ISSN 0018151X, High Temperature, 2011, Vol. 49, No. 6, pp. 900–905. © Pleiades Publishing, Ltd., 2011. Original Russian Text © B.I. Basok, A.A. Avramenko, V.V. Gotsulenko, 2011, published in Teplofizika Vysokikh Temperatur, 2011, Vol. 49, No. 6, pp. 931–936.
HEAT AND MASS TRANSFER AND PHYSICAL GASDYNAMICS
ThermalConvective Instability in a Liquid Layer in a System with Rotating Pivots B. I. Basok, A. A. Avramenko, and V. V. Gotsulenko Institute for Technical Thermal Physics, National Academy of Sciences, Ukraine, Kiev email:
[email protected] Received April 21, 2010
Abstract—The stability of the main stationary liquid motion defined by Dirichlet boundary conditions on lateral surfaces of pivots was considered for a heated layer of a viscous thermal conducting liquid periodically perforated by a set of thin pivots. It was established that at any type of perforation and exceeding the critical Rayleigh number loss of stability is monotonic. In addition, the asymptotics for the critical Rayleigh number were defined at different types of perforation of the liquid layer. DOI: 10.1134/S0018151X11060046
INTRODUCTION It is a wellknown fact that mechanical equilibrium is possible in nonuniformly heated liquid at certain conditions in the gravitation force field. However, if the nonuniformity of temperature is considerable, then the equilibrium state loses stability and further convective motion appears. In the layer of a viscous thermal conducting liquid, the liquid is in a mechanical equilibrium state at low enough temperature gradients and, in this case, heat is transferred only at the expense of thermal conductiv ity. The equilibrium state is violated with increase in the temperature difference, and twodimensional dis sipative periodic structures appear in the form of con vective shafts [1]. With further increase in tempera ture, convective shafts become unstable, and as a result of following bifurcations, they may proceed to three dimensional periodic structures (Benard cells) [2, 3]. The cellular structure of convection also loses stability, and finally, turbulence appears [4]. The Rayleigh problem considered below is reduced to calculating the critical Rayleigh number Rac the exceeding of which causes convective shafts and loss of stability of the laminar flow. It is necessary to note that the exact analytic solution of this problem can be obtained only with a very abstract statement with two free nondeformable boundaries [1]. In this case Ra c = 27 π 4 ≈ 657. For a problem with other more 4 realistic boundary conditions, the critical Rayleigh number turns out to be greater. For example [1], for a layer with two solid boundaries Rac ≈ 1708. However, qualitatively the structure of bifurcation does not change [1]. The development of computation technics and the appearance of hydrodynamic software promoted
detailed study of the influence of many complicating factors on convective stability: magnetic field and rotation [5], nonlinear dependence of viscosity on temperature [6], and others. Numerical solutions using computers for different variants of the thermal convection problem and some numerical experiments are presented in [7, 8]. It is necessary to note that even nonlinear boundary conditions considered in multifold and particularly densely perforated domains or domains with fast oscil lating boundary in fact cannot be numerically simu lated. This is connected with the fact that, at any dis cretization of the initial differential equations (method of grids, finite elements, and others), a sepa rate boundary condition leading to an additional alge braic in the discretization equation is set on each con nection component. As a result the number of alge braic equations becomes so large that their solution is difficult even using modern computers. In order to analyze such kinds of problems, different averaging procedures (limiting passage to a problem where per foration parameters approach to zero) [9] have been developed. This work is devoted to the Rayleigh problem in the densely perforated layer of a viscous thermal conduc tive liquid. In a number of works [10–12], such a sys tem was called the generalized Couette cell. It is nec essary to note that the Couette cell is a classic model object of hydrodynamics representing two cylinders with a common axis, the space between which is filled with liquid and one (or both) of which rotates at a given speed. According to study [4] with increasing rotation speed of the inner cylinder of the Couette cell, turbulent chaotic flow appears after 3–4 bifurca tions; this invalidates the hypothesis about the appear ance of turbulence suggested by Landau [4]. In this way the generalized Couette cell represents a cylinder
900
THERMALCONVECTIVE INSTABILITY
901
xs
x2
ε x2 ˜k T ε s/2
rε
T = T2
˜ Ω
s/2
x1
s/2
T1 > T2
s/2
l x1
T = T1 x1
xs
Fig. 1. Perforation scheme of the liquid layer.
perforated by a set of thin pivots, the space between which is filled with liquid and a certain temperature gradient is applied to its base. Separate boundary con ditions for the speed and temperature of the liquid are set on the lateral surfaces of pivots. Practically, the Couette cell can be used for inten sification of thermaldiffusion processes, particularly for mixing several poorly mutually moistening liquids as well as for obtaining different disperse systems. Cylindrical perforation of vertical burning chambers of industrial plants allows controlling the mode of vibrational burning [13]. Injection of fuel into com bustion chambers of liquidpropellant reactive engines by the scheme with postcombustion is performed in the form of twisted rotating streams of liquid, which also leads to consideration of a model of the general ized Couette cell.
