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Steady and oscillatory convection in rotating fluid layers heated and salted from below
International Journal of Non-Linear Mechanics 78 (2016) 121–130
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International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
Steady and oscillatory convection in rotating fluid layers heated and salted from below Roberta De Luca a, Salvatore Rionero a,b,n a b
University of Naples Federico II, Department of Mathematics and Applications “Renato Caccioppoli”, Via Cinzia, 80126 Naples, Italy Accademia Nazionale dei Lincei, Via della Lungara 10, 06165 Rome, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 24 July 2015 Received in revised form 9 October 2015 Accepted 28 October 2015 Available online 6 November 2015
Double convection in rotating horizontal layers filled by a Navier–Stokes fluid mixture, heated and salted from below, is investigated. Onset of linear instability – for any value of the fluid and salt Prandtl numbers Pr, P1 – either via the Routh–Hurwitz conditions or via steady or oscillatory convection, is characterized. Introducing a new field connecting the perturbation fields to the temperature and salt concentration, in the cases P 1 ¼ 1 or Pr ¼1 or P 1 P r ¼ 1, stability conditions in algebraic closed form are obtained. Linear stability is recovered as non-linear global asymptotic stability via the Auxiliary System Method. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Double-convection Auxiliary System Method Global stability Navier–Stokes
1. Introduction Because of the numerous applications in the real world phenomena (industrial processes, water pollution, geology, volcanism, etc.), many efforts have been devoted to analyze the stability of the thermal conduction solution of multicomponent Navier–Stokes fluid mixture in the absence of rotation [3,4,12–14,17,18] and in the more realistic case of the presence of rotation [1,2,5–7, 9–11,15,16,19,20]. However, as far as we know, in the case of double convection in rotating layers either the onset of linear instability or the non-linear energy stability analysis is not completely investigated. In fact – as concerns the linear instability – the onset of convection via steady or oscillatory state is not completely characterized via algebraic closed form. Further – as concerns the non-linear energy stability – the coincidence between the linear and non-linear stability conditions is obtained generally under severe restrictions on the initial data. In the present paper we reconsider the problem in the case of rotating layers heated and salted from below, aimed to characterize via algebraic closed forms the onset of instability via steady or oscillatory convection. In particular, our scope is to show that: (1) in the cases P 1 ¼ 1 or Pr ¼1 or P 1 P r ¼ 1, the onset of convection can be characterized via n Corresponding author at: University of Naples Federico II, Department of Mathematics and Applications “Renato Caccioppoli”, Via Cinzia, 80126 Naples, Italy. E-mail addresses: [email protected] (R. De Luca), [email protected], [email protected] (S. Rionero).
http://dx.doi.org/10.1016/j.ijnonlinmec.2015.10.011 0020-7462/& 2015 Elsevier Ltd. All rights reserved.
algebraic closed form by introducing a new unknown field; (2) linear stability can be recovered as non-linear global asymptotic stability via the Auxiliary System Method (see [13–17]).
2. Mathematical model Let L be a horizontal layer of depth d filled by a Navier–Stokes fluid mixture in which a chemical specie (salt) S is dissolved in and let Oxyz be an orthogonal frame of reference with fundamental unit vectors i; j; k (k pointing vertically upwards). We suppose that L is uniformly heated from below and rotates uniformly about the vertical axis with constant angular velocity ω ¼ ωk. The equat ions governing the fluid motion, in the Boussinesq approximation, are [1,7]: 8 ρ0 ðvt þ v ∇vÞ ¼ ∇P þ ρ0 νΔv 2ρ0 ωk v ρ0 ½1 AðT T 0 Þ þA1 ðC C 0 Þgk; > > > > < ∇ v ¼ 0; > > T t þv ∇T ¼ kΔT; > > : C þv ∇C ¼ k ΔC; t
1
ð2:1Þ ρ0
with ρ0 being constant density, P ¼ p 2 j ω xj ; x ¼ ðx; y; zÞ, v the fluid velocity, T the temperature, C the salt concentration, p the pressure, T 0 the reference temperature, g ¼ gk the gravity, C 0 the reference salt concentration, ν the kinematic viscosity, A the thermal expansion coefficient, A1 the salt expansion coefficient, k the thermal diffusivity, k1 the salt diffusivity. 2
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(2)
To (2.1) we append the boundary conditions 8 Tðx; y; 0; tÞ ¼ T l ; Tðx; y; d; tÞ ¼ T u ; T l 4 T u > < Cðx; y; 0; tÞ ¼ C l ; Cðx; y; d; tÞ ¼ C u ; C l 4 C u > : v k ¼ 0; on z ¼ 0; d:
derivatives, can be expanded in a Fourier series absolutely uniformly convergent in Ω; 8 t A R þ
ð2:2Þ
and let us denote by BðΩÞ the set of the functions
The boundary value problem (2.1)–(2.2) admits the thermal conduction solution m 0 ¼ ðp; v; T ; C Þ given by 8 δT δC > > z; C ¼ C l z; v ¼ 0; T ¼ T l > > > d d < δT ¼ T l T u ; δ C ¼ C l C u ; > > ρ ω2 z2 > ρ gz2 > > AδT A1 δC þ 0 ; : pðzÞ ¼ p 0 ρ0 gz½1 AðT l T 0 Þ þA1 ðC l C 0 Þ 0 2d 2
ð2Þ0 p 0 ¼ const: 40:
Setting
φ such that
ð1Þ0 φ : ðx; tÞ A Ω R þ -φðx; tÞ A R, φ A W 2;2 ðΩÞ; 8 t A R þ , φ is periodic y directions of period 2aπx , 2aπy respectively h i in thehx and i ∂φ ∂φ and ∂z ¼ ∂z ¼ 0; z¼0
z¼1
φ, together with all the first derivatives and second spatial
derivatives, can be expanded in a Fourier series absolutely uniformly convergent in Ω; 8 t A R þ .
ð2:3Þ
Since the sequence f sin nπ zgn A N is a complete orthogonal system
p ¼ p þ π;
v ¼ v þ u; T ¼ T þ θ;
C ¼ C þΦ
ð2:4Þ
and introducing the non-dimensional scalings 8 2 > d ν ν2 ρ > > > t ¼ t n ; u ¼ un ; π ¼ π n 2 0 ; > > k d d > > > n ♯ > n > x ¼ x d; θ ¼ θ T ; > > > !1=2 < ν3 δT n ♯ ♯ Φ ¼ Φ Φ ; T ¼ ; > 3 > > Agkd > > > !1=2 > > > > ν3 δ C > ♯ > Φ ¼ ; > > 3 : A1 gk1 d Eq. (2.1) (omitting the asterisks) reduces to 8 1 > P r ut þ u ∇u ¼ ∇π þ Δu þ T u k þ Rθ R1 Φ k; > > > < ∇ u ¼ 0; > θt þ P r u ∇θ ¼ Rw þ Δθ; > > > : P ðΦ þ P u ∇ΦÞ ¼ R w þ ΔΦ; 1
Ψ, together with all the first derivatives and second spatial
t
r
where 8 3 > Agd δT > > > R2 ¼ > > ν k > > > 3 > > A gd δCP 1 1 > 2 > > R1 ¼ > > ν k > < 4 4 ω2 d T2¼ > 2 > ν > > > ν > > > Pr ¼ > > k > > > > k > > > : P1 ¼ k 1
for L2 ð0; 1Þ under the boundary conditions ½Ψ z ¼ 0 ¼ ½Ψ z ¼ 1 ¼ 0, by virtue of periodicity, it turns out that 8 Ψ A AðΩÞ, there exists a sequence fΨ~ ðx; y; tÞg (Ψ~ being of “plane form”) such that nAN
n
n
8 1 1 X X > > > Ψn ¼ Ψ~ n sin nπ z; >Ψ ¼ > < n¼1
ð2:5Þ
n¼1
> > > Δ Ψ ¼ a2 Ψ ; > > : 1
ΔΨ ¼
1 X
1 X ∂Ψ ∂Ψ~ n ¼ sin nπ z; ∂t ∂t n¼1
ξn Ψ~ n sin nπ z;
ð2:8Þ
n¼1
with Δ1 ¼ ∂2 =∂x2 þ∂2 =∂y2 and
ξn ¼ a2 þ n2 π 2 ;
a2 ¼ a2x þ a2y ;
ð2:9Þ
the series appearing in (2.8) being absolutely uniformly conver-
ð2:6Þ
1
thermal Rayleigh number; salt Rayleigh number;
gent in Ω. Analogously, since the sequence f cos nπ zgn A N is a complete orthogonal for L2 ð0; 1Þ under the boundary conditions h i h system i ∂φ ∂φ ∂z z ¼ 0 ¼ ∂z z ¼ 1 ¼ 0, by virtue of periodicity, it turns out that ~ n ðx; y; tÞgn A N (φ ~ n being of 8 φ A BðΩÞ, there exists a sequence fφ “plane form”) such that 8 1 1 1 X X X ~n ∂φ ∂φ > > ¼ cos nπ z; φ¼ φn ¼ φ~ n cos nπ z; > > < ∂t ∂t n¼1 n¼1 n¼1 ð2:10Þ 1 X > > 2 > ξn φ~ n cos nπ z: > : Δ1 φ ¼ a φ; Δφ ¼ n¼1
Taylor number; Setting fluid Prandtl number;
ζ ¼ ð∇ uÞ k ¼
salt Prandtl number:
To (2.6) the boundary conditions (free-free case) are appended ∂u ∂v ¼ ¼w¼θ¼Φ¼0 ∂z ∂z
on z ¼ 0; 1;
ð2:7Þ
with u ¼ ðu; v; wÞ. We assume (as usually done, in stability problems in layers) that: (i) the perturbations ð∇π ; u; v; w; θ; ΦÞ are periodic in the x and y directions, respectively of periods 2π =ax ; 2π =ay ; (ii) Ω ¼ ½0; 2π =ax ½0; 2π =ay ½0; 1 is the periodicity cell; (iii) u; v; w, θ, Φ are such that together with all their first derivatives and second spatial derivatives are square integrable in Ω; 8 t A R þ and can be expanded in a Fourier series uniformly convergent in Ω.
∂v ∂u ; ∂x ∂y
ð2:11Þ
the horizontal components of u are given by 1 ∂2 w ∂ζ 1 ∂2 w ∂ζ u¼ 2 þ ; v¼ 2 ð2:12Þ a ∂x∂z ∂y a ∂y∂z ∂x P ∂vn ∂un and – in view of u ¼ 1 n ¼ 1 un ; ζ n ¼ ∂x ∂y – it follows that 8 1 ∂ζ 1 ∂ 2 wn ∂ζ n 2 > wn > ; þ n ; vn ¼ 2 > un ¼ 2 ∂∂x∂z < ∂y ∂y∂z ∂x a a ð2:13Þ 1 > > > : ∇ un ¼ a2 Δ1 wn þ wn ¼ 0 z Remark 2.1. We remark that, in view of (2.10)1, one has Z Z u dΩ v dΩ 0: Ω
Ω
ð2:14Þ
Remark 2.2. Let us denote by J J and 〈; 〉 the L ðΩÞnorm and the scalar product, respectively. Multiplying (2.6)1 for u and integrating over Ω, one has 2
Let us denote by AðΩÞ the set of functions
Ψ such that:
(1) Ψ : ðx; tÞ A Ω R þ -Ψ ðx; tÞ A R, Ψ A W 2;2 ðΩÞ; 8 t A R þ , Ψ is periodic in the x and y directions of period 2aπx , 2aπy respectively and ðΨ Þz ¼ 0 ¼ ðΨ Þz ¼ 1 ¼ 0;
P r 1 d J u J 2 ¼ J ∇u J 2 þ 〈Rθ; w〉 〈R1 Φ; w〉: 2 dt
ð2:15Þ
R. De Luca, S. Rionero / International Journal of Non-Linear Mechanics 78 (2016) 121–130
Then, the Holder and Poincarè inequalities lead to P r 1
dJuJ r2R J θ J J w J þ 2R1 J Φ J J w J 2α J u J 2 ; dt 2
ð2:16Þ
where α ð 4 0Þ is the positive constant appearing in the Poincarè inequality. From (2.16) one obtains dJuJ rP r R J θ J þR1 J Φ J P r α J u J : dt
ð2:17Þ
Integrating (2.17), one has J u J r J u0 J e P r α t þ
Z
t
Pr e 0
P r α ðt τÞ
R J θ J þ R1 J Φ J dτ
ð2:18Þ
where u0 ¼ uðx; 0Þ. Hence if J θ J ; J Φ J , go exponentially to zero as t-1, then also J u J goes exponentially to zero as t-1.
