Derivation of the NavierStokes Equations


Jousef Murad


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classical mechanics the father of physics and perhaps of scientific thought was initially developed in the 1600s by the famous natural philosophers of the 17th century such as Isaac Newton building on the data and observations of astronomers including Tycho Brahe hey Galileo and Johannes Kepler classical mechanics concerns itself with a mathematical description of the motion of physical bodies tying together the concepts of force momentum velocity and energy to describe the behavior of macroscopic objects though it was developed nearly 400 years ago many of the basic tenets of classical mechanics hold for common situations excluding microscopic particle dynamics high velocity motion and large-scale mechanics classical mechanics holds accurately four scales from 1 picometer which is 2 to the power of minus 12 meters to 10 to the 30 meters due to its consistent success classical mechanics has been widely studied by physicists and mathematicians alike even though it must rely on quantum mechanics for small scale motion and special relativity for high velocity travel it is considered a mostly complete and solved set of theories however there is still one problem in classical mechanics which remain unsolved the solution in fact with a solution is guaranteed to exist to the general case of the navier-stokes equations for fluid dynamics is unknown and now ladies and gentlemen have fun with an introduction to the derivation of the navier-stokes equations hey guys welcome to the derivation of the navier-stokes equations and I hope that you like the small intro to classical mechanics and let's jump right into the presentation so here a brief recap of the CFT introduction that I've already made a little bit of history concerning the navier-stokes equations so the motion of fluid is an exploratory topic for human beings and the evolution of mathematical models emerged at the end of the 19th century after the Industrial Revolution the initial appropriate description of the viscous fluid motion had been indicated in the paper Principia by Sir Isaac Newton in 1687 in which dynamic behavior of fluids under constant viscosity was investigated later Daniel Bernoulli in 1738 and Leonid EULA in 1755 subsequently derived the equation of inviscid flow which is now expressed as a so-called oil last enlisted equations even though Claude Louie navier in 1827 August a Louie Cauchy in 1828 simio Denis Poisson 1829 and además have been known 1843 had carried out studies to explore the mathematical model of fluid flow they had overlooked the whiskers slash frictional force in 1845 Sir jock Stokes had derived the equation of motion of a viscous flow by adding Newtonian viscous terms thereby the navier-stokes equations had been brought to the final form which has been used to generate numerical solutions for fluid flow ever since so the fundamental equations for fluid mechanics are the conservation of mass which I have already described in another video in today's video we will talk about the conservation of momentum / the navier stokes equations and we have the conservation of energy which might come soon in case you want to have a derivation of the energy equations this is important if we deal with compressible fluids quick recap of the conservation of mass as you can see right here it's in the integral form of course we have the surface integral in the second term but as you know from the mass conservation derivation we can transform the surface integral into the volume integral by using gawss's theorem or the so called divergence theorem here with the conservation of momentum with the term dealing with the change in time the nonlinear convective term then we have pressure and viscous forces as well as extra forces rewriting the equation we get the following form and this is nothing else than Newton's second law of motion please keep that in mind because at the end of the video this will be important again so stick with me so here the note again that in the beginners car for CFD I have missed the velocity term just in case you will watch the video or already watched it so the conservation of mass just to have the quick recap is that the temporal change of mass inside of our control volume is the flows into the control volume minus the flows out of the control volume for the conservation of momentum we have something similar we have that the temporal change of momentum inside the control volume is equal to the momentum flow into the control volume minus the momentum flow out of the control volume plus forces acting on our control volume so we will have a look at a small infinitesimal control volume as depicted on the left again and momentum is defined as mass times velocity so we take the definition of mass which is nothing else then d rho by dt times DX dy DZ and add the velocity to it namely V as a vector so we have the same approach as for the mass conservation but with a definition of the momentum right now so recap for the mass continuity equation again so that you see what approach we took we had a look at what's coming into our control volume on the left side and we had a look how the component in X direction changes along the x direction so for the momentum conservation we just add the velocity and you end up with the following form as you can see on the slide so we work with that in every direction right now so here we have that P times u times u changes its value in