⎡ ⎤ Ω ε = Ω\ ⎢ ∪ ( Cε + εk ) × [0; ]⎥ , ⎣k∈Θε ⎦ where J ε = Θε , is the power (i.e., number of ele × ( 0; ) . Here is a ments) of the quantity Θε, Ω = Ω positive number (height of the Couette cell). In this way Ω ε is an open positive quantity in R3 periodically perorated by thin pivots Tεk = Tεk × [0, ] . Since each of cylinders Tεk is obtained as a result of homothetic transformation by two first coordinates, i.e.,
MATHEMATICAL FORMALIZATION OF THE GENERALIZED COUETTE CELL
ε × {0} , Γ2ε = Ω ε × {} , Γ3 = ∂Ω × [0, ] , where =Ω k k ∂ Tε is the lateral surface of cylinder Tε . The obtained quantity Ω ε ⊂ R 3 is called the generalized Couette cell [14]. The perforation density of domain Ω by thin piv ots Tεk ( k ∈ Θε ) is defined by perforation parameter σ ε = ε 2log (1 rε ) , where rε is the radius of pivots Tεk . The limit value of perforation parameter C 0 = lim σ ε is
Tεk = {( x1, x2, x3 ) : ( x1, x2 ) ∈ Cε + εk, 0 ≤ x3 ≤ } , then the mentioned perforation is periodic and its period is represented by the cell Λ = εY × [0; ] . Let us denote the boundary of domain Ωε as Γ ε Apparently Γ ε = Γ1ε ∪ Γ ε2 ∪ Γ3 ∪ ∂ Tεk, k∈Θε
This chapter is devoted to description of the struc ture of possible perforations of a horizontal liquid layer by thin pivots; i.e., the “geometry” of the considered porous medium is introduced. Let us denote a limited with a sufficiently open quantity on a plane as Ω smooth boundary ∂Ω . (Fig. 1). Assume that Y = (–1/2; 1/2}] × (–1/2; 1/2] is the unit area and C is a circle with radius r < 1/2 with centers in the coordinate origin so that C ⊂⊂ Y. Let {ε} be the sequence of positive num bers like ε = 1/N where N → ∞ (here N are natural numbers). Let us introduce the following quantities:
{
}
, Θ ε = k = ( k1, k2 ) ∈ N 2 : Cε + εk ⊂⊂ Ω Tεk = C ε + ε k, C ε = ε C, Tε = ∪ Tεk , k∈Θ ε
Tεk
= C ε + ε k ∀k ∈ Θ ε, Tε = Tε × [0, ] , ε =Ω Tε , Ωε = Ω ε × ( 0; ) . Ω
Obviously HIGH TEMPERATURE
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Γ1ε
ε→0
also introduced. When 0 < C0 < ∞ , the perforation is called critical, at C0 = 0, perforation occurs by the set of thick pivots and at C 0 = +∞ , perforation occurs by thin pivots. At ε → 0, N → ∞ the structure of the porous material considerably depends on the values of perforation parameter C0. PROBLEM STATEMENT AND SOLUTION ALGORITHM The aim of this work is definition of a stability domain in the form of critical Rayleigh number Rac exceeding of which produces thermal convection in a
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densely perforated liquid layer (i.e., at N → ∞, ε = 1/N → 0). If one increases the temperature gradient of the base of the outer cylinder of the generalized Cou ette cell (increasing Rayleigh number Rac), then the motion of the liquid promoted by rotation of the inner cylinders at some value of Rayleigh number Rac loses stability because of formation of an additional convec tive motion that excites the main flow. In