Blog - fluid flows and infinite dimensional manifolds (part 3)

This page is a blog article in progress, written by Tim van Beek. To see discussions of this article while it was being written, visit the Azimuth Forum. For the final polished article, go to the Azimuth Blog.

In part 2 we defined what an ideal incompressible fluid is. And how the equation of motion, Euler’s equation, can be seen as a geodesic equation on $\mathrm{SDiff}(M)$, the infinite dimensional manifold of volume preserving diffeomorphisms. Last time I have promised to talk about the pressure function. I have also mentioned that Arnold used the geometric setup to put a bound on weather forecasts. I will try to fulfill both promises in the next blog post!

But last time I also mentioned that the ideal fluid has serious drawbacks as a model. This is an important topic, too, so I would like to explain this a little bit further first, in this blog post.

So, this time we will talk a little bit about how we can get viscosity, and therefore turbulence, back into the picture. This will lead us to the Navier-Stokes equation. Can ideal fluids, which solve Euler’s equation, model fluids with a very small viscosity? This depends on what happens to solutions when one lets the viscosity go to zero in the Navier-Stokes equation, so I will show you a result that answers this question in a specific context.

I will throw in a few graphics that illustrate the laminar to turbulent transition at boundaries, starting with the one above. These are all from:

• Milton Van Dyke, *An Album of Fluid Motion*, Parabolic Press, 12th edition, 1982.

The equation of motion of an incompressible, (homogeneous), ideal fluid is Euler’s equation:

$\partial_t u + \nabla_u u = - \nabla p$

Ideal fluids are very nice mathematically. Especially potential flows in two dimensions can be studied using holomorphic functions! One could say that a whole “industry” evolved around the treatment of these kinds of fluid flows. It was even taught to some extend to engineers, before computers took over. A very nice, somewhat nostalgic reading is this:

• L. M. Milne-Thomson, *Theoretical Aerodynamics*, 4th edition, Dover Publications, New York, 2011. (Reprint of the 1958 edition.)

The assumption of “incompressibility” is not restrictive for most applications involving fluid flows of water and air, for example. Maybe you are a little bit surprised that I mention air, because the compressibility of air is a part of every day live, for example when you pump up a cycle tire. It is, however, not necessary to include this property when you model air fluid flows that velocities that are significantly lower than the speed of sound in air. The rule of thumbs for engineers seems to be that one needs to include compressibility for speeds around 0.3 Mach, see compressible aerodynamics (Wikipedia).

However, the concept of “ideal” takes viscosity out of the picture and therefore also turbulence and the drag that a body immersed in fluid feels. As I mentioned last time, this is called the **D’Alembert’s paradox**.

The simplest way to introduce viscosity is by considering a **Newtownian fluid**. This is a fluid where the viscosity is a constant, and the relation of velocity differences and resulting shear forces is strictly linear. This leads to the the Navier-Stokes equation for incompressible fluids:

$\partial_t u + \nabla_u u - \nu \Delta u = - \nabla p$

If you think about plastics or honey, for example, you will notice that the viscosity actually depends on the temperature, and maybe also on the pressure and other parameters, of the fluid. The science that is concerned with the exploration of these effects is called **rheology**. This is an important research topic and the reason why producers of, say, plastic sheets, sometimes keep physicists around. But let’s stick to Newtownian fluids for now.

Since we get Euler’s equation if we set $\nu = 0$ in the above equation, the question is, if in some sense or another solutions of the Navier-Stokes equation converge to a solution of Euler’s equation in the limit of vanishing viscosity? If you had asked me, I would have guessed: No. The mathematical reason is that we have a transition from a second order partial differential equation to a first order one. This is usually called a **singular perturbation**. The physical reason is that a nonzero viscosity will give rise to phenomena like turbulence and energy dissipation that cannot occur in an ideal fluid. Well, the last argument shows that we cannot expect convergence for long times if eddies are present, so there certainly is a loophole.

