GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 6, 1997, 567-578

ON THE BIFURCATION OF FLOWS OF A

HEAT-CONDUCTING FLUID BETWEEN TWO ROTATING PERMEABLE CYLINDERS

L. SHAPAKIDZE

Abstract. Sufficient conditions are found for the bifurcation of flow of a viscous heat-conducting fluid between two rotating permeable cylinders.

This paper deals with second stationary flows generated in a heat-con-ducting fluid contained between two permeable cylinders rotating in the same direction. Among other papers where similar problems are treated mention should be made of [1–4] for the case of a noncompressible fluid and [5] for the case of a heat-conducting fluid. The permeability of the cylinders changes the character of the obtained operator equations, which results in nonsymmetricity of the kernels of the corresponding integral equations. This fact necessitates to another method of investigation of this problem and this is what we do here.

1. Let a homogeneous viscous heat-conducting fluid fill up the hollow space between two rotating permeable cylinders heated up to different temperatures. The radii, angular velocities, and temperatures of the internal and outer cylinders are denoted by R_{1}, Ω_{1}, θ_{1} and R_{2}, Ω_{2}, θ_{2}, respectively. It is assumed that there are no external mass forces, the velocity of the flow across the cross-section of the hollow space between the cylinders is zero, and the fluid inflow through one cylinder is equal to the fluid outflow through the other. The scales of length, velocity, and temperature will be denoted by R_{1}, Ω_{1}R_{1}, θ_{1}, while the density scale will be understood as the fluid density at the temperature θ_{1}. Under these assumptions, if we write the Navier–Stokes equations and heat conductivity equation in terms of cylindrical coordinates (r,φ,z) with the axis z coinciding with the axis

567 of the cylinders, then they will admit the following exact solution with the velocity vector V⃗_{0}(v_{0r}, v_{0φ}, v_{0z}), temperature T_{0}, and pressure Π_{0}:

1991 Mathematics Subject Classification. 76E15.

Key words and phrases. Flow, bifurcation, perturbation, axisymmetric flow, permeable cylinders.

1072-947X/97/1100-0567$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

\begin{matrix}v_{0r} = \frac{κ_{0}}{r} , v_{0φ} = \begin{cases}ar^{κ+1} + b/r, & κ ≠ −2, \\ \frac{a_{1} ln r + 1}{r}, & κ = −2, \end{cases} \\ v_{0z} = 0, T_{0} = c_{1} + c_{2}r^{κ_{1}}, \\ Π_{0} = ∫_{1}^{r} \{[1 − βθ_{1}(c_{1} + c_{2}r^{κ_{1}})](ar^{κ} + \frac{b}{r^{2}} )^{2}r + \frac{κ_{0}^{2}}{r^{3}} \}dr; \end{matrix}  (1.1)

here

a = \frac{ΩR^{2} − 1}{R^{κ+2} − 1}, b = 1 − a, a_{1} = \frac{ΩR^{2} − 1}{ln R},

c_{1} = \frac{θ − R^{κ_{1}}}{1 − R^{κ_{1}}} , c_{2} = \frac{1 − θ}{1 − R^{κ_{1}}} , κ_{0} = \frac{s}{Ω_{1}R_{1}^{2}} , κ = \frac{s}{ν} ,

κ_{1} = \frac{s}{χ}, θ = \frac{θ_{2}}{θ_{1}} , R = \frac{R_{2}}{R_{1}} ,

s is the radial flow per cylinder length unit; β, ν and χ are, respectively, the thermal expansion, kinematic viscosity, and heat conductivity coefficients.

Our task here consists in finding axisymmetric stationary flows which differ from (1.1), are periodic with respect to z with period 2π/α_{0}, and are such that the velocity flow across the cross-section of the cylinder cavity is zero.