x direction with d by DX of Rho times u times u times DX here we have all the Momentum's acting on the x faces so we have three Momentum's acting on each x face so we have six in total with it change along each component respectively the same goes for the y direction or Y faces as well as for that that faces in case we want to have a closer look at the equations just pause the video what I would advise you to do is to take some notes from time to time so you will retain the information better so we move on this is not the only contributions for the changes in momentum so what is actually missing we have normal and shear stresses that are missing and we have body forces that are missing normal and shear stresses so let's talk about them first and there's a convention and the convention is that the first index indicates which surface the stresses acting on and the second index indicates the direction of the coordinate so what does it actually mean that means that if we have an arrow on the top set surface for instance showing into the Y direction we have tau Z Y and the first index is that because it's on the top surface but the error is shown into the Y direction similarly we have tau X Y because it's acting on the x2 FS but pointing into the y direction we have also body forces and these are forces acting throughout the volume of a body for example we have gravitation electric and magnetic forces and some more the body force is indicated by the vector K which can be K X KY and KZ with the units in Newton per cubic meter so here you can see the normal and shear stresses indicated by tau xx tau XY tau XZ and the changes along the direction the same goes for the Y surfaces as well as the set surfaces so we assemble the equations now according to the formula that we have already described so we first take the x-direction for clarification I have colored all the arrows showing in X Direction positive or negative in blue and these are the components that will go into the X equation so if we assemble everything we get the following formula so on the left hand side we have the term as it is on the right hand side we have the momentum coming into our system minus the momentum going out of our system the surface we are dealing with the same goes for Y and the same goes for that then we have of course our body forces and we add to that our normal and shear stresses respectively that means we have for the X direction the following term for the y direction the second term and for the set direction the third term now as you can see these terms cancel out because they are identical and what we end up with is the equations for the X Y and set direction so these are theoretically all the equations we need but we need to answer the following questions question number one is which term includes the thermo dynamical pressure for instance when we look at gases and secondly is there a relationship between stresses tau and velocity components UV and W and yes there is if we assume that we deal with a Newtonian fluid we can say that tau equals mu times dou u by dou Z for instance where mu is the proportionality constant and the u by DZ the gradient just a quick recap maybe you know it already from your lectures what is the stress tensor and how it is defined so with the stress center first of all is a symmetric tensor with six independent components because we have the symmetry condition applied that means that only these six components are independent and this component is identical to this component to Z tau Z X is identical to tau XZ and tau Y Z is identical to tau Z Y so the stress tensor can be split into two parts the first part is the so called hydrostatic stress tensor or volumetric stress tensor this part of the stress tensor changes the volume of the body and we have the so called stress deviator tensor which distorts the body and we can write it like that where the spruce deviator is sij and the stress tensor is P Delta IJ now you might ask what is this Delta IJ this is the so-called Kronecker Delta and it equals 1 if the indices are the same and it equals to zero if the indices are not the same so if we have Delta 1 1 please stays in the equation if we have Delta 2 2 and Delta 3 3 the same happens but if we have Delta 1 2 for instance then the pressure in the stress tensor vanishes for the pressure in an inviscid flow we have no shear stresses and only the normal stresses are acting it's a good approach to take the pressure as the average of the normal stresses and we take the negative direction because the pressure is acting and we have the minus because the pressures acting on our volume meaning that the vector shows to our control volume and not away from it so we can also separate the stress as you can see right here with the equation as you know it from the sides before and we can just rewrite it in terms of s IJ equals Sigma IJ minus P Delta IJ and applied to our notation we can say this is nothing else than tau xx equals to Sigma X X minus P and the same for the Y in that direction and we put it into our equations that we have already derived and what we end up with is the following equations the question now is where this our material law come into play this is the so called Stokes hypothesis and we take these equations now as a postulate of course you can derive these equations and if you are more interested into how to derive the Stokes hypothesis I can either do a video on that or you can find more information in the book from schlichting from 1968 where