The Navier-Stokes Equations: Part II

Picking up where the last one left off, still dedicated to BK

Hopefully we remember Newton's second law: Force = mass x acceleration. Why can't we apply this to a fluid? Well the mass is no longer a number, that's a density at each point (a scalar field). The force is a vector at each point and as well as forces acting on the fluid the fluid can exert a force on itself. And finally the acceleration can be measured from the Eulerian or Lagrangian perspective and that is also a vector field in either case. But let's try our best. We established the material derivative in the previous post so we can take the material derivative of the velocity field and multiply it by the density at each point to get the right hand side of Newton's 2nd law. What about the force?

In the most general sense we're just going to consider some body force acting on the fluid and leave it there. This is an 'external force' and some common examples are gravity, the Lorentz force arising from electric and magnetic fields (e.g. the plasma in the sun is in a magnetic field) or the coriolis effect arising from fluids on a rotating body, such as the Earth's atmosphere. We won't delve into this any deeper in this post.

What about the 'internal forces'? Well one of these comes from the pressure. This can be easily thought of as the fluid "pressing on itself". At rest this happens in all directions equally so cancels out but when I apply a pressure say to a column of gas with a piston, the gas flows away from the piston: it moves to equalise pressure. Specifically, pressure gradients drive flow. So we expect to the gradient of pressure to appear with the forces. [and in using the word gradient I am absolved of explaining this in any more depth].

With all these terms in place I can now unveil the Euler equations! The letter የ (rho) is the density, Du/Dt is the material derivative of the velocity so is the acceleration, p is the pressure. the triangle is the gradient and f is the force.

And you're thinking: I came here for the Navier-Stokes equations, not another thing named after Euler. That's because the Euler equations only model fluids with no viscosity (inviscid fluids). This works well for gases but not for honey or treacle. I want to try to motivate where the viscosity term comes from. We can think of it as arising from the fluid 'rubbing against itself' and so it must arise from differences in the fluid velocity. If a fluid parcel and its neighbour are moving identically, they won't resist each other. 

Let's derive an expression for the viscosity.[[according to the idea of linear response the viscosity should be linear in the velocity: I don't understand this myself but expect a post when I do]] So the viscosity is a vector (as it describes a force). It can't be proportional to the velocity field as it is arises from local differences i.e. spatial derivatives [[there's a higher level argument involving Lorentz transformations: the force shouldn't change when I move at the same speed as the fluid and view it as stationary]]. The only option containing 1 derivative is the curl but the the curl is invariant under the coordinate transformation x→-x whereas all other terms in the Euler equations pick up a minus sign under this transform so this isn't a contender. The options for vectors we can build out of 2 derivatives are the Laplacian or the gradient of the divergence [[or the curl of the curl which is the difference between these 2!]] but the divergence of an incompressible fluid is 0 and incompressible ≠ inviscid so we're left with the Laplacian (written as triangle squared)! Which does measure the local curvature so feels like something that could represent small discrepancies in velocity. Let's add this in to get the Navier-Stokes equations in all their glory!

We may recall that some of these terms appear in the heat/diffusion equation which describes the fact that heat dissipates in order to smooth out differences. Well the same goes with viscosity except we can think of it as diffusing momentum: if one region of the fluid has a lot of momentum it'll rub against neighbouring regions more and so its momentum will be transferred.

I hope now you are happy with what each of the terms in the Navier-Stokes equations refers to! I have one more post about fluids in the pipeline and it involves e.coli. I've really appreciated the suggestions from BK, RA and FC and welcome anymore ideas for posts! Also I'm going to figure out if I can sort a mailing list so that I don't have to pester people whenever I write a new post.