2. To find solutions V′, Π′, T′ of our problem in the form V⃗′ = V⃗_{0} + v⃗(v_{r}, v_{φ}, v_{z}), T′ = T_{0} + c_{2}P T, Π′ = Π_{0} + Π/λ, we obtain the following system of perturbation equations:

\begin{matrix}Δv_{r} − \frac{v_{r}}{r^{2}} − \frac{∂Π}{∂r} = λ[(v⃗, ∇)v_{r} − \frac{v_{φ}^{2}}{r} − 2ω_{1}v_{φ} + \\ + \frac{κ_{0}}{r} (\frac{∂v_{r}}{∂r} − \frac{v_{r}}{r} ) + Ra ω_{2}T ], \\ Δv_{φ} − \frac{v_{φ}}{r^{2}} =λ[(v⃗, ∇)v_{φ}+ \frac{v_{r}v_{φ}}{r} −g_{1}(r)v_{r}+ \frac{κ_{0}}{r} (\frac{∂v_{φ}}{∂r} + \frac{v_{φ}}{r} )], \\ Δv_{z} − \frac{∂Π}{∂z} = λ[(v⃗, ∇)v_{z} + \frac{κ_{0}}{r} \frac{∂v_{z}}{∂r} ], \\ ΔT = λP [(v⃗, ∇)T + \frac{κ_{0}}{r} \frac{∂T}{∂r} + g_{2}(r)v_{r}], \\ \frac{∂v_{r}}{∂r} + \frac{v_{r}}{r} + \frac{∂v_{z}}{∂z} = 0; \end{matrix} (2.1)

v⃗|_{r=1,R} = 0, T |_{r=1,R} = 0,  (2.2)

where Ra = βc_{2}θ_{1}P is the Rayleigh number, P = \frac{ν}{χ} is the Prandtl number,

λ = \frac{Ω_{1}R_{1}^{2}}{ν} is the Reynolds number, ω_{1} = \frac{v_{0φ}}{r}, ω_{2} = ω_{1}^{2}r, κ_{1} = κP,

g_{1}(r) = \begin{cases}−(κ + 2)ar^{κ}, & κ ≠ −2, \\ −\frac{a_{1}}{r^{2}} , & κ = −2, \end{cases} g_{2}(r) = κr^{κP −1},

(V⃗ , ∇) = v_{r} \frac{∂}{∂r} + v_{z} \frac{∂}{∂z} , Δ = \frac{∂^{2}}{∂r^{2}} + \frac{1}{r} \frac{∂}{∂r} + \frac{∂^{2}}{∂z^{2}} ,

and the components v_{r}, v_{φ}, v_{z},T must satisfy the following conditions:

∫_{1}^{r} v_{z}(r, z)r dr = 0, V⃗, T are periodic with respect to z with period 2π/α_{0};

v_{r}, v_{φ}, T are odd functions, and v_{z} is an even function with respect to z.

Problem (2.1)–(2.2) is written in terms of the Boussinesq approximation [6] assuming that the flow velocity through the cylinder walls is such that it is not influenced by perturbations arising in the fluid between the two cylinders.

To flow (1.1) there corresponds a trivial solution of problem (2.1)–(2.2) and we assume that for small λ this system has a unique solution v⃗ = T = 0.

The linearized problem corresponding to system (2.1)–(2.2)

\begin{matrix}Δu_{r} − \frac{u_{r}}{r^{2}} − \frac{∂Π_{1}}{∂r} = λ[ − 2ω_{1}u_{φ} + \frac{κ_{0}}{r} (\frac{∂u_{r}}{∂r} − \frac{u_{r}}{r} ) + Ra ω_{2}T_{1}], \\ Δu_{φ} − \frac{u_{φ}}{r^{2}} = λ[ − g_{1}(r)u_{r} + \frac{κ_{0}}{r} (\frac{∂u_{φ}}{∂r} + \frac{u_{φ}}{r} )], \\ Δu_{z} − \frac{∂Π_{1}}{∂z} = λ \frac{κ_{0}}{r} \frac{∂u_{z}}{∂r} , \\ ΔT_{1} = λP [\frac{κ_{0}}{r} \frac{∂T_{1}}{∂r} + g_{2}(r)u_{r}], \\ \frac{∂u_{r}}{∂r} + \frac{u_{r}}{r} + \frac{∂u_{z}}{∂z} = 0; \end{matrix} (2.3)

