Differential geometry on diffeomorphism groups and Lagrangian stability of viscous flows

A differential geometrical formulation of the motion of an incompressible viscous fluid is presented. The geodesic equation on a manifold is extended with an additional term consisting of an endomorphism of the tangent space, and a class of curves defined by the resultant equation is introduced. Based on this extension, the motion of an incompressible viscous fluid is formulated as curves of this class in a diffeomorphism group, and the expression for the variational equation of the curves is derived. The expression is then shown to coincide with the governing equations for the Lagrangian displacement, which interprets the physical meaning of variation vector fields of the curves. The variational equation is reduced to a more simplified form which can be used to study evolution of the distances between fluid particles advected by a given basic flow.