3. Linear stability ^ Þ the solution of the linear system ^ θ^ ; Φ Denoting by ð∇π^ ; u; ^ n Þ the nth-Fourier linked to (2.6)–(2.7), and by ð∇π^ n ; u^ n ; θ^ n ; Φ ^ Þ, one has 8 n A N, ^ θ^ ; Φ component of ð∇π^ ; u; 8 ∂u^ n > ^ n k; > ¼ ∇π^ n þ Δu^ n þ T u^ n k þ Rθ^ n R1 Φ P r 1 > > > ∂t > > > > ∇ u^ n ¼ 0; > < ∂θ^ ^ n þ Δθ^ n ; > n ¼ Rw > > ∂t > > > > ^ > > ^ n: > P 1 ∂Φ n ¼ R1 w ^ n þ ΔΦ : ∂t
ð3:1Þ
with 8 > > ð0Þ > > ∇π n ¼ ð∇π n Þðt ¼ 0Þ ; > > > > > > > > > uð0Þ ¼ ðu Þ > > n ðt ¼ 0Þ ; > < n > > > θð0Þ ¼ ðθn Þðt ¼ 0Þ ; > > > n > > > > > > > > Φð0Þ ¼ ðΦn Þðt ¼ 0Þ ; > > : n ð0Þ
1 X
∇π ð0Þ ¼ ð∇π Þðt ¼ 0Þ ¼ uð0Þ ¼ ðuÞðt ¼ 0Þ ¼
θð0Þ ¼ ðθÞðt ¼ 0Þ ¼
n¼1 1 X
Φð0Þ ¼ ðΦÞðt ¼ 0Þ ¼
Φð0Þ n ;
ð0Þ
ð3:4Þ
∂ζ n ; ∂z
Z¼
n¼1
P r ξn
B B P r T =ξn Ln ¼ B B 0 @ 0
P r T n2 π 2 P r ξn
0 a2 P r R=ξn
R
ξn
R1 =P 1
0
0
1
C a2 P r R1 =ξn C C: C 0 A ξn =P 1
ð3:9Þ
The spectral equation of Ln is ð3:10Þ
2
ð3:12Þ
The conditions, necessary and sufficient for the stability of the thermal conduction solution, require that all the roots of (3.10) have negative real part 8 ðn; a2 Þ A N R þ . These conditions are well known in the literature and are given, for algebraic equations of any degree, by the Routh–Hurwitz conditions (cfr. [8]) which, for Eq. (3.10), specify as: I1n o0;
I3n o 0;
I4n 4 0; I1n I2n I3n 4 I23n þ I21n I4n ;
8 ðn; a2 Þ A N R þ :
ð3:13Þ
Remark 3.1. We remark that:
Therefore, setting Zn ¼
with 0
Φn
ð3:3Þ
∇π ð0Þ ; uð0Þ ; θ ; Φ being assigned initial data such that ∇ uð0Þ ¼ 0. The z-components of the curl and double curl of (3.1)1 are (omitting the hats)
1 X
Φn
ð3:8Þ
being 8 3 > ½P 1 P r ðP r þ 2Þ þ 2P r þ 1ξn þ P 1 P 2r n2 π 2 T 2 > > f 2 ðn; a2 ; T 2 Þ ¼ ; > > 2 > P1 Pr a > > > < 3 ½P r ðP 1 þ 1Þ þ 2ξn þ P r ðP 1 þ1Þn2 π 2 T 2 ; f 3 ðn; a2 ; T 2 Þ ¼ > ð1 þ P 1 P r Þa2 > > > > > > ξ3 þ n2 π 2 T 2 > > f 4 ðn; a2 ; T 2 Þ ¼ n : : a2
n¼1
8 ∂wn 1 ∂ζ n > > ¼ Δζ n þ T ; < Pr ∂t ∂z ∂ Δ w ∂ζ > 1 n > : Pr ¼ ΔΔwn T n þ Δ1 Rθn R1 Φn : ∂t ∂z
In view of (2.8), (3.6) can be written as 0 1 0 1 Zn Zn C B wn C ∂B B wn C B C B C ¼ Ln B C @ θn A ∂t @ θn A
ð3:2Þ
uð0Þ n ;
θð0Þ n ;
ð3:7Þ
where Ijn ¼ Ijn ðn; a ; T Þ; ðj ¼ 1; 2; 3; 4Þ; are the principal invariants of (3.9) given by 8 ξ ð1 þ P 1 þ 2P 1 P r Þ > > ; > I1n ¼ n > > P > > " 1 # > > 2 > P r a2 2 R1 > 2 2 > > I2n ¼ R f 2 ðn; a ; T Þ ; > < P1 ξn ð3:11Þ
2 > a P r ð1 þ P 1 P r Þ 2 1 þ Pr 2 > 2 2 > > ¼ R R f ðn; a ; T Þ ; I 3n 3 > > P1 1 þ P1 Pr 1 > > > > 2 2 h i > P r a ξn 2 > > > R R21 þf 4 ðn; a2 ; T 2 Þ ; : I4n ¼ P 1
∇π ð0Þ n ;
n¼1 1 X
ð3:6Þ
under the i.b.c. 8 ð0Þ > ðu Þ ¼ uð0Þ ðθn Þðt ¼ 0Þ ¼ θn ; n ; > < n ðt ¼ 0Þ ð0Þ ðΦn Þðt ¼ 0Þ ¼ Φn ; > > : Z n ¼ wn ¼ θn ¼ Φn ¼ 0; z ¼ 0; 1:
2
n¼1 1 X
from (3.1) one has 8 ∂Z n ∂2 wn > > P r 1 ¼ ΔZ n þ T ; > > > ∂t ∂z2 > > > > > 1 ∂ Δw n > ¼ ΔΔwn T Z n þ Δ1 Rθn R1 Φn ; < Pr ∂t > ∂θ n > > ¼ Rwn þ Δθn ; > > > ∂t > > > ∂Φ > > : P 1 n ¼ R1 wn þ ΔΦn ; ∂t
λ4 I1n λ3 þ I2n λ2 I3n λ þ I4n ¼ 0;
To (3.1) the following initial boundary conditions are appended 8 ð∇π^ n Þðt ¼ 0Þ ¼ ∇π ð0Þ ðu^ n Þðt ¼ 0Þ ¼ uð0Þ > n ; n ; > > > < ^ ð0Þ ð0Þ ^ nÞ ðθ n Þðt ¼ 0Þ ¼ θn ; ðΦ ¼ Φ ; ðt ¼ 0Þ n > > ∂u^ n ∂v^ n > ^ ^ > ^ n ¼ θ n ¼ Φ n ¼ 0; z ¼ 0; 1; : ¼ ¼w ∂z ∂z
123
Zn;
ð3:5Þ
(i) Eq. (3.13) imply I2n 40, 8 ðn; a2 Þ A N R þ ; (ii) in view of (3.11)1 , (3.13)1 is always verified 8 ðn; a2 Þ A N R þ ;
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ðXÞ ¼ þ 1 and g 02 ðXÞ ¼ 6X 2 6π 2 X ¼ 6a2 X 4 0; 8 X A R þ . Then a2c2 exists and is unique.□
(iii) each one of the conditions I2n 4 0;
I3n o 0;
I4n 4 0;
I1n I2n I3n 4I23n þ I21n I4n ;
ð3:14Þ
is necessary for the stability of the thermal conduction solution.
Lemma 3.2. Each one of the conditions R2 o RC 2 ;
R2 o RC 3 ;
R2 o RC 4 ;
ð3:21Þ
is necessary for the stability of the thermal conduction solution. Proof. The proof follows in view of (iii) of Remark 3.1 and (3.16).□
Denoting by 8 R2 > > min f ðn; a2 ; T 2 Þ; RC 2 ¼ 1 þ > > > P 1 ðn;a2 Þ A NR þ 2 > > < 1 þ Pr 2 R þ min f ðn; a2 ; T 2 Þ; RC 3 ¼ > 1 þP 1 P r 1 ðn;a2 Þ A NR þ 3 > > > > 2 > > min þ f 4 ðn; a2 ; T 2 Þ; : RC 4 ¼ R1 þ
Theorem 3.1. If and only if (3.13) are verified for n¼ 1, 8 a2 A R þ , the thermal conduction solution is linearly stable. Proof. Obviously one has only to show that ð3:15Þ
ðn;a2 Þ A NR
it turns out 8 I 4 0; > > < 2n I3n o 0; > > : I 4 0; 4n
that 8 ðn; a2 Þ A N R þ 3 R2 o RC 2 ; 8 ðn; a2 Þ A N R þ 3 R2 o RC 3 ;
ð3:16Þ
8 ðn; a2 Þ A N R þ 3 R2 o RC 4 :
Since the functions f i ðn; a2 ; T 2 Þ, ði ¼ 2; 3; 4Þ, are increasing functions of n, it follows that 8 > R2 > > RC 2 ¼ 1 þ min f 2 ð1; a2 ; T 2 Þ; > > P 1 a2 A R þ > > < 1 þ Pr 2 ð3:17Þ RC 3 ¼ R þ min f ð1; a2 ; T 2 Þ; > 1 þP 1 P r 1 a2 A R þ 3 > > > > 2 2 > 2 > : RC 4 ¼ R1 þ minþ f 4 ð1; a ; T Þ: a2 A R
I3 o 0;
I4 4 0;
I1 I2 I3 4 1; I23 þ I21 I4
8 a2 A R þ ;
ð3:22Þ
with Ij ¼ Ij1 ; ðj ¼ 1; 2; 3; 4Þ, implies (3.13). Following the procedure introduced in Section 3 of [16], we remark that (3.13) is equivalent to the variational problem 8 I1n I2n I3n > 4 1; > 2 > ðn;a2min Þ A NR þ I2 > > 3n þ I1n I4n < I2n 4 0; max I3n o 0; min ð3:23Þ ðn;a2 Þ A NR þ ðn;a2 Þ A NR þ > > > > > min þ I4n 4 0: : ðn;a2 Þ A NR
By virtue of min
ðn;a2 Þ A NR þ
f i ðn; a2 ; T 2 Þ ¼ minþ f i ð1; a2 ; T 2 Þ; a2 A R
ði ¼ 2; 3; 4Þ;
ð3:24Þ
and (3.15)–(3.17), it follows that the variational problem (3.23) is equivalent to (3.22).□ Remark 3.2. We remark that: (i) Eqs. (3.22) are the Routh–Hurwitz conditions guaranteeing that all the roots of
The following lemma holds. Lemma 3.1. RC 2 ; RC 3 ; RC 4 are respectively given by 8 2 3 2 > R2 ½P 1 P r ðP r þ2Þ þ2P r þ 1ða2c2 þ π 2 Þ þ P 1 P r π 2 T > > ; RC 2 ¼ 1 þ > 2 > P1 P 1 P r ac 2 > > > > > 3 2 < 1 þ P r 2 ½P r ðP 1 þ 1Þ þ 2ða2c3 þ π 2 Þ þ P r ðP 1 þ 1Þπ 2 T RC 3 ¼ R1 þ ; 2 1 þP 1 P r ð1 þ P 1 P r Þac3 > > > > > > ða2 þ π 2 Þ3 þ π 2 T 2 > > > RC ¼ R21 þ c4 ; > : 4 a2c4 ð3:18Þ where, setting X ¼ a2 þ π 2 , a2c2 ; a2c3 ; a2c4 are respectively given by the unique positive roots of the equations 8 > P 1 P 2r π 2 T 2 > > 2X 3 3π 2 X 2 ¼ 0; > > P P ðP > r r þ 2Þ þ 2P r þ 1 1 < 2 ð3:19Þ P r ðP 1 þ 1Þπ 2 T > 2X 3 3π 2 X 2 ¼ 0; > > P ðP þ 1Þ þ 2 > r 1 > > : 3 2X 3π 2 X 2 π 2 T 2 ¼ 0: Proof. We confine ourselves to prove (3.18)1 since the proof of (3.18)2– (3.18)3 can be obtained as the same manner. Let us remark that f 2 ðn; a2 ; T 2 Þ is an increasing function of n A N. Hence its minimum with respect to n is attained in n¼ 1. In view of ∂f 2 ð1; a2 ; T 2 Þ 3½P 1 P r ðP r þ2Þ þ2P r þ 1a2 ða2 þ π 2 Þ2 ½P 1 P r ðP r þ2Þ þ2P r þ 1ða2 þ π 2 Þ3 P 1 P 2r π 2 T 2 ¼ þ ; ∂a2 P 1 P r a4 P 1 P r a4
ð3:20Þ 2
I1 o 0;
it follows that f 2 ð1; a2 ; T Þ reaches its minimum with respect to a2 in a2c2 . Let us observe that, denoting by g 2 ðXÞ the function given by the left-hand side of (3.19)1, it turns out that g 2 ð0Þ o 0, limX-1 g 2
λ4 I1 λ3 þ I2 λ2 I3 λ þ I4 ¼ 0;
ð3:25Þ
have negative real part; (ii) in view of Theorem 3.1 we can confine ourselves to investigate the sign of the real part of the roots of (3.25); (iii) denoting by R~ the lowest positive value of R2 verifying I1 I2 I3 ¼ 1, it follows that the thermal conduction solution is I23 þ I21 I4 linearly stable if and only if ~ R2 o minfRC 3 ; RC 4 ; Rg; (iv) since for i ¼2,3,4 min
ðn;a2 Þ A NR
f ðn; a2 ; T 2 Þ 4 þ i
ð3:26Þ π2 2 ; 0 ; f ðn; a ; 0Þ ¼ f 1; i i 2 ðn;a2 Þ A NR þ min
ð3:27Þ one recovers for T 2 ¼ 0 the results obtained in [17] in the absence of rotation and that – as expected – rotation produces a stabilizing effect.
