The Navier-Stokes Equations II — Theory and Numerical Methods: Proceedings of a Conference held in Oberwolfach, Germany, August 18–24, 1991

328 Pages·1992·4.379 MB·English

by  Lenoid K. Antanovskii (auth.), John G. Heywood, Kyûya Masuda, Reimund Rautmann, Vsevolod A. Solonnikov (eds.)

Lecture Notes in Mathematics 1530
Editors:
.A Dotd, Heidelberg
.B Eckmann, hciri~Z
E Groningen Takens,
J. G. Heywood K. Masuda
R. Rautmann V.A. Solonnikov ).sdE(
The Navier-Stokes
Equations II - Theory and
Numerical Methods
Proceedings of a Conference held in
Oberwolfach, Germany, August 18-24, 1991
galreV-regnirpS
Berlin Heidelberg NewYork
London Paris Tokyo
Kong Hong Barcelona
Budapest
Editors
John G. Heywood
Department of Mathematics
University of British Columbia
Vancouver B. C. V6T 1Y4, Canada
KyfiMyaas uda
Department of Mathematics
Rikkyo University
3-34-1 Nishi-Ikebukuro, Toshimaku
Tokyo, Japan
Reimund Rautmann
Fachbereicb Mathematik-Informatik
Universit~it-Gesamthochschule Paderborn
Warburger Str. 001
W-4790 Paderborn, Germany
Vsevolod A. Solonnikov
.tS Petersburg Branch of .V A. Steklov
Mathematical Institute of the
Russian Academy of Sciences
Fontanka 27, St. Petersburg, Russia
Mathematics Subject Classification (1991): 00B25, 35Q30, 35Q35, 35R35, 35S10,
35S15, 35B40, 35B45, 35C15, 35D10, 60M15, 60M30, 65M06, 65M12, 65M25,
65M60, 65M70, 76D05, 76D07, 76U05
ISBN 3-540-56261-3 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-56261-3 Springer-Verlag New York Berlin Heidelberg
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Printed in Germany
Typesetting: Camera-ready by author/editor
46/3140-543210 - Printed on acid-free paper
This volume is dedicated
to the memory of
Professor Charles J. Amick
(* 1951, t 1991).
Preface
The Navier-Stokes equations have long been regarded with scientific fascination for the
wide variety of physical phenomena that come within their governance. But the rigorous
mathematical treatment of these phenomena remained out of reach until quite recently,
for the lack of a sufficiently developed basic theory of the equations. During the first half
of this century, new roads were opened to a basic theory by the pioneering works of Os-
een, Odqvist, Leray, and Hopf. This research accelerated during the fifties and sixties, and
finally, during the last twenty five years, the basic theory of the equations has developed
and matured to a point that makes possible its application to the rigorous understanding
of many widely diverse phenomena. Researchers are now undertaking the study of flows
with free surfaces, flows past obstacles, jets through apertures, heat convection, bifurca-
tion, attractors, turbulence, etc., on the basis of an exact mathematical analysis. At the
same time, the advent of high speed computers has made computational fluid dynamics
a subject of the greatest practical importance. Hence, the development of computational
methods has become another focus of the highest priority for the application of the math-
ematical theory. It is not surprising, then, that there has been an explosion of activity in
recent years, in the diversity of topics being studied, in the number of researchers who
are involved, and in the number of countries where they are located.
Inevitably, it has become difficult for researchers in one area to keep up with even basic
developments arising in another. The Navier-Stokes theory is beginning to suffer the
same branching into separate and isolated streams that has befallen twentieth century
mathematics as a whole. The organizers of the Oberwolfach meetings on the Navier-
Stokes equations of 1988 and 1991 have endeavored to bring together leading researchers
from all parts of the world, and from all areas of research that are intimately connected
with the basic mathematical theory of the equations. They also included representatives
from the engineering community, who presented experimental and numerical works of
high theoretical interest.
These proceedings contain most of the new results presented during the conference and,
in addition, some contributions quite recently given from participants to the conference's
field.
Vancouver, Tokyo, Paderborn, and St. Petersburg, June 1992.