dimensionless form, the equation set of liquid motion in domain Ωε in the Boussinesq approxima tion is denoted as [1, 15] ⎧∂Vε + ( V ⋅ ∇) V = −grad ( p ) + ΔV + Gr θ e , ε ε ε ε ε 2 ⎪ ∂t ⎪ (1) ⎨div Vε ( r, t ) = 0, ∀r ∈ Ω ε, t ≥ 0, ⎪∂θ −1 ⎪ ε + Vε ⋅ grad ( θ ε ) − Vε ⋅ e 2 = Pr Δθε ( r, t ) , ⎩ ∂t where Vε is the speed of the liquid; θε is its dimension less temperature; pε ( r, t ) is the pressure at the point defined by radiusvector r at the moment of time t; Gr = g γ (T1 − T2 ) 3 ν is the Grashof number; Pr = ν/k is the Prandtl number; is the height of the Couette cell; T1 and T2 are the temperatures of its lower and upper base, respectively; g is the gravitation accelera tion; k is the thermal conductivity coefficient; γ is the thermal shear coefficient; and ν is the kinematic vis cosity coefficient. Equation set (1) is supplemented by the following boundary conditions:
Vε Γ1ε ∪Γ 2ε ∪Γ3 = 0, θ ε Γ1ε ∪Γ 2ε ∪Γ3 = 0, Vε ∂ Τεk = α ε , k
θ ε ∂ Τεk = β εk ∀k ∈ {1,.., J ε} , α εk
ωεk
(2)
where = × rk is the linear rotation speed and is the temperature on the lateral surface of pivot Tεk . It is necessary to note that the existence of solutions for boundary problem (1), (2) requires additional kinematic conditions [12] besides smoothness condi tions on the boundary functions α εk ( r ) and β εk ( r) . First of all it is necessary to impose conditions of existence of (probably not the only) solenoid vectorfunction U(r) on functions α εk ( r ) k = 1; J ε so that its narrow
)
ing on lateral surfaces of pivots Tεk would coincide with values of function α εk ( r ) , i.e., the following conditions would be satisfied [12] div U ( r ) = 0 ∀r ∈ Ω (3) and U ∂ Τεk = α εk ∀k ∈ {1,.., J ε} . As is shown in [16], this requires fulfilling the fol lowing integral identity: Jε
∑ ∫∫ α k =0 ∂ Tεk
k ε
⋅ n ds = 0,
Study of the asymptotics of solutions ( Vε, pε, θε ) of boundary problem (1), (2) at ε → 0, N → ∞ was per formed in works [10–12, 14]. It can be shown that the averaged (limit) problem for (1), (2) at ε → 0, N → ∞ can be denoted as
⎧∂V ∞ ∞ ∞ ∞ ⎪ ∂t + V ⋅ ∇ V + M1 ( C 0 ) ⋅ V − U ⎪ ∞ ∞ ∞ ⎪= − grad p + sign ( C0 ) ΔV + Gr θ e 2, ⎪⎪ ∞ (5) ⎨div V ( r, t ) = 0, ∀r ∈ Ω, t ≥ 0, ⎪ ∞ ⎪∂θ + V ∞ ⋅ grad θ ∞ − V ∞ ⋅ e 2 + M 2 ( C 0 ) θ ∞ − b ⎪ ∂t ⎪= sign ( C ) Pr −1Δθ ∞ ( r, t ) . 0 ⎪⎩ Equation set (5) is already considered in homoge neous domain Ω and is supplemented by the Dirichlet boundary conditions
(
)
(4)
(
)
( )
( )
V∞ β εk
(
where n is the external normal to surface ∂ Tεk . Simi larly functions β εk ( r) are defined as narrowing of some function b(r) in the whole domain Ω, which possesses some smoothness properties determined for example in [12]. In this way the condition of “rotation” of thin pivots Tεk means fulfillment of exactly kinematic con dition (4). It is obvious that the case when all the cyl inders Tεk really rotate at some constant linear speed V satisfies these conditions since vectors n and α εk ≡ V are orthogonal.
