Manifolds in Fluid Dynamics

In studying fluid dynamics it is useful to employ two different perspectives of a fluid flowing through a domain D. The Eulerian point of view is to consider a fixed point x ∈ D, and observe the fluid flowing past. The Lagrangian point of view is to consider a fixed but arbitrary volume of fluid, called a parcel of fluid, and follow the parcel as it travels with the flow. One would think that over time one of these two descriptions would win out over the other, but that has not been the case. The reason for this is that while the Lagrangian description holds some mathematical and conceptual advantages, it is difficult to apply to physical flows – measurements are often taken at fixed points in the real world, matching the Eulerian description [3]. Regardless of our choice of perspective, we must make some preliminary assumptions. We assume that the domain D is a continuum; a nonempty, compact, connected metric space. This assumption can be thought of as ignoring the molecular structure that is incumbent to any physical fluid (liquid or gas). Due to the nature of molecules, if we tried to follow the flow of a single molecule of fluid it would act very chaotically, and would not be representative of the fluid as a whole. To avoid complications of this sort, we restrict ourselves to dealing with small volumes of fluid that are not “too small”; hence the need to follow a parcel through the flow, rather than just a point. In [3], Stephen Childress gives reasonable lower bounds for the diameter of a parcel: 10−7 cm for liquids, 10−3 cm for gases. The following definitions will prove useful. A Riemannian metric (or metric tensor) is a positive definite symmetric bilinear form g on the tangent bundle of a manifold. A Riemannian manifold is a manifold M together with a Riemannian metric, usually written as a pair (M, g). Given a Riemannian manifold, the tangent and cotangent bundles are naturally isomorphic, TM∼= T ∗M : v 7→ g( , v).