Russian Mathematics (Iz. VUZ)
Vol. 50, No. 3, pp.47{55, 2006
Izvestiya VUZ. Matematika
UDC 571.988:532
CORRECT DEFINITION OF INITIAL-BOUNDARY-VALUE PROBLEMS
FOR THE GENERALIZED KELVIN{VOIGHT MODEL
M.V. Turbin
Introduction
It is well known 1] that the motion of an incompressible uid in a bounded domain Rn ,
n = 2 3 on a time interval 0 T ], T > 0 is governed by the following system of equations in the
Cauchy form:
@v + v @v + grad p = Div + f (x t) 2 0 T ]
@t i @xi
div v = 0 (x t) 2 0 T ]:
(0.1)
(0.2)
Hereinafter we assume that the summation is performed by the repeated indices. In this system
v(x t) is the velocity vector of a particle at the point x at the moment t and v1 : : : vn are components of v p = p(x t) is the uid pressure at the point x at the moment t f = f (x t) is the
vector of the external forces (they are also called volume ones) applied to the uid. Denote by
Div the vector, whose coordinates are the divergence of the rows of the matrix = (ij (x)),
where is the deviator of the stress tensor. We introduce the deviator of the stress tensor into equation (0.1) in order to take into account the reactions which occur in the uid during the
motion. System (0.1), (0.2) describes the motion of all types of uids, but at the same time it
contains the deviator of the stress tensor which is not explicitly expressed in terms of unknowns
of this system. In order to express the deviator of the stress tensor in terms of the unknowns of
system (0.1), (0.2), one usually uses the correlations between the deviator of the stress tensor, the
tensor of the deformations rates
1
@v
@v
i
j
Eij = Eij (v) = 2 @x + @x
j
i
and their derivatives. Note that establishing the correlation between the deviator of the stress
tensor, the tensor of the deformations rates, and their derivatives, we at the same time dene
the type of the uid. This correlation is named dening or rheological. Rheological correlations
belong to the category of hypotheses which should be veried for concrete uids with the help of
experimental data.
Within the latter hundred and fty years, the model of the Newtonian uid remains the main
object of interest of mathematicians in hydrodynamics. Its rheological correlation has the form
= 2 E , where is the cinematic viscosity coecient. Inserting this correlation into (0.1), (0.2),
we obtain the well-known Navier{Stokes system of equations. It describes the ow with moderate
rates of the most viscous incompressible uids which one meets in practice. But even in the middle
The work was supported by the Russian Foundation for Basic Research (no. 04-01-00081) and the Ministry
of Education of the Russian Federation and CRDF (no. VZ-010-0).
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