u⃗(u_{r}, u_{φ}, u_{z})|_{r=1,R} = 0, T_{1}|_{r=1,R} = 0  (2.4)

and the conjugate problem of (2.3)–(2.4) with respect to the scalar product

[u⃗, ψ⃗] = ∫_{1}^{R} ∫_{−π/α_{0}}^{π/α_{0}} u⃗ · ψ⃗ r dr dz

can be respectively written as

\begin{matrix}Δψ_{r} − \frac{ψ_{r}}{r^{2}} = \frac{∂Q}{∂r} + λ[ − g_{1}(r)ψ_{φ} − \frac{κ_{0}}{r} (\frac{∂ψ_{r}}{∂r} + \frac{ψ_{r}}{r} ) + P g_{2}(r)T_{2}], \\ Δψ_{φ} − \frac{ψ_{φ}}{r^{2}} = λ[ − 2ω_{1}ψ_{r} − \frac{κ_{0}}{r} (\frac{∂ψ_{φ}}{∂r} − \frac{ψ_{φ}}{r^{2}} )], \\ Δψ_{z} = \frac{∂Q}{∂z} − λ \frac{κ_{0}}{r} \frac{∂ψ_{z}}{∂r} , \\ ΔT_{2} = λP [ − \frac{κ_{0}}{r} \frac{∂T_{2}}{∂r} + \frac{Ra}{P} ω_{2}ψ_{r}], \\ \frac{∂ψ_{r}}{∂r} + \frac{ψ_{r}}{r} + \frac{∂ψ_{z}}{∂z} = 0; \end{matrix}(2.5)

ψ⃗(ψ_{r}, ψ_{φ}, ψ_{z})|_{r=1,R} = 0, T_{2}|_{r=1,R} = 0.  (2.6)

Let us consider the set M of twice continuously differentiable solenoidal pairs V⃗ {v⃗(v_{r}, v_{φ}, v_{z}), T } which are defined in the closed domain {1 ≤ r ≤ R, −∞ < z < +∞} and which are axisymmetric, vanish for r = 1, R, have a flow across the cross-section of the hollow space between the cylinders equal to zero, and are such that v_{r}, v_{φ}, T are even functions and v_{z} is an odd function with respect to z. Denote by H_{1} the Hilbert space obtained by completion of the set M with respect to the norm generated by the scalar product

(V⃗ · V⃗^{I})_{H_{1}} = − ∫_{−π/α_{0}}^{π/α_{0}} dz ∫_{1}^{R} \{(Δv_{r} − \frac{v_{r}}{r^{2}} )v_{r}^{I} +

+ (Δv_{φ} − \frac{v_{φ}}{r^{2}} )v_{φ}^{I} + Δv_{z} · v_{z}^{I} + ΔT · T^{I}\}rdr, V⃗^{I} ∈ M.

Following [7], problem (2.1)–(2.2) can be reduced to the nonlinear operator equation

V⃗ = λKV⃗ .  (2.7)

The linearized problem (2.3)–(2.4) and its conjugate problem will respectively satisfy the operator equations

U⃗ = λAU⃗,  (2.8)
Ψ⃗ = λA^{∗}Ψ⃗.  (2.9)

Applying the results of [7, 8], we easily ascertain that the operators K, A, and A^{∗} are completely continuous in the space H_{1}. The operator A is the Frechet differential of the operator K at the point V⃗ = 0, and A^{∗} is the conjugate operator of A in the space H_{1}.

To apply the bifurcation theory of nonlinear operator equations it is necessary to investigate the spectrum of the linear operator A, since, as follows from Krasnoselskii’s results [9], the bifurcation points of the nonlinear operator K can be only having the odd multiplicity (in particular, simple ones) characteristic numbers of its Frechet differential at the point V⃗ = 0.