∂Ω
= U∞
(
∂Ω
, θ∞
∂Ω
)
= b ∂Ω ,
(6)
where functions U∞(r) and b(r), ∀ r ∈ Ω are supposed to be set initially [11, 12]. According to (2)–(4) these functions can be used for formation of boundary con ditions on lateral surfaces of pivots Tεk for speed Vε and temperature θε at the selected perforation scheme of domain Ω. For perforation of domain Ω by pivots Tεk of critical sizes, i.e., at 0 < C0 < ∞, it was shown in [12] that matri ces M1 ( C0 ) and M2 ( C0 ) are diagonal; at the same time M1 ( C0 ) = M2 ( C0 ) = 2π E, where E is the unit matrix. C0 In this case (5) represents the Brinkman equation set. Upon perforation by thin pivots when C0 = +∞, it was strictly proved in [14] that M1 ( C0 = +∞) = M 2 ( C0 = +∞) = 0. When C0 = 0 equation set (5) repre sents equations of the Darcy model [14, 17]. The solutions of problems (1), (2) and (5), (6) are interconnected by the following regularity condition [11]: HIGH TEMPERATURE
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∫
2
∞
lim LU ( Vε ) − V ( r, t ) d r = 0, ε→0
Ω
∫
∞
(7)
2
lim Lb ( θε ) − θ ( r, t ) d r = 0, ε→0
Ω
where continuation operators LU ( Vε ) and Lb ( θ ε ) are defined by relationships
⎧Vε ( r, t ) ⎪ LU ( Vε ) = ⎨ ⎪U ( r, t ) ⎩ ⎧θ ε ( r, t ) ⎪ Lb ( θε ) = ⎨ ⎪b ( r, t ) ⎩
at r ∈ Ω ε, Jε
at r ∈
∪T
k ε ,
k =1
(8)
at r ∈ Ω ε, Jε
at r ∈
∪T
k ε .
k =1
Therefore, solution of boundary problem (1), (2) describing thermal convection in a cylindrical liquid layer Ωε densely perforated by pivots Tεk at ε → 0, N → ∞ is reduced to solution of averaged problem (5), (6) according to (7). Analysis of the stability of the equi librium state of the densely perforated liquid layer can be formally reduced to consideration of the stability of the corresponding solution of problem (5), (6). Let us denote steady fields of speeds and tempera tures, i.e., stationary solutions of problem (1), (2), as
Vε∗ ( r ) and θ∗ε ( r ) . It is also necessary to introduce excited unsteady solutions in order to define the stabil ity character of this solution Vε ( r, t ) = Vε* ( r ) + V ε ( r, t )
(9)
and θε ( r, t ) = θ*ε ( r ) + θ ε ( r, t ) .
Now according to discussions above, instead of substitution of functions (9) in equations (1), we will perform a similar procedure with their limits at ε → 0, N→∞ ∞ ∞ V ( r, t ) = V* ( r ) + V ( r, t ) and θ∞ ( r, t ) = θ*∞ ( r ) + θ ( r, t ) .
(10)
Further, it will be assumed [15] that system (5) has a cylindrical symmetry which allows proceeding to its twodimensional statement. Considering this and sub stituting (10) into (5), we get the following equation set HIGH TEMPERATURE
