A Stability Estimate for Robin Boundary Coefficients in Stokes Fluid Flows

In this report, we examine the unsteady Stokes equations with non-homogeneous boundary conditions. As an application of a Carleman estimate, we first establish log type stabilities for the solution of the equations from either an interior measurement of the velocity, or a boundary observation depending on the trace of the velocity and of the Cauchy stress tensor measurements on a part of the boundary. We then consider the inverse problem of determining the time-independent Robin coefficient from a measurement of the solution and of Cauchy data on a sub-boundary.

A parallel partition of unity method for the unsteady Stokes equations

Purpose The purpose of this paper is to propose a parallel partition of unity method to solve the time-dependent Stokes problems. Design/methodology/approach This paper solved the time-dependent Stokes equations using the finite element method and the partition of unity method. Findings The proposed method in this paper obtained the same accuracy as the standard Galerkin method, but it, in general, saves time. Originality/value Based on a combination of the partition of unity method and the finite element method, the authors, in this paper, propose a new parallel partition of unity method to solve the unsteady Stokes equations. The idea is that, at each time step, one need to only solve a series of independent local sub-problems in parallel instead of one global problem. Numerical tests show that the proposed method not only reaches the same convergence orders as the fully discrete standard Galerkin method but also saves ample computing time.

The Derivation and Application of Fundamental Solutions for Unsteady Stokes Equations

In this paper, two formulations in explicit form to derive the fundamental solutions for two and three dimensional unsteady unbounded Stokes flows due to a mass source and a point force are presented, based on the vector calculus method and also the Hörmander’s method. The mathematical derivation process for the fundamental solutions is detailed. The steady fundamental solutions of Stokes equations can be obtained from the unsteady fundamental solutions by the integral process. As an application, we adopt fundamental solutions: an unsteady Stokeslet and an unsteady potential dipole to validate a simple case that a sphere translates in Stokes or low-Reynolds-number flow by using the singularity method instead by the traditional method which in general limits to the assumption of oscillating flow. It is concluded that this study is able to extend the unsteady Stokes flow theory to more general transient motions by making use of the fundamental solutions of the linearly unsteady Stokes equations.