Lectures

INTRODUCTION

We introduce the class and the assignment structure. I explain the details of the final paper we should be working on the entire term. We end with a toy example problem setting the stage of analysis in this class.

Lecture Notes

Extra Example : I was asked once to present this example problem to a engineering math class once as an example of how you can use math to investigate. There is a punchline at the end...

THE NAVIER-STOKE EQUATIONS I

We embark on a two part journey of rigorously defining what these Navier-Stokes equations are and where they come from. We start with a naïve approach and ask natural questions about how one would go about modeling a region of fluid.

Lecture Notes

THE NAVIER-STOKE EQUATIONS II

We complete the governing conservation equations by making more assumptions on on how stress is related to the strain rate. The stage is set for the rest of the class! We have now a set of equations that apparently model the flow of any fluid. I mean any. The turbulent flow from a smoke stack, the flow in your arteries, the motion of honey, the list goes on. This is a bold statement. However, it has never been proven wrong and to the best of our knowledge it truly does describe all flows.

Lecture Notes

Summary Of Equations

DIMENSIONAL ANALYSIS

By non-dimensionalizing terms of the governing equations we see how measurable parameters can simplify our problems. We also see some almost comical examples of how this scaling method can almost do everything for you.

Note: I think I will redo this lecture later in the term if I have some extra time.

Lecture Notes

SIMPLE INCOMPRESSIBLE FLOWS I

Starting from the most general case, entire classes of solutions are found and the true power of making equations dimensionless is seen! This lecture is broken into two parts to show how one can start from knowing absolutely nothing about a flow and use the full equations + scaling to set up a simpler set of equation to solve. The problem? We look at flow between two flat plates separated by a small distance. We see that the scales determine the assumptions you have maybe seen in various undergraduate texts.

Lecture Notes

SIMPLE INCOMPRESSIBLE FLOWS II

We complete the parallel flow analysis in this lecture and then we solve some more example and skip the rigorous scaling simplification techniques we used in part one. This lecture also may be your first introduction to similarity variables and solutions. In the context of scaling, they arise when there are infinite/symmetric/ or time varying domains. This lack of a definitive scale to assign makes us naturally search for some ratio that looks like an order one expression for all values of space and time. Stokes problems are analyzed as well and we see dynamic modeling of a viscous liquid. The main take away from this two part lecture and over arching theme is fully defined well posed differential equations are solutions. Once you can write down that system of equations you effectively have described the function you're searching for precisely. The process of solving them is really practiced in HW1.

Lecture Notes

ADVANCED FLOWS I

We put the advanced in advanced and start going unsteady, rotational, and more! We preform only the more complicated steps and skim over the more rudimentary details you had to do for simpler flows. Pressure driven flow in a slot or pipe is completely solved right before your eyes. A closed pipe causes us to solve a partial differential equation instead of an ordinary differential equation. We get a solution only by utilizing the linearity of the simplified governing equation. We will also see just how complex things get if we consider oscillating pressure gradients (i.e. heart pumping blood thru your veins!). This part of the lecture is to show you how redefining mathematical expressions is paramount in analysis. The solution expanded fully with every term present is unreadable and uninterpretable. Pure garbage.

Lecture Notes

ADVANCED FLOWS II

Sometimes a solution can be just a set of horribly non-linear odes, that's okay! At this stage you can tag in your compooter buddy to solve the rest. We will look at Hiemenz Flow which is fundamental to assume for our next section of Stream Functions. The Von-Karmen pump is super cool! Finally you will be introduced to the concept of a composite solution and the theory of asymptotic matching. This whole thing of us neglecting term seems to be at odds with us enforcing no-slip condition at solid boundaries. This is fundamental to all fluid problems.

Lecture Notes

STREAM FUNCTIONS I

When one coordinate doesn't matter much in a flow (symmetry, fully developed, or infinity parallel in a dimension) and incompressibility holds a good method of analysis is using stream functions. If the flow is irrotational a potential flow is an available tool for you as well. We see that a streamfunction is not restricted to inviscid flow. In fact they exist for any general 2D planar flow viscous or not. We see by an example! The calculation methods are similar to what is called a potential flow and we look at this distinction.

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STREAM FUNCTIONS II

We now see more uses this concept of an ideal flow which opens up the world of potential flows. What are they? When can they be assumed? The most amazing thing about this flow is there is a calculus so simple to determine the velocity field it seems like cheating. This analysis was heavily used back in the day because it allowed engineers to simply draw velocity fields without ever calculating something. This is one of my favorite discoveries as a student because it explained this stupid graph I always saw but never knew why it looked like that.

Lecture Notes

THE VORTICITY VIEW

We view some problems from the lens of the vorticity of the medium. We will look at some previous solutions we've found and view them in the vorticity view. The entire Navier-Stokes can be put into this framework as well. In fact vorticity view gives us a much more powerful interpretation frame work for complex flows where solutions are far out of reach for the human brain. Vorticity is the dual to velocity. Where a velocity vector talks about little arrows in space, vorticity speaks about little elements of area with normal vectors in space. Kelvin and Helmholtz have some pretty general theorems you should know too.