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2 2 ⎧∂u ∂u + υ ∂u + 2π ( u − u*) = − ∂p + ⎛ ∂ u + ∂ u ⎞ , u + ⎪ ∂t ∂x1 ∂x 2 C 0 ∂x1 ⎜⎝ ∂x12 ∂x 22 ⎟⎠ ⎪ ⎪∂υ + u ∂υ + υ ∂υ + 2π ( υ − υ*) = Gr θ∞ ∗ ⎪ ∂t ∂x1 ∂x 2 C 0 ⎪ 2 2 ⎪− ∂p + ⎛ ∂ υ + ∂ υ⎞ , ⎜ 2 ⎪ ∂x 2 ⎝ ∂x ∂x 2 ⎟⎠ 1 2 ⎪ (11) ⎨∂θ∞ ∂ θ∞u ∂ θ∞υ ∗ ∞ 2 π ∗ ∗ ⎪ + + + θ −b −υ ⎪ ∂t C0 ∗ ∂x1 ∂x 2 ⎪ ⎛ 2 ∞ ∂ 2θ∗∞ ⎞ ⎪ −1 ∂ θ∗ ⎪= Pr ⎜⎜ ∂x 2 + ∂x 2 ⎟⎟ , 2 ⎠ ⎝ 1 ⎪ ⎪ ∂u ∂υ = 0, ⎪ + ⎩∂x1 ∂x 2 where r = x1e1 + x2e 2, V ∞ ( r ) = ue1 + υe 2, U(r) = u*e1 + υ* e 2. Zero divergence condition div V ( r ) = 0 implies that there is a flow function ψ ( x1, x 2, t ) : u = − ∂ψ ∂x 2 , υ = ∂ψ ∂x1 , V ∞ ( r ) = ue1 + υ e 2, that can be denoted in the following way after linearization of set (11):
( ) ( )
(
)
⎧∂ ∂θ + ΔΔψ + 2π Δψ , ⎪ Δψ = Gr ∂x1 C0 ⎪∂t ⎪∂θ 2π ∂ψ (12) + Pr −1Δθ, ⎨ + θ= ∂ ∂ t C x 0 1 ⎪ ⎪ ∂ 2θ + ∂ 2θ . Δθ ≡ ⎪ 2 2 ∂x1 ∂x 2 ⎩ Functions ψ ( x1, x2, t ) and θ ( x1, x2, t ) can be expanded to Fourier series: ∞ ∞ ⎧ ψ = ψ nm (t ) sin ( πmx1 ) sin ( πnx 2 ) x , x , t ( ) ⎪ 1 2 ⎪ n =1 m =1 ⎪ ∞ ∞ ⎪≡ nm ( x1, x 2, t ) , ψ ⎪ ⎪ n=1 m=1 (13) ⎨ ∞ ∞ ⎪ θ nm (t ) cos ( πmx1 ) sin ( πnx 2 ) ⎪θ ( x1, x 2, t ) = n =1 m =0 ⎪ ⎪ ∞ ∞ ⎪≡ θ nm ( x1, x 2, t ). ⎪⎩ n=1 m=0 Then, considering that
∑∑
∑∑
∑∑
∑∑
( = π (n = −π ( n
) , +m ) ψ + m ) θ ,
nm = −π 2 n 2 + m 2 ψ nm, Δψ ΔΔψ nm Δθ nm
4
2
2
2
2 2
nm
2
nm
∂θ nm = −πmθ nm sin ( πmx1 ) sin ( πnx 2 ) , ∂x1
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BASOK et al. λ
then exponents in exciting functions ψ nm (t ) and θ nm (t ) are real and loss of stability is characterized as mono tonic and nonoscillating; this is described by a singular saddle point in the phase field. Let us obtain the expression for the critical Ray leigh number Rac defining (Fig. 2) the neutral curve λ Ra =Ra c = 0 (boundary of the stability domain). It is assumed that in (16) λ = 0 and from det(A) = 0 we get
λ1(Ra)
Rac
Ra λ2(Ra)
(
Fig. 2. Neutral curve.
(
∂ψ nm = πmψ nm cos ( πmx1 ) sin ( πnx 2 ) , ∂x1 we come to the following equation set substituting (13) into (12):
⎧d ψ nm ⎡ 2 2 2 2π⎤ mGr ⎪ dt = − ⎢π n + m + C ⎥ ψ nm + π n 2 + m 2 θ nm, ⎣ 0⎦ ⎪ (14) ⎨ ⎡2π ⎪d θ nm 2 2 ⎤ −1 2 ⎪⎩ dt = πmψ nm − ⎣⎢C + Pr π n + m ⎦⎥ θ nm. 0 Assuming in (14) that ψ nm (t ) = α1 exp ( −λ t ) , θnm(t) = α2exp(–λt) for α1 and α2 after collapsing of common multiplier exp(–λt), we obtain (15) ( A + λ E) Z = 0, a a ⎡ ⎤ where A = 11 12 , a11 = − ⎢π2 n 2 + m 2 + 2π⎥ , a21 a22 C0 ⎦ ⎣ ⎡2π −1 2 m Gr a12 = , a21 = πm, a22 = − ⎢ + Pr π × 2 2 π n +m ⎣C 0
(
)
(
(
(n
)
)
(
(
)
)
)
+ m ⎤⎦ , Z = [α1 α 2 ] . System (15) can be solved if and only if det ( A + λ E) = 0, which is equivalent to equation 2
2
T
(16) λ 2 + Tr ( A ) λ + det ( A ) = 0, where Tr(A) = a11 + a22 is the trace of matrix A. In this way −Tr ( A ) ± Tr ( A ) − 4det ( A ) , or 2 2 2 − 2 λ1,2 = π n 2 + m 2 1 + Pr 1 + 2π C0 2 2
λ1,2 =
(
)(
)
−1 2 4 2 − 2 m Ra, ± π n 2 + m 2 1 − Pr 1 + Pr 2 4 n + m2 where Ra = GrPr = g γ 3 (T1 − T2 ) ν k is the Rayleigh number. It is necessary to note that since
(
)(
)
(
Tr 2 ( A ) − 4det ( A ) = π 4 n 2 + m 2 −1
2
m Ra > 0, + 4Pr 2 n + m2
2 2 ⎡ ⎤ Ra c ( n, m) = n + 2m ⎢π 2 n 2 + m 2 + 2π⎥ C0 ⎦ m ⎣ (17) ⎡ 2 2 ⎤ 2 π 2 × ⎢π n + m + Pr ⎥ . C0 ⎦ ⎣ The most dangerous excitations [1] correspond to min imal critical value min {Ra c ( n, m)} = min {Ra c (1, m)} =
