UK Nonlinear News Book Review
Geometric theory of incompressible flows with applications to fluid dynamics
Tian Ma and Shouhong WangPublished by the American Mathematical Society (2005), x+234 pp. ISBN: 0-8218-3693-5
As an applied mathematician with an interest in rigorous mathematical proofs of difficult aspects of fluid dynamics, I approached this book with some interest and enthusiasm. Unfortunately, having now read the book from cover to cover, I feel that I haven't really learned anything new from it. There is plenty of material, there are plenty of proofs and lots of technical details, but I left with the impression that most of the material was a relatively straightforward application of index theory and other known results, specialised to the case where the divergence of the vector field is zero. I also found quite a few typographical errors in some of the simpler descriptions, which did not inspire confidence that the more complex results would be stated without error. In simple domains, as would be encountered by a practising fluid mechanic, it was hard to see that the results went beyond the obvious. On the other hand, many of the theorems apply to flows on compact Riemannian manifolds, though it was also hard to see if any new features arose from the more complex domain, and indeed, whether such domains are likely to arise in practice (no examples were given).
The book beings with a useful "user's guide" , which states the main results: a smooth two-dimensional incompressible vector field on a compact manifold that is a subset of the sphere is structurally stable if and only if all stagnation points are of saddle or centre type, if all interior saddle points are self-connected, and if all boundary saddle points connect to boundary saddle points on the same connected component of the boundary. This result is explained and proved in the first two chapters, including a discussion of no-slip boundary conditions, where one has to be more careful about how streamlines connect to the boundary.
The third chapter dicusses the case of two-dimensional incompressible vector fields on manifolds of non-zero genus: the most common example would be flow on a torus, or equivalently, with periodic boundary conditions. In this case, there is the additional possibility of ergodic streamlines. The main result in this chapter is that no incompressible flow on a torus is structurally stable.
Chapter four turns to the stuctural stability of two-dimensional incompressible vector fields that are solutions of the Navier-Stokes equations. There is a mass of technical detail, and a discussion of an example: the two-dimensional flows that arise from the primary bifurcation in Rayleigh-Benard convection, where some fairly standard results are proved.
Chapter five discusses the conditions required for a flow that is parallel to a boundary to develop a region of recirculation. The chapter includes a "separation equation" that is supposed to locate the point at which separation of a boundary layer occurs. Unfortuately, this equation does not provide a simple condition, but instead involves an integral of the entire solution of the Navier-Stokes equations from the initial condition to the (unknown) separation time, with derivatives of the flow evaluated at the (unknown) position of the separation. It is hard to see how this condition could be useful in practice.
The final chapter presents two applications: two-dimensional flows as would occur in models of oceanic gyres, and flow in a cavity with the velocity imposed on one wall. In the first case, the result is that the number of centres minus the number of interior saddle minus half the number of boundary saddles, plus the number of continents is equal to one. In the second case, the authors noted that, when they solved the equations numerically, there is a time at which there is boundary layer separation.
Overall, I thought this book was disappointing. There isn't much in the book that would be of interest to the applied mathematician working on real (or even idealised) fluid dynamics problems, and (as far as I could tell) the main results seemed to be specialisations of existing theorems to the case of two-dimensional incompressible flows.