in quantum mechanics to develop scientific models to unify all forces of the Universe into an improved single Standard Model. SVT could possibly help that process as it could explain the origins of gravity in the Universe.

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Non-Newtonian Fluids

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A non-Newtonian fluid is a fluid or liquid that does not follow Newton's Law. This means it's viscosity can change regardless of the stress applied to it. For example, if you've ever mixed cornstarch and water together it forms a mix between a solid and a liquid. When you allow this mixture to run through your hands, it will drip just like a liquid. But if you collide your fist with it, you'll find that the liquid suddenly forms resistance around the impact area, and solidifies a bit. Ketchup is also a non-Newtonian liquid and becomes more runny when you shake it. Custard, honey, toothpaste, paint, and blood are also considered to be non-Newtonian as well. These liquids are not only dependent on their rates of viscosity, but also their deformation by shear stress. Shear stress helps assist their change in viscosity, and based on the liquid's shear stress history the changes in the substance can behave differently each time.â€‹

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There are three main types of viscosity for Newtonian and non-Newtonian fluids. It is important to know the differences in-between these types, as they can help you identify fluids better. Plus, if you would like to simulate any fluid, you'll need to calculate it's viscosity properly to get the best result.

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Viscoelastic: This happens when elastic and viscous forces work in parallel in the liquid. This most commonly happens in liquids such as whipped cream or silly putty.

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Time-Dependent Viscosity: This describes the process when viscosity increases in a liquid over time when constant stress is applied. It is also known as Rheopecty.  Or the rate of viscosity decreases during constant stress. This process is known as Thixotropic. Fluids that display these traits are: yogurt, peanut butter, gelatin, castor oils, and some forms of clays.

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Non-Newtonian Viscosity: As mentioned above, these liquids are dependent on shear thickening or shear thinning.

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Nonlinear Fluid Dynamics

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These fluids are a bit tricky to explain, but stick with me. They can best be described as chaotic fluid systems in nature. They have variables that constantly change over time, and can be rather unpredictable. They operate on a system called a nonlinear system. In fancy terms this means the change of the output is not the same or equal to the change of the input. These equations are made up of unknown functions and variables that are subject to change over time. Because of this, linear systems can be combined with non-linear ones and the equation can still be considered a non-linear one. Just because of the random variables the system uses. These systems can usually be seen as waves in a liquid.

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Non-linear dynamic systems are quite popular in the sciences and mathematics. This is because they represent aspects of nature quite well, and solving them represents a more complicated challenge than regular linear systems. Using them scientists have been able to discover that the appearance of randomness in our Universe is not random at all. Rather it is caused by chaotic systems such as these.

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Some examples of non-linear equations you can check out are:

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• The Navier–Stokes equations of fluid dynamics

• The Nonlinear Schrödinger equation

• The Algebraic Riccati equation

• The Ishimori equation

• The Korteweg–de Vries equation

• The Landau–Lifshitz–Gilbert equation

• The Liénard equation

• The Bellman equation

• The Boltzmann equation

• The Colebrook equation

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Most nonlinear dynamics exhibit one or more of these distant behaviors:

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• Amplitude death:  Oscillations in the system suddenly stop when they come into contact with another system.

• Chaos:  These systems cannot be predicted, and their final result is unknown.

• Multistability: The system displays one or more stable states while it is in motion.

• Solitons: these systems contain self-reinforcing waves.

• Limit cycles:  Orbits play a big part in dynamics. This behavior means the system contains asymptotic periodic orbits, which are orbits that do not repeat and do not close. These orbits will then destabilize fixed points and attract them into the orbits.

• Self-oscillations:  Oscillations will start to occur if the system encounters a dissipative state or system. A dissipative system is one that exchanges energy and matter.

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When you are working with non-linear dynamics, there are two methods you can use to change the outcome of your project or equation. These methods are based around real world tactics that scientists use to study these systems.

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One way to solve a nonlinear system or to experiment with one is to change and exchange the variables in the system to see what happens. In a simulation sense, this just means that you can adjust your parameters and attributes to different values. In mathematics, you can also use a finite element method for solving these systems on a grid. Which can lead you to applying the finite element method to computational fluid dynamics. In order to do this you will need a grid with a high amount of points, and to start simulating the surface of the grid with your chosen non-linear equation. Then you can increase or decrease polynomial degrees in the system to see what happens. I know this is a very brief explanation but you can read a full paper on this process: HERE.

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The second way you can explore these systems in 3D space is by analyzing how scientists work with non-linear fluids. Currently, they use nonlinear differential equations to explain unequal heat transfer, and conservation of total energy. They also apply thermodynamic analysis to these fluid flows to better understand how they process heat. These are then applied to plasma physics, optics, traffic flow, and even more studies. Most of these differential equations are solved numerically.

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If you wanted to explore these applications in 3D space you would have to visualize the heat transfer of these flows as values, and colors over the motion of the liquid. You could set a visualizer to show the most "hot" areas as red, the "colder" areas as blue, and everything else could be expressed as color in-between them.

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Simulations, Visualization, and Studies

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Here are some awesome fluid dynamic studies worth mentioning.

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There have been several studies for the vortexes and transitions in Superfluids. Particularly regarding Couette flow. This type of flow is used a lot in fluid dynamics. It is used to measure and describe how fluids react in the space between two surfaces. These surfaces can be anything, metal, other solids, gases, or even other liquids. It describes how these surfaces can influence a fluid's shear stress, viscosity, and flow direction.