4. Onset of convection via steady or oscillatory state The instability occurs or via a stationary state – associated to a zero eigenvalue and named “stationary convection” – or via an oscillatory state – associated to a pure imaginary eigenvalue and named “oscillatory or overstable or Hopf convection”. Our aim, in this section, is to analyze (3.25) in order to determine the conditions guaranteeing respectively the onset of convection via steady state or via oscillatory state. Since the real part of the eigenvalues of (3.25) depends on the Rayleigh number R2, one is led to determine the lowest value R C of R2 at which (3.25) admits a null
R. De Luca, S. Rionero / International Journal of Non-Linear Mechanics 78 (2016) 121–130
root and the lowest value R C of R2 at which (3.25) admits a pure imaginary root. Then the critical value of R2 at which instability occurs is given by R2C ¼ minfR C ; R C g
ð4:1Þ
and instability occurs via a steady state for R2 ¼ R C o R C or via an oscillatory state for R2 ¼ R C o R C . Remark 4.1. We remark that: (i) λ ¼ 0 is a solution of (3.25) if and only if I4 ¼ 0. In this case (3.25) reduces to
λðλ3 I1 λ2 þ I2 λ I3 Þ ¼ 0:
ð4:2Þ
The lowest positive value of R2 such that (3.25) admits the root λ ¼ 0 is R C ¼ RC 4 ; (ii) λ ¼ iωn , with ωn a 0, is a solution of (3.25) if and only if
ω4n I2 ω2n þ I4 ¼ 0;
ð4:3Þ
with ωn given by
125
is governed by 0 1 0 1 Z1 Z1 B C B C w1 C B w1 C ∂B B C ¼ L1 B C B C B C ∂t @ θ1 A @ θ1 A
φ1
with
0
ð5:5Þ
φ1
P r ξ1
B B Pr T B B ξ1 B L1 ¼ B B 0 B B @ 0
Pr T π2 P r ξ1 R 0
0
P r a2 P 1 R2 R21 P 1 Rξ1 ξ1 R1 2 ðP 1 1Þξ1 P1
0 a2 P r R1 Rξ1 0
ξ1 P1
1 C C C C C C: C C C A
ð5:6Þ
In the sequel we provide conditions, in algebraic closed form, guaranteeing the onset of convection via stationary or oscillatory state, in three particular cases: P 1 ¼ 1 or Pr ¼1 or P 1 P r ¼ 1.
2
ω2n ¼
I3 : I1
ð4:4Þ
Therefore only if I2 4 0;
I22 4I4 Z 0;
I3 o 0; I4 4 0;
ð4:5Þ 2
oscillatory convection can occur and occurs only if R ¼ R C , where R C is the lowest positive solution of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 I3 I2 7 I2 4I4 : ð4:6Þ ¼ 2 I1 Let us observe that, in view of (4.5)3, R C oR C ¼ RC 4 ; (iii) the solution of the general problem formalized in (4.5) for any P 1 ; P r will be solved in the cases P 1 ¼ 1 or Pr ¼ 1 or P 1 P r ¼ 1 in the following sections.
5. A new field connecting
θ and Φ
The analysis of (3.25) is simplified when at least one root is known. To this end we introduce a new field, connecting θ and Φ, aimed to prove that in the cases P 1 ¼ 1 or Pr ¼1 or P 1 P r ¼ 1, it is possible to find a real negative root of (3.25). Then the stability analysis is reduced to investigate the necessary and sufficient conditions guaranteeing that all the roots of a cubic equation have negative real part. Let us introduce the field
φ1 ¼
R1 θ1 RΦ1 : P1
ð5:1Þ
R1 φ θ1 1 : RP 1 R
Hence, (5.2) becomes ∂φ1 R1 1 Δφ ¼ 1 Δθ1 þ 1 : P1 ∂t P1 P1
~ 1 -spectral equation is have negative real part 8 a2 A R þ . The L
λ3 I~1 λ2 þ I~2 λ I~3 ¼ 0;
ð6:2Þ
where " # 8 3 2 2 > > ~ 1 ¼ ξ ð1 þ 2P r Þ; I~3 ¼ a2 P 2 R2 R2 ξ1 þ π T > ; I > 1 r 1 > a2 < " # 3 > P r a2 2 ð2 þ P r Þξ1 þ P r π 2 T 2 > > > I~ 2 ¼ R R21 : > : a2 ξ1
ð6:3Þ
The Routh–Hurwitz conditions, for the case at stake, specify as I~ 3 o 0;
I~ 1 I~2 I~3 o 0;
8 a2 A R þ :
ð6:4Þ
Remark 6.1. We remark that: (i) Eq. (6.2) admits the root λ ¼ 0 if and only if I~ 3 ¼ 0. The lowest positive value of R2 such that I~ 3 ¼ 0 is R C ¼ RC 4 ; (ii) Eq. (6.2) admits a pure imaginary root λ ¼ iωn , (ωn a 0), if and only if I~ 1 I~2 I~3 ¼ 0, i.e. ð1 þ P r ÞR2 ð1 þ P r ÞR21 2
ð5:2Þ
In view of (5.1), one has
Φ1 ¼
Let P 1 ¼ 1. Then (5.5)4 is independent of the other three equations and λ ¼ ξ1 is an L1 -eigenvalue. Then the null solution of (5.5) is stable if and only if the eigenvalues of the matrix 0 1 0 P r ξ1 P r T π 2 B C P r a2 ~ 1 ¼ B Pr T P r ξ1 R2 R21 C ð6:1Þ L R ξ1 @ ξ1 A 0 R ξ1
ð1 þ P r Þ2 ξ1 þ P 2r π 2 T 2 ¼ 0; a2 3
From (3.8) it follows that ∂φ1 R1 R ¼ Δθ1 ΔΦ1 : P1 ∂t P1
6. Onset of convection when P 1 ¼ 1
ð5:3Þ
ð5:4Þ
In view of (3.8) and (5.4), one obtains that the (linear) evolution of the first Fourier components of the independent fields fZ; w; θ; φg
ð6:5Þ
(iii) in order to find conditions guaranteeing the onset of oscillatory convection, one has to look for the lowest positive value R C of R2 verifying (6.5) in the interval 0; R C ½. Then oscillatory convection occurs for R2 ¼ R C ; (iv) if R C 2 = 0; R C ½, convection occurs via a steady state at R2 ¼ R C ¼ RC 4 . Lemma 6.1. Let P 1 ¼ 1. Then the lowest positive value of R2 such that (6.2) admits a pure imaginary root is given by 2
R C ¼ R21 þ 2
ð1 þ P r Þ2 ða c þ π 2 Þ3 þ P 2r π 2 T 2 2
ð1 þ P r Þa c
;
ð6:6Þ
126
R. De Luca, S. Rionero / International Journal of Non-Linear Mechanics 78 (2016) 121–130 2
where a c is the unique positive solution of the equation ðX ¼ a2 þ π 2 Þ 2X 3 3π 2 X 2
P 2r π 2 T 2
¼ 0: ð6:7Þ ð1 þ P r Þ2 Proof. The proof is easily obtained by remarking that the lowest value of R2 such that (6.5) holds is given by (6.6).□ Lemma 6.2. If P r o 1;
T 24
ð1 þ P r Þða2c4 þ π 2 Þ3 ; ð1 P r Þπ 2
8 R21 ;
ð6:8Þ
then
when R2 ¼ R C , Eq. (6.2) admits the root λ ¼ 0. Then, the strong principle of exchange of stability holds if and only if all the roots of the equation
λ2 I~1 λ þ I~2 ¼ 0;
are real numbers. If (6.18) admits two complex conjugate roots, the weak principle of exchange of stability holds. In view of ða2c þ π 2 Þ3 þ 4π 2 T 2 P r ð1 P r Þ ½a2 ¼ a2c 2 4 ¼ 4 ; ½I~1 4I~2 2 ½R ¼ R C a2c4 þ π 2
ð6:9Þ
P r r1;
P r 41;
ða2c4 þ π 2 Þ3 þ π 2 T 2 ; a2c4
ð6:11Þ
T 2r
π Þ ; ð1 P r Þπ 2
8 R21 ;
ð6:12Þ
or for P r Z 1;
8 T 2 ; 8 R21 ;
ð6:13Þ
(ii) the strong principle of exchange of stability holds either for (6.12) or for P r ¼ 1;
8 T 2;
8 R21 ;
ð6:14Þ
or for P r 4 1;
T 2r
ða2c4 þ π 2 Þ3 ; 4π 2 P r ðP r 1Þ
8 R21 ;
ð6:21Þ
Comparing (6.12) to (6.20) one finds that in the case (6.12) the strong principle of exchange of stability holds. Analogously comparing (6.13) to (6.20) and (6.21) one obtains respectively that in the cases (6.14) and (6.15) the strong principle of exchange of stability holds while in the case (6.16) the weak principle of exchange of stability holds.□
2
ð1 þ P r Þ2 ða c þ π 2 Þ3 þ P 2r π 2 T 2 ; R2 ¼ R C ¼ R21 þ 2 2 ð1 þ P r Þa c
ð6:22Þ
when (6.8) holds. Proof. The proof follows in view of (iii) of Remark 6.1 and Lemma 6.2.□ Critical values of the Rayleigh number for the onset of convection when P 1 ¼ 1 are collected in Table 1.
ð6:15Þ 7. Onset of convection when Pr ¼1
(iii) the weak principle of exchange of stability holds for ða2c4 þ π 2 Þ3 ; 8 R21 : 4π 2 P r ðP r 1Þ Proof. In view of (3.18)3 and (6.6), it turns out that " 2 ð1 þP r Þða c þ π 2 Þ3 þ ðP r 1Þπ 2 T 2 R C R C o ð1 þ 2P r Þ 2 ð1 þP r Þa c : P r 4 1;
ða2c4 þ π 2 Þ3 ; 8 R21 : 4π 2 P r ðP r 1Þ
Theorem 6.2 (Onset of oscillatory convection). Let P 1 ¼ 1. Then oscillatory convection occurs at
either for T 2 r ð1 þ P r Þ
ð6:20Þ
or for
(i) stationary convection occurs at
2 3
8 R21 ;
8 T 2;
ð6:10Þ
Theorem 6.1 (Onset of stationary convection). Let P 1 ¼ 1. Then:
2 ða c þ
2
½R ¼ R C
Hence in the case (6.8), (6.9) holds.□
R2 ¼ R C ¼ R21 þ
ð6:19Þ
½a ¼ ac 2 4 one has that ½I~1 4I~ 2 2 Z 0 if and only if 2
R C o RC : Proof. In view of (3.18)3 and (6.6), it follows that h i ð1 þ 2P r Þ ð1 þ P r Þða2c4 þ π 2 Þ3 þ ðP r 1Þπ 2 T 2 RC R C 4 : ð1 þ P r Þa2c4
P r o 1;
ð6:18Þ
T 24
ð6:16Þ
Let Pr ¼1. Then λ ¼ ξ1 is an eigenvalue of L1 and the spectral equation of L1 can be written as ðλ þ ξ1 Þðλ I~1 λ þ I~2 λ I~3 Þ ¼ 0; 3
ð6:17Þ
Hence, in the cases (6.12)–(6.13), it follows that R C is the critical Rayleigh number for the onset of convection and the marginal state R2 ¼ R C is stationary. Passing now to analyze the validity of the strong principle of exchange of stability, let us recall that,
2
ð7:1Þ
with I~ 1 ¼ I1 þ ξ1 ;
I~2 ¼ I2 þ ξ1 I~ 1 ;
I~3 ¼ I3 þ ξ1 I~ 2
ð7:2Þ
Ij (j¼1,2,3) being L1 principal invariants when Pr ¼1. Following, step by step, the procedure given in Section 6, the following results can be obtained.
