1. Introduction
Existence of an equilibrium surface for an isolated compressible liquid mass rotating about a fixed axis was first proved in [
1]. Our aim is to prove the existence of equilibrium figures for a rotating compressible two-layer fluid.
The problem of the rotation of an isolated incompressible liquid mass about a fixed axis as a rigid body was considered by many famous mathematicians, including Newton, Maclaurin, Jacobi, Kovalevskaya, Lyapunov, Poincare, and others [
2,
3,
4], who mainly studied the movement without capillarity. The capillary fluids were first investigated by Globa-Mikhailenko [
5], Boussinesq, and Charrueau in the beginning of 20th century. The latter gave a detailed analysis of the problem, calculated the shape of equilibrium figures, including the toroidal case, and considered some aspects of the stability [
6,
7]. These results were included in a big review on this subject, presented in the book of Appell [
8]. The stability problem for various ellipsoidal equilibrium figures is analyzed in monograph [
9].
The existence of equilibrium figures for a two-layer incompressible self-gravitating capillary liquid (oblate embedded spheroids) was obtained by V.A. Solonnikov in [
10].
Now we state, in a complete setting, the problem on unsteady motion of two compressible barotropic fluids of finite volume separated by a closed unknown interface.
At the initial instant
, let a fluid with dynamic viscosities
,
be in a bounded domain
, and in the domain
, surrounding it, let there be a fluid with dynamic viscosities
,
;
The domain is bounded by the free surface and includes the closed interface ; are given. This two-component cloud rotates about the vertical axis with an angular velocity .
For
, it is necessary to find the surfaces
,
, as well as the velocity vector field
and the density
of the fluids satisfying the diffraction problem for the Navier–Stokes system
where
is the stress tensor,
is the double strain rate tensor, and
is the identity matrix;
,
are step functions of dynamic viscosities, equal to
,
in
and
,
in
, respectively;
is the fluid pressure given by a known smooth density function;
and
are initial distributions of velocity and density of the liquids, and
is the outward normal vector to boundary union
;
are twice the mean curvatures of the surfaces
(moreover,
at points of convexity
towards
);
are surface tension coefficients on
and
, respectively; and
is the rate of evolution of
in the direction
. We assume that the Cartesian coordinate system
is introduced in the space
. The central dot denotes the Cartesian scalar product.
We mean summation over repeated indices from 1 to 3 if they are denoted by Latin letters, and from 1 to 2 if they are Greek. We mark vectors and vector spaces in bold. The notation denotes the vector with the components , .
The kinematic boundary condition excludes mass transfer across fluid boundaries. It follows from our assumption that the fluid particles do not leave the boundaries during the time.
The evolution problem for two viscous compressible immiscible liquids with an unknown interface belongs to the class of free boundary problems being intensively studied. The theory of these problems for the Navier–Stokes equations has only been in development for about three to four decades, although their setting goes back to the classical works of the 19th century.
The main difficulty of such nonlinear problems is due to the fact that the surfaces of the fluids are unknown. Another obstacle is surface tension. So, most of the authors study the problem without the capillarity. The case where surface tension on a free boundary is taken into account is essentially more difficult to investigate because the capillary forces generate noncoercive boundary conditions. The latter do not allow us to apply the methods developed for the classical problems with fixed boundaries. Interface conditions in system (
1) follow from the continuity of a velocity vector field and momentum conservation when passing across the interface between the media. Similar conditions appear on the free boundary. One can find the explanations, for example, in the textbook of Pukhnachov [
11] or in [
12].
First, local (in time) solvability was proved for a problem similar to (
1) in the whole space
, with a closed interface between the fluids. The result was obtained both in the Sobolev–Slobodetskiǐ and Hölder classes of functions [
13]. One can prove similar results for a two-component domain bounded by a free surface if one takes into account the estimates for a model problem in a half-space [
14]. In the paper of the Japanese researchers [
15], solvability theory was developed for the problem in the anisotropic Sobolev spaces
, with different orders of summability with respect to the spatial and time variables. In the same spaces, the authors of [
16] studied a compressible two-fluid problem without surface tension and finding interface between the fluids. They showed local and global solvability of the problem under additional smallness assumptions on initial data in the global case.
The next step of the study will be to prove global unique solvability of the unsteady problem (
1) for small data. If we have equilibrium two-layer figures, we can hope to obtain the convergence of the global solution to a stationary one. Thus, the motion of a two-component compressible liquid mass
slow rotating about a fixed axis will tend during the time to the rigid rotation of the corresponding equilibrium figure
. Another question is the stability of equilibrium figures obtained. A similar study for two viscous incompressible immiscible liquids was published in our recent papers, one of which is [
17].
As we have mentioned, we suppose the liquids to be barotropic, which implies that the pressure p is a known increasing function of the density: . Let, in addition, , .
Next, we assume that equilibrium figures
,
are nearly globular domains with the radiuses
(
), respectively, and the motion of fluids is close to the state of rest, i.e., the velocity is small, and the density differs little from a step function
. This picture is schematically presented in
Figure 1. We denote the balls
by
.
We are going to prove the existence of
and
, the boundaries of the figures
and
, respectively. We follow the plan of paper [
1].
At rest, the bubble consists of the nested spherical two layers
and
with the uniform distributions of densities
and with the piecewise constant pressure:
The masses of the layers are
where
.
Steady motion of a two-layer gaseous body
uniformly rotating about the axis
with a constant angular velocity
is governed by the homogeneous stationary Navier–Stokes equations
(here, the density
and velocity
depend only on
x) and the boundary conditions
where
,
are twice the mean curvatures of
,
, respectively. The last relation follows from the boundary condition
. The pressure
depends on
and should be given by an increasing function.
It is easily seen that the velocity vector field
satisfies (
4) together with the pressure function gradient
where
is the
ith basis vector,
.
First, we consider the simple case when equality (
7) coincides with the following one
whence
and
in
with constants
, because pressure functions can differ from each other by a constant in different domains. These constants can be found from relations (
2):
Let be the unit sphere in with the center in zero, . We suppose to be given by functions on . In addition, let be rotationally symmetric, i.e., they depend only on and , and be even in .
By substituting
given by (
6) and
into boundary conditions (
5), we obtain the equations for the surface
of domain
and for the interface
between the fluids:
The rotationally symmetry implies that do not depend on . It is clear that , since the first two components of are proportional to those of the radius-vector of the circles, being horizontal sections of . Therefore, on .
Obviously, the density is given by the formulas
with arbitrary positive constants
and
. Equations (
9) take the form
One can determine the constants
by prescribing the masses of fluids to be the same as the masses of the nested spherical liquid layers (
3):
We consider the angular momentum to be one more parameter of the problem. It is given:
where
is the
ith rigid rotation vector,
. Then, the angular velocity
is a function of
.
We denote by
,
,
, the Hölder space of functions
f on the sphere
with the norm
where
is the
th derivative of
f, calculated in local coordinates on the subdomain
,
. Under
, we mean the subspace of
, consisting of rotationally symmetric functions that are even with respect to
.
Theorem 1. Let , , and let the data of problem (
4)
, (
5)
be such that condition (
26)
holds. Then, for an arbitrary β satisfying the estimatewith small enough ε, there exists a unique solution to system (
11)–(
13)
. It obeys the inequality