Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-17T23:16:10.876Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamic boundary conditions for the Navier–Stokes equations

Published online by Cambridge University Press:  14 November 2011

Maríe Grobbelaar-Van Dalsen  and
Niko Sauer
Maríe Grobbelaar-Van Dalsen
Affiliation:
Department of Mathematics, Applied Mathematics and Astronomy, University of South Africa, P.O. Box 392, Pretoria 0001, South Africa
Niko Sauer
Affiliation:
Faculty of Science, University of Pretoria, Pretoria 0002, South Africa
Get access Rights & Permissions [Opens in a new window]

Synopsis

When a symmetric rigid body performs a rotation in a fluid, the system of governing equations consists of conservation of linear momentum of the fluid and conservation of angular momentum of the rigid body. Since the torque at the interface involves the drag due to the fluid flow, the conservation of angular momentum may be viewed as a boundary condition for the field equations of fluid motion. These equations at the boundary contain a time derivative and thus are of a dynamic nature. The familiar no-slip condition becomes an additional equation in the system which not only governs the fluid motion, but also the motion of the rigid body. The unknown functions in the system of equations are the velocity and pressure fields of the fluid motion and the angular velocity of the rigid body.

In this paper we formulate the physical problem for the case of rotation about an axis of symmetry as an abstract ordinary differential equation in two Banach spaces in which the velocity field is the only unknown. To achieve this, a method for the elimination of the pressure field, which also occurs in the boundary condition, is developed. Existence and uniqueness results for the abstract equation are derived with the aid of the theory of B-evolutions and the associated theory of fractional powers of a closed pair of operators.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 1-2 , 1989 , pp. 1 - 11
Copyright
Copyright © Royal Society of Edinburgh 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Agmon, S.. Lectures on Elliptic Boundary Valve Problems (New York: Van Nostrand, 1965).Google Scholar
2Fujita, H. and Kato, T.. On the Navier–Stokes initial value problem, I. Arch. Rational Mech. Anal. 16 (1964), 269–315.CrossRefGoogle Scholar
3Giga, Y.. The non-stationary Navier–Stokes system with some first order boundary condition. Proc. Japan Acad. Ser. A 58 (1982), 101–104.CrossRefGoogle Scholar
4Dalsen, M. Grobbelaar-Van. Fractional powers of a closed pair of operators. Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), 149–158.CrossRefGoogle Scholar
5Dalsen, M. Grobbelaar-Van. Semilinear evolution equations and fractional powers of a closed pair of operators. Proc. Roy. Soc. Edinburgh Sect. A 105 (1987), 101–115.CrossRefGoogle Scholar
6Hopf, E.. Über die Anfangswertaufgabe för die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951), 213–231.CrossRefGoogle Scholar
7Kato, T. and Fujita, H.. On the nonstationary Navier–Stokes system. Rend. Sent. Mat. Univ. Padova 32 (1962), 243–260.Google Scholar
8Kiselev, A. A. and Ladyzhenskaya, O. A.. On the existence and uniqueness of the solution of the non-stationary problem for an incompressible viscous fluid. Izv. Akad. Nauk USSR 21 (1957),655–680.Google Scholar
9Ladyzhenskaya, O. A.. The Mathematical Theory of Viscous Incompressible Flow (New York: Gordon & Breach, 1963).Google Scholar
10Lions, J.-L.. Equations différentielles opérationnelles et problèmes aux limites (Berlin: Springer, 1961).CrossRefGoogle Scholar
11Lions, J.-L. and Magenes, E.. Problèmes aux limites non homogènes et applications, Vol. 1 (Paris: Dunod, 1968).Google Scholar
12Miyakawa, T.. The L p approach to the Navier–Stokes equations with the Neumann boundary condition. Hiroshima Math. J. 10 (1980), 517–537.CrossRefGoogle Scholar
13Odquist, F. K. G.. Über die Randwertaufgaben der Hydrodynamik zäher Fliissigkeiten. Math. Z. 32 (1930), 329–375.CrossRefGoogle Scholar
14Sauer, N.. Linear evolution equations in two Banach spaces (Pretoria University Department of Applied Mathematics, Research Report UPTW 17, 1980).Google Scholar
15Sauer, N.. Linear evolution equations in two Banach spaces. Proc. Roy. Soc. Edinburgh Sect. A. 91 (1982), 287–303.CrossRefGoogle Scholar
16Sauer, N.. The Friedrichs extension of a pair of operators. Quaestiones Math, (to appear).Google Scholar
17Sauer, N.. The steady state Navier–Stokes equations for incompressible flows with rotating boundaries. Proc. Roy. Soc. Edinburgh Sect. A. 110 (1988), 93–99.CrossRefGoogle Scholar
18Serrin, J.. Mathematical principles of classical fluid mechanics. In Handbuch der Physik 8/1 pp. 125–263 (Berlin: Springer, 1959).Google Scholar
19Temam, R.. Navier–Stokes Equations. Theory and Numerical Analysis (Amsterdam: North Holland, 1979).Google Scholar