Table 1 Onset of convection in the case P 1 ¼ 1. T2
Pr
a2c4
2
ac
RC
RC
Convection
10
0.1
5.1535
4.9367
R21 þ 677:08
R21 þ 1446:89
Stationary
100
0.1
6.7265
4.9534
R21 þ826.29
R21 þ1450.15
Stationary
800
0.2
12.6911
5.4048
R21 þ1526.96
R21 þ1679.82
Stationary
1000
1.2
13.7674
9.0406
R21 þ1676.12
R21 þ4720.25
Stationary
3000
0.01
20.7578
4.9414
R21 þ28210.43
R21 þ1329.36
Oscillatory
9500
0.5
31.8585
14.0438
R21 þ5223.71
R21 þ5146.65
Oscillatory
10 500
0.6
33.0558
15.9040
R21 þ5527.77
R21 þ6377.05
Stationary
R. De Luca, S. Rionero / International Journal of Non-Linear Mechanics 78 (2016) 121–130
Lemma 7.1. Let Pr ¼1. Then the lowest positive value of R2 such that (7.1) admits a pure imaginary root is given by RC ¼
2
1 þ P1
where
R21 þ 2P 21 2 ac
ð1 þP 1 Þ2 ða c þ π 2 Þ3 þ P 21 π 2 T 2 2
P 21 a c
;
ð7:3Þ
8. Onset of convection when P 1 P r ¼ 1 Let P 1 P r ¼ 1. Then λ ¼ ξ1 =P 1 is an L1 -eigenvalue and the spectral equation of L1 can be written as
λþ
is the unique positive solution of the equation ðX ¼ a þ π Þ 2
2X 3 3π 2 X 2
P 21 π 2 T 2 ð1 þ P 1 Þ2
¼ 0:
2
ð7:4Þ
Theorem 7.1 (Onset of stationary convection). Let Pr ¼1. Then:
R2 ¼ R C ¼ R21 þ
ða2c4 þ π 2 Þ3 þ π 2 T 2 ; a2c4
ð7:5Þ
either for P 1 r 1;
8 T 2 ; 8 R21 ;
P1
3 2 ðλ I~ 1 λ þ I~ 2 λ I~3 Þ ¼ 0;
ð7:6Þ
ð8:1Þ
with
ξ I~ 2 ¼ I2 þ 1 I~1 ; P1
ξ I~ 3 ¼ I3 þ 1 I~ 2 P1
ð8:2Þ
Ij (j ¼1,2,3) being L1 -principal invariants when P 1 P r ¼ 1. Following, step by step, the procedure given in Section 6, the following results can be obtained. Lemma 8.1. Let P 1 P r ¼ 1. Then the lowest positive value of R2 such that (8.1) admits a pure imaginary root is given by
or for
2
P 1 4 1;
R21 r 2
2 ða c þ
π 2 Þ3 2
ðP 1 1Þa c
RC ¼ ;
2
8T ;
P 1 o 1;
R21 r
2 ð1 þ P 1 Þ2 ða c þ π 2 Þ3 þ π 2 T 2 R21 þ 2 ; 2 P 1 ð1 þP 1 Þ P ð1 þ P Þa 1
ð7:7Þ
1
ð8:3Þ
c
2
where a c is the unique positive solution of the equation ðX ¼ a2 þ π 2 Þ
(ii) the strong principle of exchange of stability holds either for (7.7) or for ð2P 1 1Þ2 ða2c4 þ π 2 Þ3 ; 4P 1 ð1 P 1 Þa2c4
8 T 2;
ð7:8Þ
2X 3 3π 2 X 2
π2T 2 ð1 þ P 1 Þ2
¼ 0:
ð8:4Þ
Theorem 8.1 (Onset of stationary convection). Let P 1 P r ¼ 1. Then: (i) stationary convection occurs at
or for P 1 ¼ 1;
8 T 2;
8 R21 ;
ð7:9Þ
R2 ¼ R C ¼ R21 þ
(iii) the weak principle of exchange of stability holds for P 1 o 1;
π
ð2P 1 1Þ2 ða2c4 þ 2 Þ3 R21 4 ; 4P 1 ð1 P 1 Þa2c4
2
8T ;
2 1 þ P 1 2 ð1 þP 1 Þ2 ða c þ 2 Þ3 þ P 21 R þ 1 2 2P 21 P 21 a c
π
π2T 2
ð7:10Þ
;
ða2c4 þ π 2 Þ3 þ π 2 T 2 ; a2c4
ð8:5Þ
either for
Theorem 7.2 (Onset of oscillatory convection). Let Pr ¼ 1. Then oscillatory convection occurs at R2 ¼ R C ¼
ξ1
ξ I~ 1 ¼ I1 þ 1 ; P1
(i) stationary convection occurs at
127
8 R21 ;
8 T 2;
P 1 r 1;
ð8:6Þ
Table 3 Oscillatory convection in the case Pr ¼ 1.
ð7:11Þ
T2
P1
for
100
1.2 6.7265
P 1 4 1;
R21 þ 2398.12 800 1.3 12.6911 8.6223 R2 þ 1526.96 0.680 1 R21 þ 3211.30 1500 1.4 15.9976 10.8194 R21 þ 2007.33 0.612
R21 4
2ða2c4 þ π 2 Þ3 ; ðP 1 1Þa2c4
8 T 2:
ð7:12Þ
a2c4
3500 1.5 21.9710
Critical values of the Rayleigh number for the onset of convection when Pr ¼1 are collected in Tables 2 and 3.
ac
RC
RC
Oscillatory convection
5.5521
R21 þ 826.29
0.764
R21 4 6660:30
2
R21 þ 3773.69 14.9330 R21 þ 3041.48 0.556 R21 þ 5151.39
R21 4 5263:56 R21 4 4552:47 R21 4 4752:05
Table 2 Stationary convection in the case Pr ¼1. T2
P1
a2c4
2
ac
RC
Stationary convection
RC
10
0.1
5.1535
4.9367
R21 þ 677:08
55R21 þ 79578:9
8 R21
100
1.2
6.7265
5.5521
R21 þ 826.29
0.764 R21 þ 2398.12
R21 r 6660:30
800
1.3
12.6911
8.6223
R21 þ 1526.96
0.680 R21 þ 3211.30
R21 r 5263:56
1500
1.4
15.9976
10.8194
1.5
21.9710
14.9330
R21 þ 3773.69 R21 þ 5151.39
R21 r 4552:47
3500
R21 þ 2007.33 R21 þ 3041.48
0.612 0.556
R21 r 4752:05
128
R. De Luca, S. Rionero / International Journal of Non-Linear Mechanics 78 (2016) 121–130
Table 4 Stationary convection in the case P 1 P r ¼ 1. P1
T2
a2c4
2
ac
RC
RC
Stationary convection
10
1.1
5.1535
4.9855
R21 þ 677:08
0:866R21 þ 2527:73
R21 r 13810:80
100
1.2
6.7265
5.3736
R21 þ826.29
0.758 R21 þ2555.91
R21 r 7147:19
800
1.3
12.6911
7.4288
R21 þ1526.96
0.669 R21 þ3176.49
R21 r 4983:47
1500
1.4
15.9976
8.6726
R21 þ2007.33
0.595 R21 þ3536.37
R21 r 377:5:41
3500
1.5
21.9710
11.5007
R21 þ3041.48
0.595 R21 þ4697.38
R21 r 4088:64
RC
RC
Oscillatory convection R21 4 13810:80
Table 5 Oscillatory convection in the case P 1 P r ¼ 1. P1
T2
a2c4
2
ac
10
1.1
5.1535
4.9855
R21 þ 677:08
0:866R21 þ 2527:73
100
1.2
6.7265
5.3736
R21 þ826.29
0.758 R21 þ 2555.91
R21 4 7147:19
800
1.3
12.6911
7.4288
R21 þ1526.96
0.669 R21 þ3176.49
R21 4 4983:47
1500
1.4
15.9976
8.6726
R21 þ2007.33
0.595 R21 þ 3536.37
R21 4 377:5:41
3500
1.5
21.9710
11.5007
R21 þ3041.48
0.595 R21 þ 4697.38
R21 4 4088:64
Linearization principle. Decay of linear energy for any initial data implies decay of non-linear energy at any instant.
or for P 1 4 1;
2
2
a c R21 þ π 2 T 2 o
ðP 1 þ 1Þða c þ π 2 Þ3 ; P1 1
ð8:7Þ
(ii) the strong principle of exchange of stability holds for (8.7) or for P 1 o1;
a2c4 R21 þ π 2 T 2 r
P 21 ða2c4 þ π 2 Þ3 ; 4ð1 P 1 Þ
ð8:8Þ
or for P 1 ¼ 1;
8 T 2;
8 R21 ;
ð8:9Þ
(iii) the weak principle of exchange of stability holds for P 1 o1;
P 21 ða2c4 þ π 2 Þ3 : a2c4 R21 þ π 2 T 2 4 4ð1 P 1 Þ
Theorem 8.2 (Onset of oscillatory convection). Let P 1 P r ¼ 1. Then oscillatory convection occurs at 2
1
1
ð8:11Þ
c
for P 1 4 1;
a2c4 R21 þ π 2 T 2 4
ðP 1 þ 1Þða2c4 þ π 2 Þ3 : P1 1
8 1 1 X X > > > ∇π ¼ ∇π n ; u ¼ un ; > > > > n¼1 n¼1 > > > 1 1 X X > < θ¼ θn ; Φ ¼ Φn ; n¼1 n¼1 > > > > > ∇ un ¼ 0; > > > > ∂un ∂vn > > ¼ ¼ wn ¼ θn ¼ Φn ¼ 0; : ∂z ∂z
ð9:1Þ
on z ¼ 0; 1
and ð8:10Þ
2 ð1 þ P 1 Þ2 ða c þ π 2 Þ3 þ π 2 T 2 R21 þ 2 ; R2 ¼ R C ¼ 2 P 1 ð1 þ P 1 Þ P ð1 þ P Þa
Let ð∇π ; u; θ; ΦÞ – with w; θ; Φ; A AðΩÞ – be solution of (2.6)– (2.7) with
8 > ð0Þ > > > ð∇π n Þðt ¼ 0Þ ¼ ∇π n ; > > > > > > > > > > ðun Þðt ¼ 0Þ ¼ uð0Þ > n ; > < > ð0Þ > > ðθn Þðt ¼ 0Þ ¼ θn ; > > > > > > > > > ð0Þ > > > > ðΦn Þðt ¼ 0Þ ¼ Φn ; : ð0Þ
ð8:12Þ
Critical values of the Rayleigh number for the onset of convection when P 1 P r ¼ 1 are collected in Tables 4 and 5.
9. Non-linear global stability Our aim is to generalize to rotating layers the linearization principle obtained in [17], via the Auxiliary System Method [13–17].
1 X
ð∇π Þðt ¼ 0Þ ¼ ∇π ð0Þ ¼
∇π nð0Þ ;
n¼1
ðuÞðt ¼ 0Þ ¼ uð0Þ ¼ ðθÞðt ¼ 0Þ ¼ θ
ð0Þ
ðΦÞðt ¼ 0Þ ¼ Φ
¼
ð0Þ
1 X
uð0Þ n ;
n¼1 1 X
θð0Þ n ;
n¼1 1 X
¼
ð9:2Þ
Φnð0Þ ;
n¼1
ð0Þ
∇π ð0Þ ; uð0Þ ; θ ; Φ being assigned initial data such that ∇ uð0Þ ¼ 0. To (2.6)–(2.7), following the guideline of the Auxiliary System Method (AS Method, [17]), we associate 8 n A N the auxiliary system 8 > 1 ∂u n > > > P r ∂t ¼ ∇π n þ Δu n þ T u n k þ Rθ n R1 Φ n k u ∇u n ; > > > > > ∇ u n ¼ 0; > < ∂θ n > ¼ Rw n þ Δθ n P r u ∇θ n ; > > ∂t > > > > > ∂Φ n > > ¼ R1 w n þ ΔΦ n P r P 1 u ∇Φ n ; : P1 ∂t ð9:3Þ
R. De Luca, S. Rionero / International Journal of Non-Linear Mechanics 78 (2016) 121–130
under the initial boundary conditions 8 > ðu n Þðt ¼ 0Þ ¼ uð0Þ ð∇π n Þðt ¼ 0Þ ¼ ∇π ð0Þ n ; n ; > > > < ð0Þ ð0Þ ðθ n Þðt ¼ 0Þ ¼ θn ; ðΦ n Þðt ¼ 0Þ ¼ Φn ; > > > ∂u n ∂v n > ¼ ¼ w n ¼ θ n ¼ Φ n ¼ 0; z ¼ 0; 1: : ∂z ∂z
129
for arbitrary initial data. Then
ð9:4Þ
dEn o 0; 8 t Z0: ð9:14Þ dt Proof. The proof can be obtained by following, step by step, the procedure applied for the proof of Theorem 4.1 in [17].□ 9.2. Global non-linear energy stability
The following theorem holds. Theorem 9.1. Let ð∇π n ; u n ; θ n ; Φ n Þ 8 n A N be solution of (9.3)–(9.4). P1 P1 P1 P1 The series n ¼ 1 ∇π n , n ¼ 1 un, n ¼ 1 θn, n ¼ 1 Φ n are a.e. convergent in Ω and 1 X
1 X
∇π n ¼ ∇π ;
n¼1
u n ¼ u;
By virtue of the linearization principle, it easily follows that conditions guaranteeing linear stability guarantee also the nonlinear global asymptotic stability.