J. G. Heywood, K. Masuda, R. Rautmann, V. A. Solonnikov.
CONTENTS
Preface
Free boundary problems
Antanovskii, L.K.:
Analyticity of a free boundary in plane quasi-steady flow of a liquid form subject to
variable surface tension ..................................................................................................... 1
Socolowsky, J.:
On a free boundary problem for the stationary Navier-Stokes equations with a dynamic
contactl ine ........................................................................................................................ 71
Solonnikov, V.A.:, Tani, A.:
Evolution free boundary problem for equations of motion of viscous compressible
barotropic liquid ................................................................................................................ 30
Wolff, M.: Heat-conducting fluids with free surface in the case of slip-condition on the walls ....... 56
Problems in unbounded domains
Borchers, W., Miyakawa, T.:
On some coercive estimates for the Stokes problem in unbounded domains ................... 71
Chang, H." The steady Navier-Stokes problem for low Reynolds number viscous jets into a half
space ................................................................................................................................. 85
Farwig, R., Sohr, H.:
An approach to resolvent estimates for the Stokes equations in Lq-spaces ...................... 97
Galdi, G.P.: On the Oseen boundary-value problem in exterior domains ........................................... 111
Salvi, IL: The exterior problem for the stationary Navier-Stokes equations: on the existence and
regularity ........................................................................................................................ 132
Schonbek, M.E. :
Some results on the asymptotic behaviour of solutions to theN avier-Stokes equations.146
Wiegner, M.: Approximation of weak solutions of the Navier-Stokes equations in unbounded domains.
161
XI
Numerical methods
Rannacher, :.R
On Chorin's projection method for the incompressible Navier-Stokes equations ........ 167
Siili, E., Ware, A :.
Analysis of the spectrai Lagrange-Galerkin method for the Navier-Stokes equations...184
Varnhorn, W.:
A fractional step method for regularized Navier-Stokes equations ................................ 196
Wetton, B.T.R.:
Finite difference vorticity methods ................................................................................ 210
Statistical methods
Fursikov, A.V.:
The closure problem for thec hain of the Friedman-Keller moment equations in the case
of large Reynolds numbers .......................................................................................... 226
Inoue, A.: A tiny step towards a theory of functional derivativeeq uations - A strong solution of the
space-time Hopf equation .............................................................................................. 246
General qualitative theory
Grubb, G.: Initial value problems for the Navier-Stokes equations with Neumann conditions .......... 262
Mogilevskii, I.:
Estimates in C 1,12 for solution of a boundary value problem for the nonstationary Stokes
system with a surface tension in boundary condition ........................................................ 284
Schmitt, B.J., Wahl v.,W.:
Decomposition of solenoidal fields into poloidal fields, toroidal fields andt he mean flow.
Applications to the Boussinesq-equations ......................................................................... 291
Walsh, O.: Eddy solutions of the Navier-Stokes equations ................................................................. 306
Xie, W.: On a three-norm inequality for the Stokes operator in nonsmooth domains .................. 310
List of participants 317
ANALYTICITY FO A FREE NI YRADNUOB PLANE QUASI-STEADY WOLF
FO A LIQUID FORM TCEJBUS OT ELBAIRAV ECAFRUS NOISNET
Leonid K. ANTANOVSKII
Lavrentyev Institute of Hydrodynamics
630090 Novosibirsk, Russia t
Abstract - The plane quasi-steady flow of an incompressible
viscous fluid completely bounded by a free surface and driven
solely by variable surface tension, is analyzed. The mathemati-
cal problem is decomposed into an auxiliary elliptic problem
for Stokes equations in a fixed region with an imposed dynamic
condition, whose solution leads to a Cauchy problem for the
free boundary governed by a kinematic condition. Using the bi-
analytic stress-stream function and time-dependent conformal
mapping of the unit disk onto the flow domain sought, the auxi-
liary problem is reduced to a Fredholm boundary integral equa-
tion for the normal velocity of the free boundary. The existen-
ce theorem is obtained in a class of analytic curves prescrib-
ing the free-boundary position.