3. Theorem. Let the following conditions be fulfilled: κRa > 0 and the functions ω_{k}(r), g_{k}(r) (k = 1, 2) are positive throughout the interval (1, R). Then for all α_{0}, except some countable set, the operator A has at least one positive simple characteristic number λ_{0} which is the bifurcation point of the nonlinear operator K. This characteristic number is less than the moduli of all other characteristic numbers of the operator A. Proof. Using a Fourier series expansion, the solution of the linear problem (2.3)–(2.4) can be represented as a linear combination of solutions of the form

{u_{r}, u_{φ}, Π_{1}, T_{1}} = {u(r), v(r), p_{1}(r), τ(r)} cos αz,
u_{z} = w sin αz, α = nα_{0} (n = 1, 2, …),

which leads us to the spectral problem

\begin{matrix}[L − \frac{κ}{r} (\frac{d}{dr} − \frac{1}{r}) − α^{2}](L − α^{2})u = λ(2α^{2}ω_{1}v − α^{2}Ra ω_{2}τ), \\ − [L − \frac{κ}{r} (\frac{d}{dr} + \frac{1}{r}) − α^{2}]v = λg_{1}(r)u, \\ − (L − \frac{κP}{r} \frac{d}{dr} + \frac{1}{r^{2}} − α^{2})τ = −λP g_{2}(r)u; \end{matrix} (3.1)

u|_{r=1,R} = v|_{r=1,R} = \frac{du}{dr}|_{r=1,R} = τ|_{r=1,R} = 0,  (3.2)

where

L = \frac{d}{dr} (\frac{d}{dr} + \frac{1}{r}), w(r) = −\frac{1}{αr} \frac{d}{dr} (ru),

p_{1} = −\frac{1}{α}(\frac{d^{2}}{dr^{2}} + \frac{1 − κ}{r} \frac{d}{dr} − α^{2})w.

We introduce the integral operators

G_{k}f = ∫_{1}^{R} G_{κ}^{k}(r, ρ)f(ρ)ρ dρ (k = 1, 2, 3),

where G_{κ}^{k} are the Green functions of the operators on the left-hand sides of system (3.1) at the boundary conditions (3.2).

Denote by H_{1}^{0} the Hilbert space L_{2} with the weight σ(r) = r on the segment [1, R] with the scalar product

(ψ_{1}, ψ_{2})_{H_{1}^{0}} = ∫_{1}^{R} ψ_{1}(r)ψ_{2}(r)r dr.

Lemma 1. The kernels G_{κ}^{k} (k = 1, 2, 3) are nonsymmetric and oscillatory.

The lemma can be easily proved by the methods of Krein [10]. The kernels G_{κ}^{1} and G_{κ}^{2} are proved to be oscillatory in [11]. As for G_{κ}^{3}, the fact that it is oscillatory follows from the representation

−(L − \frac{κP}{r} \frac{d}{dr} + \frac{1}{r})τ = \frac{r^{κP}}{ω_{0}} \frac{d}{dr} r^{1−κP} ω_{0}^{2} \frac{d}{dr} \frac{τ}{ω_{0}} ,

where ω_{0} = I_{\frac{κP}{2}} is the modified Bessel function which is a solution of the equation

(L − \frac{κP}{r} \frac{d}{dr} + \frac{1}{r^{2}} − α^{2})ω_{0} = 0.

By inverting the operators on the left-hand sides of system (3.1) we obtain

\begin{aligned}u & = λ(2α^{2}G_{1}ω_{1}(r)v − Ra α^{2}G_{1}ω_{2}(r)τ), \\ v & = λG_{2}g_{1}(r)u, \\ τ & = −λP G_{3}g_{2}(r)u. \end{aligned}  (3.3)

The spectral problem (3.3) is equivalent to an integral equation

u = μBu,  (3.4)

where μ = 2α^{2}λ^{2}, B = B_{1} + B_{2},

B_{1} = G_{1}ω_{1}(r)G_{2}g_{1}(r), B_{2} = \frac{1}{2} Ra P G_{1}ω_{2}(r)G_{3}g_{2}(r).

