consider this lake flowing if i gave you the velocity and pressure at every single point would you be able to predict how the lake flows over time this question points to the navier-stokes equations a set of equations that can describe any fluid you can think of from water to air to honey to so on you might have heard of these equations because they are part of the millennium prize problems a collection of seven incredibly important and difficult mathematical problems a correct solution to any of these seven problems is awarded a million dollar cash prize today let's talk about this particular problem the existence and smoothness of the navier-stokes equation the navier-stokes equations are probably one of the most important partial differential equations for fluid mechanics they're used to forecast the weather model airplanes and rockets and predict water currents despite their extensive usage there's still not that much we know about these equations mathematically speaking but first let's talk about what these equations actually are for the purpose of this video i'm going to make a few assumptions about the fluid we're talking about the first assumption is that the fluid is newtonian there was a really good answer on stack exchange that explained what it meant for a fluid to be newtonian but let me just summarize it for you when we call a fluid newtonian it means that the rate at which we apply some shear stress has no effect on his viscosity an example from this answer on stack exchange is ketchup when we have difficulty removing ketchup from a bottle something we all do is hit the bottom of the bottle and the ketchup comes out with ease looking at ketchup's shear rate versus this costly graph we can see that hitting the bottle which increases the shear rate decreases its viscosity which lets the fluid out the second assumption is that the fluid is incompressible it means exactly what you think if i compress the fluid that is at pressure there is no considerable variation in the volume of the fluid the last assumption is that the fluid is isothermal this just means that as the fluid flows there's no loss again of heat and this takes us to the navier-stokes equations as daunting as they may seem they're actually based on quite well-known physics properties note that when we talk about fluids we typically talk about an infinite decimal volume of fluid and so we're trying to describe the motion of each individual molecule rather than the fluid as a whole this first equation right here tells us that mass is conserved within the fluid the operator here is called the divergence of a vector field the vector field in this case u is the velocity vector field of the fluid a vector field is what you get when you assign every single point in space to a vector vector fields can describe many things from fluids to electric fields to gravitational fields and so on the divergence of a vector field is an operator that tells us how much a point tends to divert vectors away from it consider this vector field it appears that vectors seem to be moving away from the origin this indicates positive divergence similarly this vector field seems to have vectors that flow into the origin this indicates negative divergence numerically we write the divergence as a dot product between the gradient factor and its vector field this fantastic video by 3v1bond explains why so if you're interested check it out in terms of fluids the divergence of a vector field indicates how much or how little a point acts as a source of a fluid if we imagine water in some area it's impossible for the water to simply disappear it could either change forms but mass is never destroyed and thus the divergence across the fluid has to be zero and hence the first navier-stokes equation the second equation is just a rewritten version of newton's second law if you remember newton's second law tells us that the sum of forces acting on a body can be written as its mass times the acceleration considering this for a single molecule of a fluid let's see how we can derive the second navier-stokes equation first of all since we're considering each individual point let's replace mass with density the mathematical reason behind doing this is that to consider each individual point we have to divide by volume and mass by volume is equal to density next let's consider acceleration well we have the velocity of vector field u and we know that the acceleration is just the derivative of the velocity vector field so we can replace a with du by dt a quick side note you might see this part written as del u by del t plus u dot grad u this is essentially the same thing when we expand d u by dt we use the chain rule which gives us this other expression now we need to consider what are all the forces acting on this molecule of a fluid we can break this down into internal forces which are forces exerted by the molecule of the fluid itself and external forces which are forces exerted by some external object the first internal force we take into account is pressure consider drinking from a straw sucking at the end of a straw creates an area of low pressure which forces the drink to move from the area of high pressure which is below to the area of lower pressure this force is dependent on the change in pressure or the pressure gradient which we can write as grad of p the second force we factor in is friction or viscosity if i pour water into a cup the water moves quite fast whereas if i'm pouring honey it moves quite slow this is because there's a lot more friction between the molecules of honey compared to the molecules of water the mathematical way of writing this force for newtonian fluids is this expression right here i'm not going to go into depth as to how this equation arises but i've left a few resources you can use to read up about this and finally to account for the external forces we just denote any external force by the letter f it's pretty common that gravity is the only external force so we sometimes write rho times g instead of f and this is why the navier-stokes equations can model any fluid there's simply two fundamental laws of physics written out for fluids so why do these equations have a million dollar price attached to it well to answer that let's just look at the actual problem from the clay math institute it goes as follows to give reasonable leeway to solvers while retaining the heart of the problem we ask for a proof of one of the following four statements the first two questions ask for a smooth solution to the navier-stokes equations let's break down what this means when we say that a solution is smooth mathematically it means that the solution is differentiable chaotic doesn't mean that it's random it simply means that if i slightly change the initial condition it results in a large change in the outcome and this is where the question gets incredibly difficult sure the navi stokes equations lets us take some state of a fluid and let us predict what will happen in the future but the chaotic nature of the fluids with turbulence makes it incredibly hard to make long-term predictions this is why we can't predict the weather for more than seven days this is also why we can't predict when we're going to have turbulence on an airplane despite this the navy stokes equations are incredibly useful they're used extensively to model aerodynamic objects for example a plane or a car like i mentioned before the weather forecast also relies on the navy stokes equations it still remains one of the most important partial differential equations ever thanks for watching [Music] [Music] you