) (1 − Pr ) 2
−1 2
)
)
n≥1,m≥0
Ra c (1, m ) = Ra c. Considering that
m≥0
2 ⎡ 2 ⎤ Ra c (1, m) = 1 + m π 1 + m 2 + 2π⎥ 2 ⎢ C0 ⎦ m ⎣ ⎡ 2 ⎤ 2 × ⎢π 1 + m + 2π Pr ⎥ , C0 ⎦ ⎣ for value m , we obtain the equation
(
(
)
)
∂Ra c (1, m) ⎡ ⎤ = 0 ⇔ 2m 6 + ⎢2π (1 + Pr ) + 3⎥ m 4 − 1 ∂m C ⎣ 0 ⎦ m = m 2
⎛ ⎞ − ⎜ 2π ⎟ Pr − 2π (1 + Pr ) = 0. C0 ⎝ C0 ⎠ For example, at Pr = 1 it can be shown that
m = − 1 + 1 9 + 16π, from where 4 4 C0 1 ⎡ ⎤ 2 ⎛ ⎞ 1 16 π ⎥ π2 Ra c ( C 0 ) = ⎢3 + ⎜ 9 + ⎟ C0 ⎠ ⎥ 16 ⎢ ⎝ ⎣ ⎦ 2
1 ⎡ ⎤ 2 ⎛ ⎞ 16 π (18) ⎢3πC 0 + πC 0 ⎜ 9 + ⎥ + 8 C 0 ⎠⎟ ⎢ ⎥ ⎝ ⎦ ×⎣ 1 ⎡ ⎤ 2 ⎢− 1 + ⎛⎜ 9 + 16π ⎞⎟ ⎥ C 02. C0 ⎠ ⎥ ⎢ ⎝ ⎣ ⎦ It is necessary to note that condition λ = λ (Ra ) ≥ 0 ⇔ R a ≤ Ra c ( C0 ) . In this way the final stability condition is
(19) Ra ≤ Ra c ( C0 ) . The following asymptotics for the critical Rayleigh number are defined from (18):
Ra c ( C0 ) =
ξ0 ξ ξ ξ + 1 + 2 + 3 + O (1) C02 C03 2 C0 C01 2 at C0 → 0,
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(
(
(
)
)
ξ1 = 4π3 2 π 2 + 1 ,
ξ 0 = 4π2,
where
)
ξ 2 = π 7π 2 + π 4 + 1 , ξ3 = 1 π1 2 41π 2 + 20π 4 − 7 ; 8 ⎛ ⎞ χ χ Ra c ( C 0 ) = χ 0 + 1 + 22 + O ⎜ 13 ⎟ at C 0 → ∞, (21) C0 C0 ⎝ C0 ⎠
27π , χ = 18π3, χ2 = 1 4 8 π6 − 208 π4 + 128 π2 + 64. 3 9 3 In particular from (21) at C 0 = +∞ we come to the classic result [1] obtained by Rayleigh; 4 χ 0 = Ra c C0 =+∞ = 27π is the critical Rayleigh number 4 in the problem of stability of the equilibrium stationary state in the nonperforated heated liquid layer. At C0 = 0, equation set (5) transforms to equations of the Darcy–Boussinesq model [14, 17]. In this case the dissipative component with a Laplacian operator disappears and the type of motion equation changes. The behavior of the critical Rayleigh number at C0 → 0 is described qualitatively by asymptotic equality (20). However, because of the singularity of Ra c ( C0 ) at C0 = 0, it is not possible to compare the critical Rayleigh numbers calculated according to the considered model and the Darcy–Boussinesq model. where
χ0
4
=
CONCLUSIONS For a periodically heated liquid layer perforated by a system of very thin pivots (generalized Couette cell), the boundary of the stability domain (critical value of the Rayleigh number Ra c ( C0 ) ) of the main stationary flow defined by the Dirichlet boundary conditions on lateral surfaces of pivots was established. In the spec tral problem defining linear stability, all eigenvalues are real, which characterizes monotonic loss of stabil ity of the main stationary flow upon exceeding the critical Rayleigh number. Asymptotics for Rayleigh number Ra c ( C0 ) in cases when the limit perforation parameter C0 → 0 (perfora tion by thick pivots) and C0 → ∞ (perforation by thin pivots) were found. The correlations obtained allow defining the excitation domain of thermal convection in the problem under consideration at different types of perforation characterized by parameter C0.
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