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:
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 ⊂ ℝ 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 c T , ∂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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Brebbia, C. A., The solution of time-dependent problems using boundary elements, Ed. Whiteman J. R., The Mathematics of Finite Elements and Applications, 5 (1985), 229–255.
Costabel, M., Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal, (1988), 613–626.
Costabel, M., Onishi, K., Wendland, W. L., A boundary element collocation method for the Neumann problem of the heat equation. Inverse and Ill-posed Problems, (1987), 369–384.
Costabel, M., Boundary integral operators for the heat equation. Integral Equ. Oper. Theory (1990).
Costabel, M., Wendland, W. L., Strong eUipticity of boundary integral operators, J. Reine Angew. Math., 372(1986), 39–63.
Fabes, E. B., Lewis, J. E., Riviere, N. M., Boundary value problems for the Navier-Stokes equations, Amer. J. Math., 99 (1977), 626–668.
Fischer, T. M., An integral equtions procedure for the exterior 3D slow viscous flow. Integral Eqn. Oper. Th., 5 (1982), 490–505. 273–297.
Hebeker, F. K., Efiicient boimdary element methods for 3D exterior viscous flows. Num. Meth. PDE, 2 (1986), 273–297.
Hebeker, F. K., Characteristics and boundary elements for 3D Navier Stokes flows, The Mathematics of Finite Elements and Applications, Ed. J. R. Whiteman, 6, (1988), 305–312.
Hebeker, F. K., On Lagrangean and unsteady boundary element methods for in compressible Navier Stokes problems, The Navier Stokes Equations - Theory and Numerical Methods, Ed. R. Rautmann, (Oberwolfach, September 19–23, 1988), to appear.
Hebeker, F. K., Hsiao, G. C., On a boundary integral equation approach to a nonstationary problem of isothermal viscous compressible flows, Preprint 1134, Fb Mathematik, (May, 1988), Technische Hochschule Darmstadt.
Hsiao, G. C., Kress R., On an integral equation for the 2D exterior Stokes problem, Applied Numer. Math., 1(1985), 77–93.
Hsiao, G. C., Saranen, J., Integral equation solution of some heat conduction problems, interal Equations and Inverse Problems, Eds. V. Petkov and R. Lazarov (1991), 107–113.
Hsiao, G. C., Saranen, J., Coercivity of single layer heat operator, SI AM Math. Anal, Submitted.
Hsiao, G. C., Wendland, W. L., A finite element method for some integral equations of the first kind, J. Math. Anal. Appl, 58 (1977), 449–481.
Hsiao, G. C., Wendland, W. L., The Aubin-Nitsche lemma for integral equations, J. Integral Eqn., 3 (1981), 299–315.
Ladyzhenskaja, O. A., The Mathematical Theory of Viscous Incompressible Flows, 1969, New York.
Ladyzhenskaja, O. A., Solonnikov, V. A., Uralzewa, N. N., Linear and Quasilinear Equations of Parabolic Type, 1968, Providence.
Leis, R., Initial Boundary Value Problems in Mathematical Physics, 1986, John Wily & Sons, and B. G. Teubner, Stuttgart, New York.
Lions, J. L., Magenes, E., Non-homogeneous Boundary Value Problems and Applications, Vol. 2, 1972, Berlin.
Nedelec, J. C., Planchard, J., Une methode variationelle d’elements finis pour la resolution numerique d’un probleihe exterior dans R, RARI0, R-8, 7 (1973), 105–129.
Noon, P. J., The single layer heat potential and Galerkin boundary element methods for the heat equation, Ph.D. thesis, 1988, 108 pp. University of Maryland.
Piskorek, A., Zabrodski, E., Uber die instationaären hydrodynamischen Potentiale, ZAMM, 60 (1980), T267–269.
Solonnikov, V. A., Estimates of the solutions of a nonstationary linearized system of Navier Stokes equations, AMS Transl. Ser. 2, 75, (1968).
Solonnikov, V. A., Estimates for solutions of nonstationaxy Navier Stokes equations, J, Soc, Math, 8 (1977), 467–529.
Temam, R., Navier Stokes Equations, 1977, Amsterdam.
Zhu, J., A boundary integral equation method for the stationary Stokes problem in 3D, Boundary Elements, Ed. C. A. Brebbia, 5 (1983), 283–292, Berlin.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hebeker, F.K., Hsiao, G.C. (1993). On Volterra Boundary Integral Equations of the First Kind for Nonstationary Stokes Equations. In: Kane, J.H., Maier, G., Tosaka, N., Atluri, S.N. (eds) Advances in Boundary Element Techniques. Springer Series in Computational Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-51027-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-51027-4_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-51029-8
Online ISBN: 978-3-642-51027-4
eBook Packages: Springer Book Archive