10. Discussion
n¼1
1 X
θ n ¼ θ;
n¼1
1 X
Φ n ¼ Φ:
ð9:5Þ
Double convection in rotating layers, heated and salted from below, is investigated. In particular:
n¼1
Proof. The proof can be obtained by following, step by step, the procedure applied for the proof of Theorem 3.1 in [17].□ 9.1. Linearization principle Let us introduce the energy En of the nth-Fourier component of the perturbation fields h i En ¼ 12 P r 1 J un J 2 þ J θn J 2 þ P 1 J Φn J 2
ð9:6Þ
and the energy E^ n of the solutions of (3.1) h i ^ n J2 E^ n ¼ 12 P r 1 J u^ n J 2 þ J θ^ n J 2 þP 1 J Φ
ð9:7Þ
with J un J 2 ¼ J un J 2 þ J vn J 2 þ J wn J 2 ; ^ n J 2: J u^ n J 2 ¼ J u^ n J 2 þ J v^ n J 2 þ J w
ð9:8Þ
Denoting by Q n ; Q^ n , the quadratic forms Q n ¼ ξn ðu2n þv2n þ w2n þ θn þ Φn Þ þ 2Rθn wn 2
2
(1) the lowest Rayleigh number R C at which stationary convection can occur, in an algebraic closed form is obtained; (2) an algebraic system for finding the lowest Rayleigh number R C at which oscillatory convection can occur, for any values of the fluid and salt Prandtl numbers, is introduced; (3) via the introduction of a new field, conditions guaranteeing the onset of steady or oscillatory convection are given in algebraic closed form in the cases P 1 ¼ 1 or Pr ¼1 or P 1 P r ¼ 1; (4) conditions guaranteeing that R C o R C , i.e. conditions for letting convection occurs via an oscillatory state, are provided; (5) linear stability is recovered as non-linear global asymptotic stability via the AS Method.
Acknowledgments This paper has been performed under the auspices of G.N.F.M. of INdAM. One of the authors (R. De Luca) acknowledges Progetto Giovani GNFM 2015 “Dinamica di sistemi complessi infinito dimensionali con applicazioni in Fluidodinamica, Economia e Biologia”. The accuracy of an anonymous referee is gratefully acknowledged by the authors.
ð9:9Þ References
and 2
2 2 ^ 2 Þ þ 2Rθ^ n w ^ 2n þ θ^ n þ Φ ^n Q^ n ¼ ξn ðu^ n þ v^ n þ w n
ð9:10Þ
it follows that the time derivative of En along the solutions of (9.3)–(9.4) is given by Z dEn ¼ Q n ðun ; vn ; wn ; θn ; Φn Þ dΩ dt Ω
ð9:11Þ
while the time derivative of E^ n along the solutions of (3.1)–(3.2) is given by Z dE^ n ^ n Þ dΩ: ^ n ; θ^ n ; Φ ¼ Q^ n ðu^ n ; v^ n ; w dt Ω
ð9:12Þ
According to [17], the linearization principle is based on the following theorem. Theorem 9.2. Let dE^ n dt
! o 0; ðt ¼ 0Þ
ð9:13Þ
[1] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Edition, Publications Inc. New York, 1981. [2] F. Capone, R. De Luca, Ultimately boundedness and stability of triply diffusive mixtures in rotating porous layers under the action of Brinkman law, Int. J. Non-linear Mech. 47 (7) (2012) 799–805. [3] F. Capone, R. De Luca, I. Torcicollo, Longtime behaviour of vertical throughflows for binary mixtures in porous layers, Int. J. Non-linear Mech. 52 (2013) 1–7. [4] F. Capone, V. De Cataldis, R. De Luca, I. Torcicollo, On the stability of vertical constant throughflows for binary mixtures in porous layers, Int. J. Non-linear Mech. 59 (2014) 1–8. [5] F. Capone, S. Rionero, Inertia effect on the onset of convection in rotating porous layers via the “auxiliary system method”, Int. J. Non-linear Mech. 52 (2013) 1–7. [6] J. Flavin, S. Rionero, Qualitative estimates for Partial Differential Equations. An Introduction, CRC Press, Boca Raton, FL, 1996. [7] D.D, Joseph, Stability of Fluid Motions I,II, Springer Tracts in Natural Philosophy, vols. 27–28, Springer, Berlin, 1976. [8] D.R. Merkin, Introduction to the Theory of Stability. Texts in Applied Mathematics, vol. 24, Springer-Verlag, New York, 1997, xxþ 319 pp. [9] G. Mulone, S. Rionero, On the nonlinear stability of the rotating Bènard problem via the Lyapunov Direct Method, J. Math. Anal. Appl. 144 (1989) 109–127. [10] G. Mulone, S. Rionero, The rotating Bènard problem: new stability results for any Prandtl and Taylor numbers, Contin. Mech. Thermodyn. 9 (1997) 347–363. [11] D.A. Nield, A. Bejan, Convection in Porous Media, IV Ed, Springer-Verlag, Berlin, Heidelberg, New York, 2012. [12] K.R. Rajagopal, G. Saccomandi, L. Vergori, Stability analysis of the Rayleigh– Bènard convection for a fluid with temperature and pressure dependent viscosity, Z. Angew. Math. Phys. 60 (4) (2009) 739–755.
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[13] S. Rionero, Absence of subcritical instabilities and global nonlinear stability for porous ternary diffusive–convective fluid mixtures, Phys. Fluids 24 (2012) 104101. [14] S. Rionero, Multicomponent diffusive–convective fluid motions in porous layers: ultimately boundedness, absence of subcritical instabilities and global nonlinear stability for any number of salts, Phys. Fluids 25 (2013) 054104. [15] S. Rionero, Soret effects on the onset of convection in rotating porous layers via the “auxiliary system method”, Ric. Mat. 62 (2) (2013) 183–208. [16] S. Rionero, Onset of convection in rotating porous layers via a new approach, AIMS 19 (7) (2014) 2279–2296.
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Contents lists available at ScienceDirect
International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
Steady and oscillatory convection in rotating fluid layers heated and salted from below Roberta De Luca a, Salvatore Rionero a,b,n a b
University of Naples Federico II, Department of Mathematics and Applications “Renato Caccioppoli”, Via Cinzia, 80126 Naples, Italy Accademia Nazionale dei Lincei, Via della Lungara 10, 06165 Rome, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 24 July 2015 Received in revised form 9 October 2015 Accepted 28 October 2015 Available online 6 November 2015
Double convection in rotating horizontal layers filled by a Navier–Stokes fluid mixture, heated and salted from below, is investigated. Onset of linear instability – for any value of the fluid and salt Prandtl numbers Pr, P1 – either via the Routh–Hurwitz conditions or via steady or oscillatory convection, is characterized. Introducing a new field connecting the perturbation fields to the temperature and salt concentration, in the cases P 1 ¼ 1 or Pr ¼1 or P 1 P r ¼ 1, stability conditions in algebraic closed form are obtained. Linear stability is recovered as non-linear global asymptotic stability via the Auxiliary System Method. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Double-convection Auxiliary System Method Global stability Navier–Stokes
1. Introduction Because of the numerous applications in the real world phenomena (industrial processes, water pollution, geology, volcanism, etc.), many efforts have been devoted to analyze the stability of the thermal conduction solution of multicomponent Navier–Stokes fluid mixture in the absence of rotation [3,4,12–14,17,18] and in the more realistic case of the presence of rotation [1,2,5–7, 9–11,15,16,19,20]. However, as far as we know, in the case of double convection in rotating layers either the onset of linear instability or the non-linear energy stability analysis is not completely investigated. In fact – as concerns the linear instability – the onset of convection via steady or oscillatory state is not completely characterized via algebraic closed form. Further – as concerns the non-linear energy stability – the coincidence between the linear and non-linear stability conditions is obtained generally under severe restrictions on the initial data. In the present paper we reconsider the problem in the case of rotating layers heated and salted from below, aimed to characterize via algebraic closed forms the onset of instability via steady or oscillatory convection. In particular, our scope is to show that: (1) in the cases P 1 ¼ 1 or Pr ¼1 or P 1 P r ¼ 1, the onset of convection can be characterized via n Corresponding author at: University of Naples Federico II, Department of Mathematics and Applications “Renato Caccioppoli”, Via Cinzia, 80126 Naples, Italy. E-mail addresses: [email protected] (R. De Luca), [email protected], [email protected] (S. Rionero).
http://dx.doi.org/10.1016/j.ijnonlinmec.2015.10.011 0020-7462/& 2015 Elsevier Ltd. All rights reserved.
algebraic closed form by introducing a new unknown field; (2) linear stability can be recovered as non-linear global asymptotic stability via the Auxiliary System Method (see [13–17]).
2. Mathematical model Let L be a horizontal layer of depth d filled by a Navier–Stokes fluid mixture in which a chemical specie (salt) S is dissolved in and let Oxyz be an orthogonal frame of reference with fundamental unit vectors i; j; k (k pointing vertically upwards). We suppose that L is uniformly heated from below and rotates uniformly about the vertical axis with constant angular velocity ω ¼ ωk. The equat ions governing the fluid motion, in the Boussinesq approximation, are [1,7]: 8 ρ0 ðvt þ v ∇vÞ ¼ ∇P þ ρ0 νΔv 2ρ0 ωk v ρ0 ½1 AðT T 0 Þ þA1 ðC C 0 Þgk; > > > > < ∇ v ¼ 0; > > T t þv ∇T ¼ kΔT; > > : C þv ∇C ¼ k ΔC; t
1
ð2:1Þ ρ0
with ρ0 being constant density, P ¼ p 2 j ω xj ; x ¼ ðx; y; zÞ, v the fluid velocity, T the temperature, C the salt concentration, p the pressure, T 0 the reference temperature, g ¼ gk the gravity, C 0 the reference salt concentration, ν the kinematic viscosity, A the thermal expansion coefficient, A1 the salt expansion coefficient, k the thermal diffusivity, k1 the salt diffusivity. 2
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(2)
To (2.1) we append the boundary conditions 8 Tðx; y; 0; tÞ ¼ T l ; Tðx; y; d; tÞ ¼ T u ; T l 4 T u > < Cðx; y; 0; tÞ ¼ C l ; Cðx; y; d; tÞ ¼ C u ; C l 4 C u > : v k ¼ 0; on z ¼ 0; d:
derivatives, can be expanded in a Fourier series absolutely uniformly convergent in Ω; 8 t A R þ
ð2:2Þ
and let us denote by BðΩÞ the set of the functions
The boundary value problem (2.1)–(2.2) admits the thermal conduction solution m 0 ¼ ðp; v; T ; C Þ given by 8 δT δC > > z; C ¼ C l z; v ¼ 0; T ¼ T l > > > d d < δT ¼ T l T u ; δ C ¼ C l C u ; > > ρ ω2 z2 > ρ gz2 > > AδT A1 δC þ 0 ; : pðzÞ ¼ p 0 ρ0 gz½1 AðT l T 0 Þ þA1 ðC l C 0 Þ 0 2d 2
ð2Þ0 p 0 ¼ const: 40:
Setting
φ such that
ð1Þ0 φ : ðx; tÞ A Ω R þ -φðx; tÞ A R, φ A W 2;2 ðΩÞ; 8 t A R þ , φ is periodic y directions of period 2aπx , 2aπy respectively h i in thehx and i ∂φ ∂φ and ∂z ¼ ∂z ¼ 0; z¼0
z¼1
φ, together with all the first derivatives and second spatial
derivatives, can be expanded in a Fourier series absolutely uniformly convergent in Ω; 8 t A R þ .