Lemma 1 implies that the kernels of the integral operators B_{1} and B_{2} are nonsymmetric oscillatory ones.

Similarly, in finding a solution of the conjugate problem (2.5)–(2.6) in the form

{ψ_{r}, ψ_{φ}, Q, T_{2}} = {u_{1}(r), v_{1}(r), q(r), τ_{1}(r)} cos αz,
ψ_{z} = w_{1}(r) sin αz,

we come to the problem of defining the eigenfunctions:

\begin{matrix}(L − α^{2})[L + \frac{κ}{r} (\frac{d}{dr} + \frac{1}{r}) − α^{2}]u_{1} = λα^{2}g_{1}(r)v_{1} − λα^{2}P g_{2}(r)τ_{1}, \\ [L + \frac{κ}{r} (\frac{d}{dr} − \frac{1}{r}) − α^{2}]v_{1} = −2λω_{1}(r)u_{1}, \\ (L + \frac{κP}{r} \frac{d}{dr} + \frac{1}{r^{2}} )τ_{1} = λRa ω_{2}(r)u_{1}, \end{matrix}(3.5)

u_{1}|_{r=1,R} = \frac{du_{1}}{dr} |_{r=1,R} = v_{1}|_{r=1,R} = τ_{1}|_{r=1,R} = 0,  (3.6)

where

q = −\frac{1}{α}(\frac{d^{2}}{dr^{2}} + \frac{1 + κ}{r} \frac{d}{dr} − α^{2})w_{1}, w_{1} = −\frac{1}{αr} \frac{d}{dr} (ru_{1}).

Thus the linearized problem (2.3)–(2.4), equivalent to the operator equation (2.8) in the Hilbert space H_{1}, can be reduced, after separation of variables, to the integral equation (3.4). The characteristic numbers of the operators A and B are related by the relation μ = 2α^{2}λ^{2}.

Lemma 2. If κRa > 0 and the functions ω_{k}(r), g_{k}(r) (k = 1, 2) are positive throughout the interval (1, R), then the operator B is u_{0}-positive in the cone of non-negative functions.

The proof of this lemma follows from the results of [12] and Lemma 1. Similar statements for the corresponding operator represented as the sum of oscillatory operators can be found in [13].

Lemma 2 implies that for any value of α_{0} the operator B has at least one positive simple characteristic number μ_{0} [12]. In particular, this means that the rank of μ_{0} (i.e., dim(Ker (B − μ_{0}I)), where I is the identical operator) is equal to unity (see [7]).

Lemma 3. Let μ > 0 be the characteristic number of the operator B whose rank is equal to unity. Then λ = ±\sqrt{μ/2α^{2}} is the characteristic number of the operator A whose rank is also equal to unity.

To prove a similar lemma for the case of solid cylinders and a noncompressible fluid [1, 2] it is essential to assume that the operator B is symmetric, since the operators contained in it are symmetric. Then the corresponding operator B is a symmetric oscillatory operator. In the presence of the parameter s, i.e., when the cylinder walls are permeable, the symmetricity of the operator B is violated and the corresponding operator B is a nonsymmetric oscillatory one [11]. In the case of a heat-conducting fluid and permeable cylinder walls the corresponding operator B is, as shown above, a nonsymmetric, u_{0}-positive one in the cone of non-negative functions.

Proof. We calculate the scalar product (U⃗ ·Ψ⃗), where U⃗, Ψ⃗ are the eigenvectors of the operators A and A^{∗}. Multiplying the equations of system (2.3) by ψ_{r}, ψ_{φ}, ψ_{z}, T_{2}, respectively, and taking into account (3.1) and (3.5), also performing integration by parts and some simple transformations we obtain

(U⃗ · Ψ⃗)_{H_{1}} = λ \frac{π}{α_{0}} ∫_{1}^{R} \{(g_{1}(r)u − \frac{κ_{0}}{r} (\frac{dv}{dr} + \frac{v}{r} ))v_{1} +