The precise formulation of the last statement depends on the boundary conditions one chooses. One way is this: Let $u$ be a smooth solution of Euler’s equation in $\mathbb{R}^3$ with sufficiently fast decay at infinity (this is our boundary condition), then the kinetic energy $E$

$E = \frac{1}{2} \int \| u \|^2 \; \mathrm{d x}$

is conserved for all time. This it not the only conserved quantity of Euler’s equation, of course. Now let $u$ be a smooth solution of the Navier-Stokes equation in $\mathbb{R}^3$ with sufficiently fast decay at infinity, then we have

$\frac{d E}{d t} = - \nu \int \| \nabla \times u \|^2 \mathrm{d x}$

So the presence of viscosity turns a conserved quantity into a decaying quantity. Since the 20th century, engineers have taken these effects into account following the idea of “boundary layers” introduced by Ludwig Prandtl as I already mentioned last time. Actually the whole technique of singular perturbation theory has been developed following this ansatz. This has become a mathematical technique to get asymptotic expansions of solutions of complicated nonlinear partial differential equations.

The idea is that the concept of “ideal” fluid is good except at boundaries, where effects due to viscosity cannot be ignored. This is true for a lot of fluids like air and water, which have a very low viscosity. Therefore one tries to match a solution describing an ideal fluid flow far away from the boundaries with a specific solution for a viscous fluid with prescribed boundary conditions valid in a thin layer on the boundaries. This works quite well in applications. One of the major textbooks about this topic has been around for over 60 years now and has reached its 10th edition in German. It is:

• H. Schlichting and K. Gersten: *Boundary-Layer Theory*, 8th edition, Springer, Berlin, 2000.

Since I am also interested in numerical models and applications in engineering, I should probably read it. (I don’t know when the 10th edition will be published in English.)

However, this approach does not tell us under what circumstances we can expect convergence of solutions $u_{\nu}$ to the viscous Navier-Stokes equations with viscosity $\nu \gt 0$, to a solution $u_{o}$ of Euler’s equation with zero viscosity. That is, are there such solutions, boundary and initial conditions and a topology on an appropriate topological vector space, such that

$\lim_{\nu \to 0} u_{\nu} = u_{0} \; \text{?}$

I asked this question over at mathoverflow, see here, and got some interesting answers. Obviously, a lot of brain power has gone into this question, and there are both interesting positive and negative results. As an example, I will cite a very interesting positive result. I learned about it from this book:

• Andrew J. Majda and Andrea L. Bertozzi: *Vorticity and Incompressible Flow*, Cambridge University Press, Cambridge, 2001.

It is proposition 3.2. There are three assumptions that we need to make in order for things to work out:

• First, we need to fix an interval $[0, T]$ for the time. As mentioned above, we should not expect that we can get convergence for an unbounded time interval like $[0, \infty]$.

• Secondly, we need to assume that solutions $u_{\nu}$ of the Navier-Stokes equation and a solution $u_0$ of Euler’s equation exist and are *smooth*.

• Thirdly we will dodge the issue of boundary layers by assuming that the solutions exist on the whole of $\mathbb{R}^3$ with sufficiently fast decay. As I already mentioned above, a viscous fluid will of course show very different behaviour at a boundary than an ideal (= nonviscous) one. Our third assumption means that there is no such boundary layer present.

We will denote the $L^{\infty}$ norm by $\| \cdot \|_0$ and use the big Oh notation.

Given our three assumptions, proposition 3.2 says that:

$\mathrm{sup}_{0 \le t \le T} \| u_{\nu} - u_0 \|_0 = O(\nu)$

It also gives the convergence of the derivatives:

$\int_0^T \| \nabla (u_{\nu} - u_0) \|_0 d t = O(\nu^{1/2})$

This result illustrates that the boundary layer ansatz may work, because the ideal fluid approximation could be a good one away from boundaries and for fluids with low viscosity like water and air.

So, when John von Neumann joked that “ideal water” is like “dry water”, I would have said: “Well, that is half right”.