ð2:3Þ
Since the sequence f sin nπ zgn A N is a complete orthogonal system
p ¼ p þ π;
v ¼ v þ u; T ¼ T þ θ;
C ¼ C þΦ
ð2:4Þ
and introducing the non-dimensional scalings 8 2 > d ν ν2 ρ > > > t ¼ t n ; u ¼ un ; π ¼ π n 2 0 ; > > k d d > > > n ♯ > n > x ¼ x d; θ ¼ θ T ; > > > !1=2 < ν3 δT n ♯ ♯ Φ ¼ Φ Φ ; T ¼ ; > 3 > > Agkd > > > !1=2 > > > > ν3 δ C > ♯ > Φ ¼ ; > > 3 : A1 gk1 d Eq. (2.1) (omitting the asterisks) reduces to 8 1 > P r ut þ u ∇u ¼ ∇π þ Δu þ T u k þ Rθ R1 Φ k; > > > < ∇ u ¼ 0; > θt þ P r u ∇θ ¼ Rw þ Δθ; > > > : P ðΦ þ P u ∇ΦÞ ¼ R w þ ΔΦ; 1
Ψ, together with all the first derivatives and second spatial
t
r
where 8 3 > Agd δT > > > R2 ¼ > > ν k > > > 3 > > A gd δCP 1 1 > 2 > > R1 ¼ > > ν k > < 4 4 ω2 d T2¼ > 2 > ν > > > ν > > > Pr ¼ > > k > > > > k > > > : P1 ¼ k 1
for L2 ð0; 1Þ under the boundary conditions ½Ψ z ¼ 0 ¼ ½Ψ z ¼ 1 ¼ 0, by virtue of periodicity, it turns out that 8 Ψ A AðΩÞ, there exists a sequence fΨ~ ðx; y; tÞg (Ψ~ being of “plane form”) such that nAN
n
n
8 1 1 X X > > > Ψn ¼ Ψ~ n sin nπ z; >Ψ ¼ > < n¼1
ð2:5Þ
n¼1
> > > Δ Ψ ¼ a2 Ψ ; > > : 1
ΔΨ ¼
1 X
1 X ∂Ψ ∂Ψ~ n ¼ sin nπ z; ∂t ∂t n¼1
ξn Ψ~ n sin nπ z;
ð2:8Þ
n¼1
with Δ1 ¼ ∂2 =∂x2 þ∂2 =∂y2 and
ξn ¼ a2 þ n2 π 2 ;
a2 ¼ a2x þ a2y ;
ð2:9Þ
the series appearing in (2.8) being absolutely uniformly conver-
ð2:6Þ
1
thermal Rayleigh number; salt Rayleigh number;
gent in Ω. Analogously, since the sequence f cos nπ zgn A N is a complete orthogonal for L2 ð0; 1Þ under the boundary conditions h i h system i ∂φ ∂φ ∂z z ¼ 0 ¼ ∂z z ¼ 1 ¼ 0, by virtue of periodicity, it turns out that ~ n ðx; y; tÞgn A N (φ ~ n being of 8 φ A BðΩÞ, there exists a sequence fφ “plane form”) such that 8 1 1 1 X X X ~n ∂φ ∂φ > > ¼ cos nπ z; φ¼ φn ¼ φ~ n cos nπ z; > > < ∂t ∂t n¼1 n¼1 n¼1 ð2:10Þ 1 X > > 2 > ξn φ~ n cos nπ z: > : Δ1 φ ¼ a φ; Δφ ¼ n¼1
Taylor number; Setting fluid Prandtl number;
ζ ¼ ð∇ uÞ k ¼
salt Prandtl number:
To (2.6) the boundary conditions (free-free case) are appended ∂u ∂v ¼ ¼w¼θ¼Φ¼0 ∂z ∂z
on z ¼ 0; 1;
ð2:7Þ
with u ¼ ðu; v; wÞ. We assume (as usually done, in stability problems in layers) that: (i) the perturbations ð∇π ; u; v; w; θ; ΦÞ are periodic in the x and y directions, respectively of periods 2π =ax ; 2π =ay ; (ii) Ω ¼ ½0; 2π =ax ½0; 2π =ay ½0; 1 is the periodicity cell; (iii) u; v; w, θ, Φ are such that together with all their first derivatives and second spatial derivatives are square integrable in Ω; 8 t A R þ and can be expanded in a Fourier series uniformly convergent in Ω.
∂v ∂u ; ∂x ∂y
ð2:11Þ
the horizontal components of u are given by 1 ∂2 w ∂ζ 1 ∂2 w ∂ζ u¼ 2 þ ; v¼ 2 ð2:12Þ a ∂x∂z ∂y a ∂y∂z ∂x P ∂vn ∂un and – in view of u ¼ 1 n ¼ 1 un ; ζ n ¼ ∂x ∂y – it follows that 8 1 ∂ζ 1 ∂ 2 wn ∂ζ n 2 > wn > ; þ n ; vn ¼ 2 > un ¼ 2 ∂∂x∂z < ∂y ∂y∂z ∂x a a ð2:13Þ 1 > > > : ∇ un ¼ a2 Δ1 wn þ wn ¼ 0 z Remark 2.1. We remark that, in view of (2.10)1, one has Z Z u dΩ v dΩ 0: Ω
Ω
ð2:14Þ
Remark 2.2. Let us denote by J J and 〈; 〉 the L ðΩÞnorm and the scalar product, respectively. Multiplying (2.6)1 for u and integrating over Ω, one has 2
Let us denote by AðΩÞ the set of functions
Ψ such that:
(1) Ψ : ðx; tÞ A Ω R þ -Ψ ðx; tÞ A R, Ψ A W 2;2 ðΩÞ; 8 t A R þ , Ψ is periodic in the x and y directions of period 2aπx , 2aπy respectively and ðΨ Þz ¼ 0 ¼ ðΨ Þz ¼ 1 ¼ 0;
P r 1 d J u J 2 ¼ J ∇u J 2 þ 〈Rθ; w〉 〈R1 Φ; w〉: 2 dt
ð2:15Þ
R. De Luca, S. Rionero / International Journal of Non-Linear Mechanics 78 (2016) 121–130
Then, the Holder and Poincarè inequalities lead to P r 1
dJuJ r2R J θ J J w J þ 2R1 J Φ J J w J 2α J u J 2 ; dt 2
ð2:16Þ
where α ð 4 0Þ is the positive constant appearing in the Poincarè inequality. From (2.16) one obtains dJuJ rP r R J θ J þR1 J Φ J P r α J u J : dt
ð2:17Þ
Integrating (2.17), one has J u J r J u0 J e P r α t þ
Z
t
Pr e 0
P r α ðt τÞ
R J θ J þ R1 J Φ J dτ
ð2:18Þ
where u0 ¼ uðx; 0Þ. Hence if J θ J ; J Φ J , go exponentially to zero as t-1, then also J u J goes exponentially to zero as t-1.
3. Linear stability ^ Þ the solution of the linear system ^ θ^ ; Φ Denoting by ð∇π^ ; u; ^ n Þ the nth-Fourier linked to (2.6)–(2.7), and by ð∇π^ n ; u^ n ; θ^ n ; Φ ^ Þ, one has 8 n A N, ^ θ^ ; Φ component of ð∇π^ ; u; 8 ∂u^ n > ^ n k; > ¼ ∇π^ n þ Δu^ n þ T u^ n k þ Rθ^ n R1 Φ P r 1 > > > ∂t > > > > ∇ u^ n ¼ 0; > < ∂θ^ ^ n þ Δθ^ n ; > n ¼ Rw > > ∂t > > > > ^ > > ^ n: > P 1 ∂Φ n ¼ R1 w ^ n þ ΔΦ : ∂t
ð3:1Þ
with 8 > > ð0Þ > > ∇π n ¼ ð∇π n Þðt ¼ 0Þ ; > > > > > > > > > uð0Þ ¼ ðu Þ > > n ðt ¼ 0Þ ; > < n > > > θð0Þ ¼ ðθn Þðt ¼ 0Þ ; > > > n > > > > > > > > Φð0Þ ¼ ðΦn Þðt ¼ 0Þ ; > > : n ð0Þ
1 X
∇π ð0Þ ¼ ð∇π Þðt ¼ 0Þ ¼ uð0Þ ¼ ðuÞðt ¼ 0Þ ¼
θð0Þ ¼ ðθÞðt ¼ 0Þ ¼
n¼1 1 X
Φð0Þ ¼ ðΦÞðt ¼ 0Þ ¼
Φð0Þ n ;
ð0Þ
ð3:4Þ
∂ζ n ; ∂z
Z¼
n¼1
P r ξn
B B P r T =ξn Ln ¼ B B 0 @ 0
P r T n2 π 2 P r ξn
0 a2 P r R=ξn
R
ξn
R1 =P 1
0
0
1
C a2 P r R1 =ξn C C: C 0 A ξn =P 1
ð3:9Þ
The spectral equation of Ln is ð3:10Þ
2
ð3:12Þ
The conditions, necessary and sufficient for the stability of the thermal conduction solution, require that all the roots of (3.10) have negative real part 8 ðn; a2 Þ A N R þ . These conditions are well known in the literature and are given, for algebraic equations of any degree, by the Routh–Hurwitz conditions (cfr. [8]) which, for Eq. (3.10), specify as: I1n o0;
I3n o 0;
I4n 4 0; I1n I2n I3n 4 I23n þ I21n I4n ;
8 ðn; a2 Þ A N R þ :
ð3:13Þ
Remark 3.1. We remark that:
Therefore, setting Zn ¼
with 0
Φn
ð3:3Þ
∇π ð0Þ ; uð0Þ ; θ ; Φ being assigned initial data such that ∇ uð0Þ ¼ 0. The z-components of the curl and double curl of (3.1)1 are (omitting the hats)
1 X
Φn
ð3:8Þ
being 8 3 > ½P 1 P r ðP r þ 2Þ þ 2P r þ 1ξn þ P 1 P 2r n2 π 2 T 2 > > f 2 ðn; a2 ; T 2 Þ ¼ ; > > 2 > P1 Pr a > > > < 3 ½P r ðP 1 þ 1Þ þ 2ξn þ P r ðP 1 þ1Þn2 π 2 T 2 ; f 3 ðn; a2 ; T 2 Þ ¼ > ð1 þ P 1 P r Þa2 > > > > > > ξ3 þ n2 π 2 T 2 > > f 4 ðn; a2 ; T 2 Þ ¼ n : : a2
n¼1
8 ∂wn 1 ∂ζ n > > ¼ Δζ n þ T ; < Pr ∂t ∂z ∂ Δ w ∂ζ > 1 n > : Pr ¼ ΔΔwn T n þ Δ1 Rθn R1 Φn : ∂t ∂z
In view of (2.8), (3.6) can be written as 0 1 0 1 Zn Zn C B wn C ∂B B wn C B C B C ¼ Ln B C @ θn A ∂t @ θn A
ð3:2Þ
uð0Þ n ;
θð0Þ n ;
ð3:7Þ
where Ijn ¼ Ijn ðn; a ; T Þ; ðj ¼ 1; 2; 3; 4Þ; are the principal invariants of (3.9) given by 8 ξ ð1 þ P 1 þ 2P 1 P r Þ > > ; > I1n ¼ n > > P > > " 1 # > > 2 > P r a2 2 R1 > 2 2 > > I2n ¼ R f 2 ðn; a ; T Þ ; > < P1 ξn ð3:11Þ
2 > a P r ð1 þ P 1 P r Þ 2 1 þ Pr 2 > 2 2 > > ¼ R R f ðn; a ; T Þ ; I 3n 3 > > P1 1 þ P1 Pr 1 > > > > 2 2 h i > P r a ξn 2 > > > R R21 þf 4 ðn; a2 ; T 2 Þ ; : I4n ¼ P 1
∇π ð0Þ n ;
n¼1 1 X
ð3:6Þ
under the i.b.c. 8 ð0Þ > ðu Þ ¼ uð0Þ ðθn Þðt ¼ 0Þ ¼ θn ; n ; > < n ðt ¼ 0Þ ð0Þ ðΦn Þðt ¼ 0Þ ¼ Φn ; > > : Z n ¼ wn ¼ θn ¼ Φn ¼ 0; z ¼ 0; 1:
2
n¼1 1 X
from (3.1) one has 8 ∂Z n ∂2 wn > > P r 1 ¼ ΔZ n þ T ; > > > ∂t ∂z2 > > > > > 1 ∂ Δw n > ¼ ΔΔwn T Z n þ Δ1 Rθn R1 Φn ; < Pr ∂t > ∂θ n > > ¼ Rwn þ Δθn ; > > > ∂t > > > ∂Φ > > : P 1 n ¼ R1 wn þ ΔΦn ; ∂t
λ4 I1n λ3 þ I2n λ2 I3n λ þ I4n ¼ 0;
To (3.1) the following initial boundary conditions are appended 8 ð∇π^ n Þðt ¼ 0Þ ¼ ∇π ð0Þ ðu^ n Þðt ¼ 0Þ ¼ uð0Þ > n ; n ; > > > < ^ ð0Þ ð0Þ ^ nÞ ðθ n Þðt ¼ 0Þ ¼ θn ; ðΦ ¼ Φ ; ðt ¼ 0Þ n > > ∂u^ n ∂v^ n > ^ ^ > ^ n ¼ θ n ¼ Φ n ¼ 0; z ¼ 0; 1; : ¼ ¼w ∂z ∂z
123
Zn;
ð3:5Þ
(i) Eq. (3.13) imply I2n 40, 8 ðn; a2 Þ A N R þ ; (ii) in view of (3.11)1 , (3.13)1 is always verified 8 ðn; a2 Þ A N R þ ;
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ðXÞ ¼ þ 1 and g 02 ðXÞ ¼ 6X 2 6π 2 X ¼ 6a2 X 4 0; 8 X A R þ . Then a2c2 exists and is unique.□
(iii) each one of the conditions I2n 4 0;
I3n o 0;
I4n 4 0;
I1n I2n I3n 4I23n þ I21n I4n ;
ð3:14Þ
is necessary for the stability of the thermal conduction solution.