Introduction. The quasi-steady approximation can be applied for
description of viscous flows in domains of small size in one,
or several directions, such as films, capillaries, and drop-
lets, where the surface-tension and viscosity forces are domi-
nant over the inertia and gravity ones. This approach leads to
a non-standard problem for the Stokes system, which can be for-
mally reduced to a Cauchy problem for the free boundary, invol-
ving the so-called "normal velocity" operator [2-4]. To define
this operator, it is necessary to solve an auxiliary problem
for Stokes equations, in which the free boundary is prescribed
as a known smooth curve, and the kinematic condition is tempo-
rarily ignored.
Introducing the time-dependent conformal mapping of the unit
tpresent address: Microgravity Advanced Research and Support
center, Via Diocleziano 328, 80125 Naples, Italy
disk of a parametric plane onto the flow domain, a boundary
integral equation for solution of the auxiliary problem is con-
structed. This procedure is based on the preliminary considera-
tion of two Schwartz problems for analytic functions in the
unit disk in combination with an appropriate normalization of
the Stokes solution to avoid appearance of a rigid-body veloci-
ty 4. As a result, a Fredholm equation is constructed direct-
ly for the normal velocity of the free boundary which defines
its evolution by virtue of the kinematic condition.
The intention is to obtain the Cauchy problem in the special
case of the flow of a liquid form completely bounded by the
free surface and driven solely by capillarity. The existence
theorem is formulated in the class of analytic curves prescrib-
ing the free boundary position. In particular, analyticity of
the free boundary takes place regardless to variable (non-
smooth) surface tension which is assumed to be a given function
of the parametric plane.
For steady-state flows governed by the Navier-Stokes equati-
ons with constant surface tension, there was proved infinite
smoothness of the free boundary in 18 (2D case), and analyti-
city in i (2D case) and 6 (3D case). The solvability and
large-time regularity theorems for an unsteady surface-tension-
driven flow were obtained in 21, 22. The quasi-steady evolu-
tion of the shape of a liquid form was analyzed theoretically
and numerically in 12, 14.
.I Formulation. Let a fluid of constant viscosity ~ and variab-
le surface tension ~ occupy a domain ~ bounded by the free sur-
face F = a~. The quasi-steady problem implies finding the doma-
in ~(t), velocity v(z,t), and pressure p(z,t), depending on
time t and a point z • ~(t), as the solution of the following
system of equations,
Vp = ~Av, V.v = 0 in ~, (i.i)
pn - ~(n-Vv + vv'n) = ?F.(~VFZ) on F, (1.2)
V = v.n on F, (1.3)
n
= Q, at t = 0. (1.4)
Here n V is the normal speed of F along its inward normal n; ~,,
the given initial position of ~; the dot is used to denote the
inner product.
Due to the well-known formula AFz = CFn (C F is the curvature
of F), the dynamic (force balance) condition (1.2) yields the
Laplace formula p = GC F at equilibrium. In the general case of
variable G, the tangential stress VEG induces a flow caused by
capillarity. For instance, thermocapillary convection is the
result of dependence of G on temperature [i0] for which one is
able to formulate a boundary-value problem and solve it in as-
sociation with the equations (1.1)-(1.4). To concentrate our
attention on the hydrodynamic part of the problem, the given
function G = G(z,t) will be assumed for definiteness. The case
of constant surface tension is also of interest [12, 14].
2. Stress-stream function. For description of plane flows, let
us identify each two-dimensional vector with a complex number.
In particular, we introduce z = x + iy, v = v + iv instead
x y
of z = (x,y), v = (Vx,Vy) . Defining the differential form
F(dz) = i[p(z,t)dz + 2~@v(z't)dz],
a~
we rewrite the dynamic condition (1.2) for the solenoidal velo-
city field as F(dz) = d G(z,t)~--~ where s is arc length. We
have adopted that the inward normal is connected with the tan-
.Oz
gential vector by the formula n = x~-~ which corresponds to the
usual choice of the orientation of a~ (the positive direction
of the contour a~ leaves a on the left). Herein and throughout
the overbar denotes the complex conjugate.
By the definition, a complex-valued function w = ~ + i@ is
bianalytic if it satisfies the equation a2w/az 2 = 0 [8]. The
latter is equivalent to the well-known Goursat representation
w(z,t) = Wo(Z,t ) + Wl(Z,t)z where Wo(Z,t ) and Wl(Z,t ) are ana-