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Couette Flow is also used in studies to show how Superfluids like Helium react in nature and their vortex motions.  Their transition lines and flow patterns are of great interest, and can be checked out in this paper: HERE.

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The Center of Mathematical Sciences at The University of Cambridge does a lot of different studies into fluid flows. Especially for it's application into astrophysical fluid dynamics and nonlinear dynamics. Their two departments DAMTP(Department of Pure Mathematics and Mathematical Statistics)  and DPMMS (Department of Applied Mathematics and Theoretical Physics) carries out a huge range of experiments for these fluids for specific astrophysical interests, planetary formation, and others.

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Earlier I briefly mentioned the nonlinear Schrödinger equation. The Schrödinger equation and it's variations have been studied countless times. It is often used to explain wave interaction in classical fluid dynamics, but can also be applied to superfluid interactions.  Here is a study you can check out if you are interested: HERE.

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Visual Effect Simulations

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Earlier in this article I mentioned me and a friend did a Houdini HUG presentation on Jupiter. Me and David were somewhat successful in building a planetary atmosphere. Well while we were researching, we came across a lot of this science. The reason is that these forms of fluid flow are quite often found in planetary atmospheres. So when building something like that in Houdini, if you wish to have the best result possible, you will need to apply these concepts into your build. The best way to do that is with FLIP simulations, or particles. The hardest element to control will be your shear stress levels. Currently, there really isn't a go-to parameter for that in Houdini, so you'll have to customize that as you go.

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But honestly, you could create anything you want with these theories, not just a planet. Any liquid or gas. There are already examples out there of people using the Schrödinger equation to simulate smoke. You can read the paper on that: HERE

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Putting that aside, let's talk about some elements in Houdini that could help you implement these theories.

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• When calculating the mass of your particle fluids it is equal to volume times density.

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• If you are using outside forces to affect the motion of your particle fluid, consider turning off Ignore Mass. Just so the forces don't affect your particle separation.

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• If you need to add bubbles or "gas pockets" into your fluid you will have to use the Sink tool as well as a piece of geometry to represent your gas.

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• By default FLIP fluids have no viscosity. This allows you to input any viscosity changes you would like. So if you need to add Viscoelastic, Time Dependent Viscosity, or non-Newtonian Viscosity into your simulation, you can absolutely do that. However, keep in mind the viscosity is dependent on the scale of your particles.

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• Turn on 32-bit solve for faster viscosity solves.

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• If you have a slow moving viscous liquid turn off Reseeding to make sure all particles move the same way.

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• To stop particles from creeping out of your viscous liquid turn off No Detection.

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• Always use the Surface Visualization on the FLIP Object to visualize viscous fluids better.

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• There are Viscous Fluids Shelf Tools that pretty much has all your basic needs for these fluids. They contain temperature attributes that can control your viscosity, which makes them great for Superfluids. However, these tools are mostly centered around lava.

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• You can also compress fluids in Houdini. The Fluid Compress geometry node compresses fluid simulations. But often you'll have to use a fluid tank for this node to work. However, this node is often used to make simulations operate faster and to give you better visualization options, instead of compressing them from a scientific perspective.

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• Another downside to FLIP is that it is currently unable to calculate accurate shear stress. However, you can fake it loosely by creating a gradient of the velocity to mimic a shearing coefficient. Then you can combine that gradient with surface tension to create it.

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References:

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Non-Newtonian fluid:

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IMPACT DYNAMICS OF NEWTONIAN AND NON-NEWTONIAN
FLUID DROPLETS ON SUPER HYDROPHOBIC SUBSTRATE:

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Nonlinear Fluid Dynamics Description of non-Newtonian Fluids:

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Dynamic wetting of shear thinning fluids: https://aip.scitation.org/doi/10.1063/1.2432107

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NON-NEWTONIAN FLUIDS: http://www.thermopedia.com/content/986/

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Non-Newtonian Fluids: Dilatant vs. Pseudoplastic:

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Sample records for viscosity fluids introduced:

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VISCOSITY OF NEWTONIAN AND NON-NEWTONIAN FLUIDS: https://www.rheosense.com/applications/viscosity/newtonian-non-newtonian

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Wave–Vortex Interactions in Fluids and Superfluids:

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Superfluidity:

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Mechanism of Superfluid Flow:

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Vortex Lines and Transitions in Superfluid Hydrodynamics:

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Introduction to quantum turbulence:

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Skoltech Superfluidity:

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VFX Artist Shows What the Speed of Sound LOOKS like!:

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Liquid Lakes Form Europa's Blemishes:

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Critical transport and vortex dynamics in a thin atomic Josephson junction: https://arxiv.org/pdf/1905.08893.pdf

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Nonlinear system:

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Nonlinear Fluid Flow and Heat Transfer:

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Nonlinear Fluid Dynamics from Gravity: https://arxiv.org/abs/0712.2456

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A Numerical Approach to Solving Nonlinear Differential Equations on a Grid with Potential Applicability to Computational Fluid Dynamics:

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Fluid Dynamics and Behavior of Nonlinear Viscous Fluid Dampers

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Astrophysical Fluid Dynamics and Nonlinear Dynamics

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