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On Volterra Boundary Integral Equations of the First Kind for Nonstationary Stokes Equations

  • Friedrich K. Hebeker
  • George C. Hsiao
Part of the Springer Series in Computational Mechanics book series (SSCMECH)

Abstract

We are concerned here with slow viscous incompressible flow in a container or around a rigid body. The physical situation is described mathematically by the initial- boundary value problem for the nonstationary Stokes equations:
$$\upsilon _t - \nu \Delta \upsilon + \nabla _p = f\,\,{\rm in}\,\Omega _T \,{\rm or}\,\Omega _T^c ,$$
(1.1)
$${\rm div}\,\upsilon = 0\,\,{\rm in}\,\Omega _T \,{\rm or}\,\Omega _T^c ,$$
(1.2)
$$\upsilon = - \upsilon _\infty \,\,{\rm on}\,\partial \Omega _T ,$$
(1.3)
$$\upsilon ,p \to 0\,\,{\rm as}\left| x \right| \to \infty ,t > 0,$$
(1.4)
$$\upsilon = \upsilon _0 \,\,{\rm as}\,t = 0,x \in \Omega \,\,{\rm or}\,x \in \Omega ^c $$
(1.5)
Here denotes υ(t) the vector of uniform onset flow at infinity, w = υ + υ the velocity field, and p the scalar pressure field of the flow, which evolves from an initial state υ0. Further, \(\nu \sim {1 \over { R}e}\) is the dynamic viscosity of the medium. In a body-fixed frame, the flow region is Ω ⊂ 3 or \(\Omega ^C = \mathbb{R}^3 /\bar \Omega\), the boundary ∂Ω of which is assumed as sufficiently smooth (∂Ω ∈ C , e.g.). Further, Ω T = Ω × (0, T) (and so Ω T c , ∂ΩT) where (0, T) denotes a fixed finite time interval. The vector field ƒ includes the exterior forces (and virtually some small convective terms).

Keywords

Boundary Element Method Boundary Integral Equation Pseudodifferential Operator Jump Relation Boundary Integral Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Friedrich K. Hebeker
    • 1
    • 2
  • George C. Hsiao
    • 1
    • 2
  1. 1.Institute of Supercomputing and Applied MathematicsIBM Scientific CenterHeidelbergGermany
  2. 2.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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