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)


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:
$$\upsilon _t - \nu \Delta \upsilon + \nabla _p = f\,\,{\rm in}\,\Omega _T \,{\rm or}\,\Omega _T^c ,$$
$${\rm div}\,\upsilon = 0\,\,{\rm in}\,\Omega _T \,{\rm or}\,\Omega _T^c ,$$
$$\upsilon = - \upsilon _\infty \,\,{\rm on}\,\partial \Omega _T ,$$
$$\upsilon ,p \to 0\,\,{\rm as}\left| x \right| \to \infty ,t > 0,$$
$$\upsilon = \upsilon _0 \,\,{\rm as}\,t = 0,x \in \Omega \,\,{\rm or}\,x \in \Omega ^c $$
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).


Boundary Element Method Boundary Integral Equation Pseudodifferential Operator Jump Relation Boundary Integral Operator 
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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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