+(2ω_{1}(r)v − Ra ω_{2}(r)τ)u_{1} − \frac{1}{α^{2}} \frac{κ_{0}}{r} (\frac{du_{1}}{dr} + \frac{u_{1}}{r} )(L − α^{2})u −

−(\frac{κ_{0}}{r} \frac{dτ}{dr} + g_{2}(r))P τ_{1}\}r dr =

= λ \frac{π}{α_{0}} ∫_{1}^{R} \{(g_{1}(r)u − \frac{κ_{0}}{r} (\frac{dv}{dr} + \frac{v}{r} ))v_{1} + (2ω_{1}(r)v − Ra ω_{2}(r)τ)u_{1} +

+\frac{1}{α^{2}} \frac{κ_{0}}{r} (\frac{d}{dr} − \frac{1}{r})(L − α^{2})u · u_{1} − (\frac{κ_{0}}{r} \frac{dτ}{dr} + g_{2}(r)u)P τ_{1}\}r dr =

= \frac{π}{α_{0}} ∫_{1}^{R}\{\frac{1}{α^{2}} (L−α^{2})^{2}u · u_{1}−(L−α^{2})v · v_{1}−(L+ \frac{1}{r^{2}} −α^{2})τ · τ_{1}\}r dr.

Denote by H_{2}^{0} the Hilbert space of square-summable vector-functions V⃗ (u, v, τ) with the scalar product

(V⃗ · V⃗^{I})_{H_{2}^{0}} = ∫_{1}^{R}(u · u_{1} + v · v_{1} + τ · τ_{1})r dr, V⃗^{I}(u_{1}, v_{1}, τ_{1}) ∈ H_{2}^{0}.

Since the characteristic number μ > 0 of the operator B is simple, one can easily verify that V⃗ (u, v, τ) ∈ H_{2}^{0}, where (u, v, τ) is a solution of problem (3.1)–(3.2), is also a simple eigenvector of this system.

Let us consider the linear space N of the vector-functions defined on the segment [1, R] and satisfying the following conditions: u are continuously differentiable functions on the segment [1, R] up to the fourth order

inclusive with the condition u|_{r=1,R} = \frac{du}{dr}|_{r=1,R} = 0; v, τ are continuously differentiable functions up to the second order inclusive with the boundary condition v|_{r=1,R} = τ|_{r=1,R} = 0.

Since the operators r(L − α^{2})^{2}, −r(L − α^{2}), −r(L + \frac{1}{r^{2}} − α^{2}) are positive definite, by closing the linear space N in the norm generated by the scalar product

(V⃗ ·V⃗^{I})_{H_{2}} =∫_{1}^{R}[\frac{1}{α^{2}} (L−α^{2})^{2}u · u^{I}−(L−α^{2})v · v^{I}−(L+ \frac{1}{r^{2}} −α^{2})τ · τ^{I}]r dr,

V⃗ (u, v, τ), V⃗^{I}(u^{I}, v^{I}, τ^{I}) ∈ H_{2},

we obtain the complete energetic Hilbert space H_{2} (see [14]).

Rewrite problems (3.1)–(3.2) and (3.3)–(3.4) in the space H_{2} in the operator form:

V⃗ = λK_{1}V⃗ , V⃗_{1} = λK_{1}^{∗}V⃗_{1}, V⃗ , V⃗_{1} ∈ H_{2},

where K_{1} and K_{1}^{∗} are completely continuous operators acting in the space H_{2} and satisfying an additional requirement that the integral identities

(K_{1}V⃗ · Φ⃗)_{H_{2}} = λ ∫_{1}^{R} \{[2ω_{1}v + \frac{κ_{0}}{rα^{2}} (\frac{d}{dr} − \frac{1}{r})(L − α^{2})u −

−Ra ω_{2}τ]Φ_{r} − [\frac{κ_{0}}{r} (\frac{d}{dr} + \frac{1}{r})v − g_{1}(r)u]Φ_{φ} −