Lemma 3.2. Each one of the conditions R2 o RC 2 ;
R2 o RC 3 ;
R2 o RC 4 ;
ð3:21Þ
is necessary for the stability of the thermal conduction solution. Proof. The proof follows in view of (iii) of Remark 3.1 and (3.16).□
Denoting by 8 R2 > > min f ðn; a2 ; T 2 Þ; RC 2 ¼ 1 þ > > > P 1 ðn;a2 Þ A NR þ 2 > > < 1 þ Pr 2 R þ min f ðn; a2 ; T 2 Þ; RC 3 ¼ > 1 þP 1 P r 1 ðn;a2 Þ A NR þ 3 > > > > 2 > > min þ f 4 ðn; a2 ; T 2 Þ; : RC 4 ¼ R1 þ
Theorem 3.1. If and only if (3.13) are verified for n¼ 1, 8 a2 A R þ , the thermal conduction solution is linearly stable. Proof. Obviously one has only to show that ð3:15Þ
ðn;a2 Þ A NR
it turns out 8 I 4 0; > > < 2n I3n o 0; > > : I 4 0; 4n
that 8 ðn; a2 Þ A N R þ 3 R2 o RC 2 ; 8 ðn; a2 Þ A N R þ 3 R2 o RC 3 ;
ð3:16Þ
8 ðn; a2 Þ A N R þ 3 R2 o RC 4 :
Since the functions f i ðn; a2 ; T 2 Þ, ði ¼ 2; 3; 4Þ, are increasing functions of n, it follows that 8 > R2 > > RC 2 ¼ 1 þ min f 2 ð1; a2 ; T 2 Þ; > > P 1 a2 A R þ > > < 1 þ Pr 2 ð3:17Þ RC 3 ¼ R þ min f ð1; a2 ; T 2 Þ; > 1 þP 1 P r 1 a2 A R þ 3 > > > > 2 2 > 2 > : RC 4 ¼ R1 þ minþ f 4 ð1; a ; T Þ: a2 A R
I3 o 0;
I4 4 0;
I1 I2 I3 4 1; I23 þ I21 I4
8 a2 A R þ ;
ð3:22Þ
with Ij ¼ Ij1 ; ðj ¼ 1; 2; 3; 4Þ, implies (3.13). Following the procedure introduced in Section 3 of [16], we remark that (3.13) is equivalent to the variational problem 8 I1n I2n I3n > 4 1; > 2 > ðn;a2min Þ A NR þ I2 > > 3n þ I1n I4n < I2n 4 0; max I3n o 0; min ð3:23Þ ðn;a2 Þ A NR þ ðn;a2 Þ A NR þ > > > > > min þ I4n 4 0: : ðn;a2 Þ A NR
By virtue of min
ðn;a2 Þ A NR þ
f i ðn; a2 ; T 2 Þ ¼ minþ f i ð1; a2 ; T 2 Þ; a2 A R
ði ¼ 2; 3; 4Þ;
ð3:24Þ
and (3.15)–(3.17), it follows that the variational problem (3.23) is equivalent to (3.22).□ Remark 3.2. We remark that: (i) Eqs. (3.22) are the Routh–Hurwitz conditions guaranteeing that all the roots of
The following lemma holds. Lemma 3.1. RC 2 ; RC 3 ; RC 4 are respectively given by 8 2 3 2 > R2 ½P 1 P r ðP r þ2Þ þ2P r þ 1ða2c2 þ π 2 Þ þ P 1 P r π 2 T > > ; RC 2 ¼ 1 þ > 2 > P1 P 1 P r ac 2 > > > > > 3 2 < 1 þ P r 2 ½P r ðP 1 þ 1Þ þ 2ða2c3 þ π 2 Þ þ P r ðP 1 þ 1Þπ 2 T RC 3 ¼ R1 þ ; 2 1 þP 1 P r ð1 þ P 1 P r Þac3 > > > > > > ða2 þ π 2 Þ3 þ π 2 T 2 > > > RC ¼ R21 þ c4 ; > : 4 a2c4 ð3:18Þ where, setting X ¼ a2 þ π 2 , a2c2 ; a2c3 ; a2c4 are respectively given by the unique positive roots of the equations 8 > P 1 P 2r π 2 T 2 > > 2X 3 3π 2 X 2 ¼ 0; > > P P ðP > r r þ 2Þ þ 2P r þ 1 1 < 2 ð3:19Þ P r ðP 1 þ 1Þπ 2 T > 2X 3 3π 2 X 2 ¼ 0; > > P ðP þ 1Þ þ 2 > r 1 > > : 3 2X 3π 2 X 2 π 2 T 2 ¼ 0: Proof. We confine ourselves to prove (3.18)1 since the proof of (3.18)2– (3.18)3 can be obtained as the same manner. Let us remark that f 2 ðn; a2 ; T 2 Þ is an increasing function of n A N. Hence its minimum with respect to n is attained in n¼ 1. In view of ∂f 2 ð1; a2 ; T 2 Þ 3½P 1 P r ðP r þ2Þ þ2P r þ 1a2 ða2 þ π 2 Þ2 ½P 1 P r ðP r þ2Þ þ2P r þ 1ða2 þ π 2 Þ3 P 1 P 2r π 2 T 2 ¼ þ ; ∂a2 P 1 P r a4 P 1 P r a4
ð3:20Þ 2
I1 o 0;
it follows that f 2 ð1; a2 ; T Þ reaches its minimum with respect to a2 in a2c2 . Let us observe that, denoting by g 2 ðXÞ the function given by the left-hand side of (3.19)1, it turns out that g 2 ð0Þ o 0, limX-1 g 2
λ4 I1 λ3 þ I2 λ2 I3 λ þ I4 ¼ 0;
ð3:25Þ
have negative real part; (ii) in view of Theorem 3.1 we can confine ourselves to investigate the sign of the real part of the roots of (3.25); (iii) denoting by R~ the lowest positive value of R2 verifying I1 I2 I3 ¼ 1, it follows that the thermal conduction solution is I23 þ I21 I4 linearly stable if and only if ~ R2 o minfRC 3 ; RC 4 ; Rg; (iv) since for i ¼2,3,4 min
ðn;a2 Þ A NR
f ðn; a2 ; T 2 Þ 4 þ i
ð3:26Þ π2 2 ; 0 ; f ðn; a ; 0Þ ¼ f 1; i i 2 ðn;a2 Þ A NR þ min
ð3:27Þ one recovers for T 2 ¼ 0 the results obtained in [17] in the absence of rotation and that – as expected – rotation produces a stabilizing effect.
4. Onset of convection via steady or oscillatory state The instability occurs or via a stationary state – associated to a zero eigenvalue and named “stationary convection” – or via an oscillatory state – associated to a pure imaginary eigenvalue and named “oscillatory or overstable or Hopf convection”. Our aim, in this section, is to analyze (3.25) in order to determine the conditions guaranteeing respectively the onset of convection via steady state or via oscillatory state. Since the real part of the eigenvalues of (3.25) depends on the Rayleigh number R2, one is led to determine the lowest value R C of R2 at which (3.25) admits a null
R. De Luca, S. Rionero / International Journal of Non-Linear Mechanics 78 (2016) 121–130
root and the lowest value R C of R2 at which (3.25) admits a pure imaginary root. Then the critical value of R2 at which instability occurs is given by R2C ¼ minfR C ; R C g
ð4:1Þ
and instability occurs via a steady state for R2 ¼ R C o R C or via an oscillatory state for R2 ¼ R C o R C . Remark 4.1. We remark that: (i) λ ¼ 0 is a solution of (3.25) if and only if I4 ¼ 0. In this case (3.25) reduces to
λðλ3 I1 λ2 þ I2 λ I3 Þ ¼ 0:
ð4:2Þ
The lowest positive value of R2 such that (3.25) admits the root λ ¼ 0 is R C ¼ RC 4 ; (ii) λ ¼ iωn , with ωn a 0, is a solution of (3.25) if and only if
ω4n I2 ω2n þ I4 ¼ 0;
ð4:3Þ
with ωn given by
125
is governed by 0 1 0 1 Z1 Z1 B C B C w1 C B w1 C ∂B B C ¼ L1 B C B C B C ∂t @ θ1 A @ θ1 A
φ1
with
0
ð5:5Þ
φ1
P r ξ1
B B Pr T B B ξ1 B L1 ¼ B B 0 B B @ 0
Pr T π2 P r ξ1 R 0
0
P r a2 P 1 R2 R21 P 1 Rξ1 ξ1 R1 2 ðP 1 1Þξ1 P1
0 a2 P r R1 Rξ1 0
ξ1 P1
1 C C C C C C: C C C A
ð5:6Þ
In the sequel we provide conditions, in algebraic closed form, guaranteeing the onset of convection via stationary or oscillatory state, in three particular cases: P 1 ¼ 1 or Pr ¼1 or P 1 P r ¼ 1.
2
ω2n ¼
I3 : I1
ð4:4Þ
Therefore only if I2 4 0;
I22 4I4 Z 0;
I3 o 0; I4 4 0;
ð4:5Þ 2
oscillatory convection can occur and occurs only if R ¼ R C , where R C is the lowest positive solution of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 I3 I2 7 I2 4I4 : ð4:6Þ ¼ 2 I1 Let us observe that, in view of (4.5)3, R C oR C ¼ RC 4 ; (iii) the solution of the general problem formalized in (4.5) for any P 1 ; P r will be solved in the cases P 1 ¼ 1 or Pr ¼ 1 or P 1 P r ¼ 1 in the following sections.
5. A new field connecting
θ and Φ
The analysis of (3.25) is simplified when at least one root is known. To this end we introduce a new field, connecting θ and Φ, aimed to prove that in the cases P 1 ¼ 1 or Pr ¼1 or P 1 P r ¼ 1, it is possible to find a real negative root of (3.25). Then the stability analysis is reduced to investigate the necessary and sufficient conditions guaranteeing that all the roots of a cubic equation have negative real part. Let us introduce the field
φ1 ¼
R1 θ1 RΦ1 : P1
ð5:1Þ
R1 φ θ1 1 : RP 1 R
Hence, (5.2) becomes ∂φ1 R1 1 Δφ ¼ 1 Δθ1 þ 1 : P1 ∂t P1 P1
~ 1 -spectral equation is have negative real part 8 a2 A R þ . The L
λ3 I~1 λ2 þ I~2 λ I~3 ¼ 0;
ð6:2Þ
where " # 8 3 2 2 > > ~ 1 ¼ ξ ð1 þ 2P r Þ; I~3 ¼ a2 P 2 R2 R2 ξ1 þ π T > ; I > 1 r 1 > a2 < " # 3 > P r a2 2 ð2 þ P r Þξ1 þ P r π 2 T 2 > > > I~ 2 ¼ R R21 : > : a2 ξ1
ð6:3Þ
The Routh–Hurwitz conditions, for the case at stake, specify as I~ 3 o 0;
I~ 1 I~2 I~3 o 0;
8 a2 A R þ :
ð6:4Þ
Remark 6.1. We remark that: (i) Eq. (6.2) admits the root λ ¼ 0 if and only if I~ 3 ¼ 0. The lowest positive value of R2 such that I~ 3 ¼ 0 is R C ¼ RC 4 ; (ii) Eq. (6.2) admits a pure imaginary root λ ¼ iωn , (ωn a 0), if and only if I~ 1 I~2 I~3 ¼ 0, i.e. ð1 þ P r ÞR2 ð1 þ P r ÞR21 2
ð5:2Þ
In view of (5.1), one has
Φ1 ¼
Let P 1 ¼ 1. Then (5.5)4 is independent of the other three equations and λ ¼ ξ1 is an L1 -eigenvalue. Then the null solution of (5.5) is stable if and only if the eigenvalues of the matrix 0 1 0 P r ξ1 P r T π 2 B C P r a2 ~ 1 ¼ B Pr T P r ξ1 R2 R21 C ð6:1Þ L R ξ1 @ ξ1 A 0 R ξ1
ð1 þ P r Þ2 ξ1 þ P 2r π 2 T 2 ¼ 0; a2 3
From (3.8) it follows that ∂φ1 R1 R ¼ Δθ1 ΔΦ1 : P1 ∂t P1
6. Onset of convection when P 1 ¼ 1
ð5:3Þ
ð5:4Þ
In view of (3.8) and (5.4), one obtains that the (linear) evolution of the first Fourier components of the independent fields fZ; w; θ; φg
ð6:5Þ
(iii) in order to find conditions guaranteeing the onset of oscillatory convection, one has to look for the lowest positive value R C of R2 verifying (6.5) in the interval 0; R C ½. Then oscillatory convection occurs for R2 ¼ R C ; (iv) if R C 2 = 0; R C ½, convection occurs via a steady state at R2 ¼ R C ¼ RC 4 . Lemma 6.1. Let P 1 ¼ 1. Then the lowest positive value of R2 such that (6.2) admits a pure imaginary root is given by 2
R C ¼ R21 þ 2
ð1 þ P r Þ2 ða c þ π 2 Þ3 þ P 2r π 2 T 2 2
ð1 þ P r Þa c
;
ð6:6Þ
126
R. De Luca, S. Rionero / International Journal of Non-Linear Mechanics 78 (2016) 121–130 2
where a c is the unique positive solution of the equation ðX ¼ a2 þ π 2 Þ 2X 3 3π 2 X 2
P 2r π 2 T 2
¼ 0: ð6:7Þ ð1 þ P r Þ2 Proof. The proof is easily obtained by remarking that the lowest value of R2 such that (6.5) holds is given by (6.6).□ Lemma 6.2. If P r o 1;
T 24
ð1 þ P r Þða2c4 þ π 2 Þ3 ; ð1 P r Þπ 2
8 R21 ;
ð6:8Þ
then
when R2 ¼ R C , Eq. (6.2) admits the root λ ¼ 0. Then, the strong principle of exchange of stability holds if and only if all the roots of the equation
λ2 I~1 λ þ I~2 ¼ 0;
are real numbers. If (6.18) admits two complex conjugate roots, the weak principle of exchange of stability holds. In view of ða2c þ π 2 Þ3 þ 4π 2 T 2 P r ð1 P r Þ ½a2 ¼ a2c 2 4 ¼ 4 ; ½I~1 4I~2 2 ½R ¼ R C a2c4 þ π 2
ð6:9Þ
P r r1;
P r 41;
ða2c4 þ π 2 Þ3 þ π 2 T 2 ; a2c4
ð6:11Þ
T 2r
π Þ ; ð1 P r Þπ 2
8 R21 ;
ð6:12Þ
or for P r Z 1;
8 T 2 ; 8 R21 ;
ð6:13Þ
(ii) the strong principle of exchange of stability holds either for (6.12) or for P r ¼ 1;
8 T 2;
8 R21 ;
ð6:14Þ
or for P r 4 1;
T 2r
ða2c4 þ π 2 Þ3 ; 4π 2 P r ðP r 1Þ
8 R21 ;
ð6:21Þ
Comparing (6.12) to (6.20) one finds that in the case (6.12) the strong principle of exchange of stability holds. Analogously comparing (6.13) to (6.20) and (6.21) one obtains respectively that in the cases (6.14) and (6.15) the strong principle of exchange of stability holds while in the case (6.16) the weak principle of exchange of stability holds.□
2
ð1 þ P r Þ2 ða c þ π 2 Þ3 þ P 2r π 2 T 2 ; R2 ¼ R C ¼ R21 þ 2 2 ð1 þ P r Þa c
ð6:22Þ
when (6.8) holds. Proof. The proof follows in view of (iii) of Remark 6.1 and Lemma 6.2.□ Critical values of the Rayleigh number for the onset of convection when P 1 ¼ 1 are collected in Table 1.
