Asymptotic uniqueness for elastic tube flows satisfying a windkessel condition

Abstract

An improvement is presented of a theorem in [1] on mean square asymptotic uniqueness for viscous flow of an incompressible fluid in an elastic tube. As previously, radial velocity components are ignored and no initial velocity profile is specified. Axial symmetry is dropped, however, and a “windkessel condition for uniqueness” is introduced to replace the former “normality relations”. The windkessel condition is x 2x 1⇒ ∫ A(x 2,t) (u 2) xdA⩽ ∫ A(x 1,t) (u 2) xdA where u is the difference of any two regular flows and A( x,t) is the cross section of the tube at position x. It has a heuristic relation to a condition that the capacity of the tube to absorb changes in square magnitude of the velocity does not increase as the flow proceeds down the tube, and it replaces any necessity to impose artificial boundary conditions at the ends of a tube section. It is found that the mean square difference between any two regular solutions decays exponentially with time, and a relation between a bound for the decay rate and the radius (and the kinematic viscosity coefficient) is studied. The derivation of this decay rate utilizes a well-known variational principle for eigenvalues. All conditions specified are intended to have direct relevance to blood flow in short, relatively straight sections of the trunk of the aorta.