−[P g_{2}(r)u + \frac{κP}{r} \frac{d}{dr} ]Φ_{z}\}r dr,

(K_{1}^{∗}V⃗_{1} · Φ⃗)_{H_{2}} = λ ∫_{1}^{R} \{[g_{1}(r)v_{1} − \frac{κ_{0}}{α^{2}} (L − α^{2})\frac{1}{r}(\frac{d}{dr} + \frac{1}{r})u_{1} −

−P g_{2}(r)τ_{1}]Φ_{r} + [2ω_{1}(r)u_{1} + \frac{κ_{0}}{r} (\frac{d}{dr} − \frac{1}{r})]Φ_{φ} +

−[Ra ω_{2}(r)u_{1} − \frac{κP}{r} \frac{dτ_{1}}{dr} ]Φ_{z}\}r dr

be fulfilled for any vectors V⃗, V⃗_{1}, Φ⃗ ∈ H_{2} (see [7]).

Performing integration by parts, we readily obtain the equality

(K_{1}^{∗}V⃗_{1}, Φ⃗)_{H_{2}} = (V⃗_{1}, K_{1}Φ⃗)_{H_{2}}.

Therefore K_{1} is the conjugate operator of K_{1}^{∗} in the space H_{2}.

We use the results of [14], in particular the theorem stating that to each element from H_{2} there may correspond only one element from H_{2}^{0}. Note that in that case to different elements from H_{2} there correspond different elements from H_{2}^{0}. Hence it is not difficult to show that if the equations

V⃗ = λK_{1}V⃗ , W⃗ = λK_{1}W⃗ + V⃗ , V⃗ , W⃗ ∈ H_{2},

where K_{1} is a completely continuous operator in H_{2}, are fulfilled, then in the space H_{2}^{0} the equations

Ṽ = λK_{1} Ṽ, W̃ = λK_{1} W̃ + Ṽ, Ṽ, W̃ ∈ H_{2}^{0},

will also have solutions.

Indeed, let us be given the equation

V⃗ = λK_{1}V⃗ , V⃗ ∈ H_{2}.

Let us consider a sequence V⃗_{n} ∈H_{2} such that

∥V⃗ − V⃗_{n}∥_{H_{2}} → 0, ∥Ṽ − Ṽ_{n}∥_{H_{2}^{0}} → 0.

The existence of such a sequence follows from the proof of the abovementioned theorem from [14].

We write the equality

V⃗ − V⃗_{n} = λK_{1}(V⃗ − V⃗_{n}) + δ_{n},  (3.7)

where V⃗_{n} ∈ H_{2}, δ_{n} = λK_{1}V⃗_{n} − V⃗_{n}.

Then we have

∥δ_{n}∥_{H_{2}} ≤ ∥V⃗ − V⃗_{n}∥_{H_{2}} + λ∥K_{1}(V⃗ − V⃗_{n})∥_{H_{2}}.

Using the inequality from [14]

∥V⃗ ∥_{H_{2}^{0}} ≤ ∥V⃗ ∥_{H_{2}}

we find that ∥δ_{n}∥_{H_{2}} → 0 implies ∥δ_{n}∥_{H_{2}^{0}} → 0.

Passing in (3.7) to the limit in H_{2}^{0}, we obtain

V⃗ − Ṽ = λ(K_{1}V⃗ − K_{1} Ṽ)

which gives

Ṽ = λK_{1} Ṽ,

where Ṽ ∈ H_{2}^{0}.

By a similar reasoning one can ascertain that if the equation

W⃗ = λK_{1}W⃗ + V⃗ , V⃗ , W⃗ ∈ H_{2},

has a solution, then the corresponding equation

W̃ = λK_{1} W̃ + Ṽ, Ṽ, W̃ ∈ H_{2}^{0},

will be fulfilled in H_{2}^{0}, which is impossible because V⃗ ∈ H_{2}^{0} is a simple eigenvector and, accordingly, λ is a simple characteristic number of problem (3.1)–(3.2). Hence it follows that (V⃗ · V⃗_{1})_{H_{2}} ≠ 0 [7]. Now we obtain

(U⃗ · Ψ⃗)_{H_{1}} = \frac{π}{α_{0}} (V⃗ · V⃗_{1})_{H_{2}} ≠ 0.