he explains this in detail so if you put everything to our equations we get the preliminary equations which look like this for x y&z respectively we can use the product rule from the for the left hand side and you end up with the following equations and if you recall from slide 18 of the mass conservation equation that we said that this equation here holds then this last part of the equation is equal to 0 and u times 0 is 0 anyway so this term goes out of the equation and what we then have is the left hand side of the navier-stokes equation looks like this and we can just rewrite it in a more compact form we put out of the equation and we say we have to you by DT plus u times 2 u by DX and so on here you can now see the final general form of the navier-stokes equations and if we deal with an incompressible flow we can just take the assumption that the divergence of our velocity field is equal to zero so that means that the terms as you can see right here drop out and we end up with our final form that looks like this and we can write it in a more pretty form which is the so called vector notation so we can rewrite it in that form that we say Rho times DV by DT plus V times the divergence times V equals K minus the pressure gradient so minus the gradient of P plus mu times the Laplace operator so laplacian of the velocity field and i have rewritten the laplacian operator below so what it actually means and we can even rewrite the laplacian operator as the divergence of the gradient so we have 2 times nabla so to speak which is nothing else than nablus square of the velocity field so if you have any questions regarding that feel free to ask in the comment section so I hope this is clear so far let's move on so for an incompressible flow we have two fundamental equations which look like this we have course our navier-stokes equations plus the mass continuity equation as you can see on the bottom so what we get two equations that build a system of for partial nonlinear differential equations of second order where we have four unknowns namely u V and W as well as our pressure P so we can even rewrite the terms in the brackets of our navies Stokes equation we have something called the lagrangian reference frame and the Alerian reference frame so you can see on the left hand side where we have the so-called substantial derivative which is nothing as a Lagrangian reference frame and on the right hand side how we derived it is the so-called Alerian frame we have the local change in red and the convective term in purple so here again as is the substantial temporal change where we is nothing else at the Lagrangian approach and on the right hand side we have the local temporal change at a fixed location plus the convective spatial change due to convection from one place to another which is nothing else than the OL Aryan frame so for more information have a look at this slide right here so if we have Lagrangian reference frame we follow our fluid particle and if we do that we get something called the pass line and if we are in our layering system we have a fixed volume and we look at the volume and how the flux is changed inside of the volume over time which is nothing else then the oil arian frame as I mentioned we can determine something called the streamlines that you might already know from your CFD simulations also please keep in mind that it's mathematically enormous ly more convenient to use the or Larian reference frame and here we have the beautiful navy a stokes equations again where we have the mass indicated by the blue row then we have the acceleration term and that's equal to the force and as you can see we have mass times acceleration equals force and again that's nothing else than Newton's second law of motion so that's it so far as future potential video topics we have something about turbulence that we can have a small beginner's guide about turbulence what turbulence is how you can define it we can even talk about the navier-stokes equation not only what the million dollar question is but also about forms of the name is dr. equation for instance the comparison between compressible incompressible flow visit versus inviscid laminar was turbulent and steady versus unsaid in what it actually means in terms of the navier-stokes equation then of course you guys also wanted to have video about the world functions once by plus that's also very interesting then a very cool topic and very easy to understand in my opinion is if we derive the stability criteria when we have a look at the so called peclet number in the convection diffusion problem that can be programmed using MATLAB and then I can explain you what you can interpret into this stability criteria also a nice video would be how to properly manage the CFD project also thought about doing a simulation Sunday we can take some either FAA or CFD simulations and have a look at them of course as always if you have any questions or ideas from your site please put them down in the comment section and of course all slides can always be downloaded from my patreon page the other sources and I would say that's it for today's video I hope that you enjoyed it and if you have any questions as I already mentioned please put them down in the comment section make sure not to forget to hit the like button as always please leave a comment down below because you know the algorithm prefers videos if they have more likes and comment and as always make sure to keep engineer on your mind see you on the next one [Music] [Music]