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THE COOLEST SOLUTION

We take an entire class just going into every detail of the solution for Jeffery-Hamel Flow. This solution is like listening to your favorite album of all time on awesome expensive headphones. I do this lecture to show how in reality as a fluid expert you should really hold onto one problem you love. One that somehow sparks joy in your heart. I have ~130 pages of hand written notes applying every method of fluid analysis and new math concepts I learn to this one problem.

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ASYMPTOTIC METHODS

That trick of posing a series is explained in more detail, but not too much detail. It is one of the best tools in your tool belt and the pinnacle of approximate mathematics. Solve ANY problem...for small epsilon. I truly believe humanity would evolve faster if we had more people getting paid to develop this theory of asymptotic more and have it become the new calculus of early college. The fact that it is a non-unique theory only gives it more power to out maneuver any computer.

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THE BOUNDARY LAYER

The boundary layer equations are derived as a special case of the Navier-Stokes. This is one of the most famous simplifications of the Navier-Stokes equations and highlights the singular nature of our governing equations. The main issue is when any flow gets close to the boundary we need to start obeying no slip. This causes a grab bag of issues. However, we can pose a very general class of problems to solve given some relaxed assumptions. The Falkner-Skan Similarity Solutions are a fantastic way to really learn this similarity solution.

Lecture Notes

LIFE AT LOW REYNOLDS

When the Reynolds number is low the governing equations have a much simpler structure and several solutions are possible. In fact the equation are so nice all solution exist and are unique. At this stage in the class I hope to stress how many of the things we did previously were just a specific cases of this flow. It goes by many names. Sometimes it's called "Creeping Flow" sometimes it's called "Lubrication Approximation", and sometimes it goes by "Stokes Flow." Whatever name it goes by the important part is the equations are linear and reversible.

Lecture Notes

Classic 1977 Paper

LIFE AT HIGH REYNOLDS

Today we talk about Turbulence since it is what we experience in our daily lives. Why is turbulence hard? How does turbulence get past the Navier-Stokes? If it is a very fast flow isn't that just ideal flow and wasn't that supposed to be easy? What is it that a turbulence model is even modeling? Hopefully this lecture gives you the language and nomenclature to study turbulence if you wish. The main take away I wish you to get from this lecture is really...

Lecture Notes

THE LUBRICATION APPROXIMATION

If the Reynolds number is moderate but two length scales are vastly different then a class of problems can be solved. The class of problems helped optimize the fundamentals of journal bearing and sliding mechanical parts. The lubrication approximation is a Stokes flow so solutions are possible and unique, but only for certain geometries can they be written down. The solutions that are available still tell most of the story. A big picture point of this lecture is to show how much easier it is to first derive a very general equation based on the Navier-Stokes, and then to go directly to that new simplified equation instead of starting from the full Navier-Stokes. Knowing when assumptions of equations hold saves you time=$$$.

Lecture Notes

SURFACE TENSION

A liquid has a cohesive force that can drive flows in some cases, this is one of natures most widely used mechanisms to just do stuff. How do the Naiver-Stokes equation changes when an interface is present. In keeping with our box of energy analogy, the major difference is now a surface energy is the dominant energy per unit volume in a system. This amazing consequence is an entire subject of study mathematically, experimentally, and numerically. We already looked at lubrication theory and hopefully you see the natural extension. This theory just allows the surface to deform by some physical laws involving surface tension. That is it for energy too by the way, all heat energy does is alter the thermodynamics field parameters.

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STABILITY AND TRANSITION I

Now we discuss this odd bridge of transitioning from this wonderful creeping stokes world to the turbulent world. What is known about this process is underwhelming and changes almost every decade. Numerical, Experimental, and Mathematically everything seems to be in disagreement. At some point in the flow something becomes too large and large amounts of vorticity must arise making the inaccessible full 3D solution a requirement of analysis. However, amazing recent progress has been made in the problem of turbulence by Susan Friedlander.

Lecture Notes

STABILITY AND TRANSITION II

We continue with this discussion of transition and arrive at the end of line for the basic fundamental story of fluids.

"When nice mathematical assumptions hold solutions will yield,

all which can be measured by non-dimensional numbers however they're revealed,

Who told?! Well scaling of physical parameters and substitution will tell!

Accost your differential terms, they're just one according to a mathematical spell.

When length is small, fluid is thick, or speed is tiny.

My equations become nice and linearly tidy.

The drag becomes proportional to speed and length give or take a couple properties

Things far away feel any disturbance, essential to this flow's qualities.

If the speeds gets out of hand. Fast! Faster! To much! We're turbulent and dicey!

The flow becomes swirly, recant a poem by Richardson, and the energy cascades nicely."

If you live in our length scale though it will probably be turbulent. Computers are helping in this investigation of what turbulence means, but work is equally needed in the experimental and mathematical side of the research.

I'll leave you with one last extra credit. Write down the full Incompressible Navier Stokes equations. Now pretend this is a system of 4 scale algebraic equations for the scales of (U,V,W,P). Now use a computer to investigate this system of algebraic equations. They are approximated and not identically zero, but whatever you've lost all fear at this point and are willing to drive this garbage math home until it conclude super wrong conclusions.. Now go into a turbulence theory book and recreate almost any plot you see using your new scaling power taken to a questionable limit. "Good Night And Good Luck."

Lecture Notes