Therefore the rank of the characteristic number λ of the operator A is equal to unity.

Next, using the arguments from [1], we show that λ_{0} = \sqrt{\frac{μ_{0}}{2α_{0}^{2}}} is the

simple characteristic number of the operator A.

Since the operator B is u_{0}-positive, the characteristic number μ_{0} is less than the moduli of all other characteristic numbers of the operator B [9]. But in that case λ_{0} is less than the moduli of all other characteristic numbers

λ = \sqrt{\frac{μ}{2α^{2}}} of the operator A.

Thus we have shown that under the conditions of the theorem the operator A has at least one simple characteristic number which is the bifurcation point of the operator K. In that case the main flow (1.1) gives rise to secondary axisymmetric stationary flow bifurcations.

One can easily verify that the conditions of the theorem are fulfilled when the temperature of the internal cylinder exceeds the temperature of the external cylinder (θ < 1) in the case of fluid inflow through the external cylinder (κ < 0), and, conversely, when the temperature of the external cylinder exceeds the temperature of the internal cylinder (θ > 1) in the case of fluid inflow through the internal cylinder (κ > 0), while the angular velocities and radii are related through the relation 0 < Ω < \frac{1}{R^{2}} .

Note that if Ra = 0 then for any κ the condition 0 < Ω < \frac{1}{R^{2}} is the sufficient one for secondary axisymmetric stationary flows to arise in the noncompressible fluid between two rotating permeable cylinders. Moreover, for each α_{0} we have a sequence of simple characteristic numbers of the operator A, each of which is the bifurcation point of the corresponding nonlinear operator [11].

References

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2. V. I. Yudovich, The bifurcation of a rotating flow of liquid. (Russian) Dokl. Akad Nauk SSSR 169(1966), No. 2, 306–309; English translation: Soviet Phys. Doklady 11(1966)(1967), 566–568.

3. Yu. P. Ivanilov and G. N. Yakovlev, The bifurcation of fluid flow between rotating cylinders. (Russian) Prikl. Mat. Mekh. 30(1966), No. 4, 688–698; English translation: J. Appl. Math. Mech. 30(1966)(1967), 910–916.

4. W. Velte, Stabilität und Verzweigung stationärer Lösungen der Navier-Stokesschen Gleichungen beim Taylor problem. Arch. Rat. Mech. Anal. 22(1966), No. 1, 1–14.

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7. V. I. Yudovich, Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid. (Russian) Prikl. Mat. Mekh. 29(1965), No. 3, 453–467; English translation: J. Appl. Math. Mech. 29(1965), 527–544.

8. I. I. Vorovich and V. I. Yudovich, Steady flow of a viscous incompressible fluid. (Russian) Mat. Sb. 53(95)(1961),393–428.

9. M. A. Krasnoselskii, Topological methods in the theory of non-linear integral equations. (Russian) Gosudarstv. Izdat Tekhn. Teor. Lit., Moscow, 1956.

10. M. G. Krein, On nonsymmetric oscillatory Green’s functions of ordinary differential operators. (Russian) Dokl. Akad. Nauk SSSR 25(1939), No. 8, 643–646.

11. L. D. Shapakidze, On the bifurcation of fluid flow between two rotating permeable cylinders. (Russian) Soobshch. Akad. Nauk Gruz. SSR 99(1980), No. 2, 325–328.

12. M. A. Krasnoselskii, Positive solutions of operator equations. (Russian) Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1962.

13. V. V. Kolesov, On stability conditions for the nonisothermic Couette flow. (Russian) Prikl. Mat. Mech. 44(1980), No. 3, 441–447; English translation: J. Appl. Math. Mech. 44(1980)(1981), No. 3, 311–315.

14. S. G. Mikhlin, Variational methods in mathematical physics. (Russian) Nauka, Moscow, 1970.

(Received 16.08.1995)

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093

Georgia