ð6:15Þ 7. Onset of convection when Pr ¼1
(iii) the weak principle of exchange of stability holds for ða2c4 þ π 2 Þ3 ; 8 R21 : 4π 2 P r ðP r 1Þ Proof. In view of (3.18)3 and (6.6), it turns out that " 2 ð1 þP r Þða c þ π 2 Þ3 þ ðP r 1Þπ 2 T 2 R C R C o ð1 þ 2P r Þ 2 ð1 þP r Þa c : P r 4 1;
ða2c4 þ π 2 Þ3 ; 8 R21 : 4π 2 P r ðP r 1Þ
Theorem 6.2 (Onset of oscillatory convection). Let P 1 ¼ 1. Then oscillatory convection occurs at
either for T 2 r ð1 þ P r Þ
ð6:20Þ
or for
(i) stationary convection occurs at
2 3
8 R21 ;
8 T 2;
ð6:10Þ
Theorem 6.1 (Onset of stationary convection). Let P 1 ¼ 1. Then:
2 ða c þ
2
½R ¼ R C
Hence in the case (6.8), (6.9) holds.□
R2 ¼ R C ¼ R21 þ
ð6:19Þ
½a ¼ ac 2 4 one has that ½I~1 4I~ 2 2 Z 0 if and only if 2
R C o RC : Proof. In view of (3.18)3 and (6.6), it follows that h i ð1 þ 2P r Þ ð1 þ P r Þða2c4 þ π 2 Þ3 þ ðP r 1Þπ 2 T 2 RC R C 4 : ð1 þ P r Þa2c4
P r o 1;
ð6:18Þ
T 24
ð6:16Þ
Let Pr ¼1. Then λ ¼ ξ1 is an eigenvalue of L1 and the spectral equation of L1 can be written as ðλ þ ξ1 Þðλ I~1 λ þ I~2 λ I~3 Þ ¼ 0; 3
ð6:17Þ
Hence, in the cases (6.12)–(6.13), it follows that R C is the critical Rayleigh number for the onset of convection and the marginal state R2 ¼ R C is stationary. Passing now to analyze the validity of the strong principle of exchange of stability, let us recall that,
2
ð7:1Þ
with I~ 1 ¼ I1 þ ξ1 ;
I~2 ¼ I2 þ ξ1 I~ 1 ;
I~3 ¼ I3 þ ξ1 I~ 2
ð7:2Þ
Ij (j¼1,2,3) being L1 principal invariants when Pr ¼1. Following, step by step, the procedure given in Section 6, the following results can be obtained.
Table 1 Onset of convection in the case P 1 ¼ 1. T2
Pr
a2c4
2
ac
RC
RC
Convection
10
0.1
5.1535
4.9367
R21 þ 677:08
R21 þ 1446:89
Stationary
100
0.1
6.7265
4.9534
R21 þ826.29
R21 þ1450.15
Stationary
800
0.2
12.6911
5.4048
R21 þ1526.96
R21 þ1679.82
Stationary
1000
1.2
13.7674
9.0406
R21 þ1676.12
R21 þ4720.25
Stationary
3000
0.01
20.7578
4.9414
R21 þ28210.43
R21 þ1329.36
Oscillatory
9500
0.5
31.8585
14.0438
R21 þ5223.71
R21 þ5146.65
Oscillatory
10 500
0.6
33.0558
15.9040
R21 þ5527.77
R21 þ6377.05
Stationary
R. De Luca, S. Rionero / International Journal of Non-Linear Mechanics 78 (2016) 121–130
Lemma 7.1. Let Pr ¼1. Then the lowest positive value of R2 such that (7.1) admits a pure imaginary root is given by RC ¼
2
1 þ P1
where
R21 þ 2P 21 2 ac
ð1 þP 1 Þ2 ða c þ π 2 Þ3 þ P 21 π 2 T 2 2
P 21 a c
;
ð7:3Þ
8. Onset of convection when P 1 P r ¼ 1 Let P 1 P r ¼ 1. Then λ ¼ ξ1 =P 1 is an L1 -eigenvalue and the spectral equation of L1 can be written as
λþ
is the unique positive solution of the equation ðX ¼ a þ π Þ 2
2X 3 3π 2 X 2
P 21 π 2 T 2 ð1 þ P 1 Þ2
¼ 0:
2
ð7:4Þ
Theorem 7.1 (Onset of stationary convection). Let Pr ¼1. Then:
R2 ¼ R C ¼ R21 þ
ða2c4 þ π 2 Þ3 þ π 2 T 2 ; a2c4
ð7:5Þ
either for P 1 r 1;
8 T 2 ; 8 R21 ;
P1
3 2 ðλ I~ 1 λ þ I~ 2 λ I~3 Þ ¼ 0;
ð7:6Þ
ð8:1Þ
with
ξ I~ 2 ¼ I2 þ 1 I~1 ; P1
ξ I~ 3 ¼ I3 þ 1 I~ 2 P1
ð8:2Þ
Ij (j ¼1,2,3) being L1 -principal invariants when P 1 P r ¼ 1. Following, step by step, the procedure given in Section 6, the following results can be obtained. Lemma 8.1. Let P 1 P r ¼ 1. Then the lowest positive value of R2 such that (8.1) admits a pure imaginary root is given by
or for
2
P 1 4 1;
R21 r 2
2 ða c þ
π 2 Þ3 2
ðP 1 1Þa c
RC ¼ ;
2
8T ;
P 1 o 1;
R21 r
2 ð1 þ P 1 Þ2 ða c þ π 2 Þ3 þ π 2 T 2 R21 þ 2 ; 2 P 1 ð1 þP 1 Þ P ð1 þ P Þa 1
ð7:7Þ
1
ð8:3Þ
c
2
where a c is the unique positive solution of the equation ðX ¼ a2 þ π 2 Þ
(ii) the strong principle of exchange of stability holds either for (7.7) or for ð2P 1 1Þ2 ða2c4 þ π 2 Þ3 ; 4P 1 ð1 P 1 Þa2c4
8 T 2;
ð7:8Þ
2X 3 3π 2 X 2
π2T 2 ð1 þ P 1 Þ2
¼ 0:
ð8:4Þ
Theorem 8.1 (Onset of stationary convection). Let P 1 P r ¼ 1. Then: (i) stationary convection occurs at
or for P 1 ¼ 1;
8 T 2;
8 R21 ;
ð7:9Þ
R2 ¼ R C ¼ R21 þ
(iii) the weak principle of exchange of stability holds for P 1 o 1;
π
ð2P 1 1Þ2 ða2c4 þ 2 Þ3 R21 4 ; 4P 1 ð1 P 1 Þa2c4
2
8T ;
2 1 þ P 1 2 ð1 þP 1 Þ2 ða c þ 2 Þ3 þ P 21 R þ 1 2 2P 21 P 21 a c
π
π2T 2
ð7:10Þ
;
ða2c4 þ π 2 Þ3 þ π 2 T 2 ; a2c4
ð8:5Þ
either for
Theorem 7.2 (Onset of oscillatory convection). Let Pr ¼ 1. Then oscillatory convection occurs at R2 ¼ R C ¼
ξ1
ξ I~ 1 ¼ I1 þ 1 ; P1
(i) stationary convection occurs at
127
8 R21 ;
8 T 2;
P 1 r 1;
ð8:6Þ
Table 3 Oscillatory convection in the case Pr ¼ 1.
ð7:11Þ
T2
P1
for
100
1.2 6.7265
P 1 4 1;
R21 þ 2398.12 800 1.3 12.6911 8.6223 R2 þ 1526.96 0.680 1 R21 þ 3211.30 1500 1.4 15.9976 10.8194 R21 þ 2007.33 0.612
R21 4
2ða2c4 þ π 2 Þ3 ; ðP 1 1Þa2c4
8 T 2:
ð7:12Þ
a2c4
3500 1.5 21.9710
Critical values of the Rayleigh number for the onset of convection when Pr ¼1 are collected in Tables 2 and 3.
ac
RC
RC
Oscillatory convection
5.5521
R21 þ 826.29
0.764
R21 4 6660:30
2
R21 þ 3773.69 14.9330 R21 þ 3041.48 0.556 R21 þ 5151.39
R21 4 5263:56 R21 4 4552:47 R21 4 4752:05
Table 2 Stationary convection in the case Pr ¼1. T2
P1
a2c4
2
ac
RC
Stationary convection
RC
10
0.1
5.1535
4.9367
R21 þ 677:08
55R21 þ 79578:9
8 R21
100
1.2
6.7265
5.5521
R21 þ 826.29
0.764 R21 þ 2398.12
R21 r 6660:30
800
1.3
12.6911
8.6223
R21 þ 1526.96
0.680 R21 þ 3211.30
R21 r 5263:56
1500
1.4
15.9976
10.8194
1.5
21.9710
14.9330
R21 þ 3773.69 R21 þ 5151.39
R21 r 4552:47
3500
R21 þ 2007.33 R21 þ 3041.48
0.612 0.556
R21 r 4752:05
128
R. De Luca, S. Rionero / International Journal of Non-Linear Mechanics 78 (2016) 121–130
Table 4 Stationary convection in the case P 1 P r ¼ 1. P1
T2
a2c4
2
ac
RC
RC
Stationary convection
10
1.1
5.1535
4.9855
R21 þ 677:08
0:866R21 þ 2527:73
R21 r 13810:80
100
1.2
6.7265
5.3736
R21 þ826.29
0.758 R21 þ2555.91
R21 r 7147:19
800
1.3
12.6911
7.4288
R21 þ1526.96
0.669 R21 þ3176.49
R21 r 4983:47
1500
1.4
15.9976
8.6726
R21 þ2007.33
0.595 R21 þ3536.37
R21 r 377:5:41
3500
1.5
21.9710
11.5007
R21 þ3041.48
0.595 R21 þ4697.38
R21 r 4088:64
RC
RC
Oscillatory convection R21 4 13810:80
Table 5 Oscillatory convection in the case P 1 P r ¼ 1. P1
T2
a2c4
2
ac
10
1.1
5.1535
4.9855
R21 þ 677:08
0:866R21 þ 2527:73
100
1.2
6.7265
5.3736
R21 þ826.29
0.758 R21 þ 2555.91
R21 4 7147:19
800
1.3
12.6911
7.4288
R21 þ1526.96
0.669 R21 þ3176.49
R21 4 4983:47
1500
1.4
15.9976
8.6726
R21 þ2007.33
0.595 R21 þ 3536.37
R21 4 377:5:41
3500
1.5
21.9710
11.5007
R21 þ3041.48
0.595 R21 þ 4697.38
R21 4 4088:64
Linearization principle. Decay of linear energy for any initial data implies decay of non-linear energy at any instant.
or for P 1 4 1;
2
2
a c R21 þ π 2 T 2 o
ðP 1 þ 1Þða c þ π 2 Þ3 ; P1 1
ð8:7Þ
(ii) the strong principle of exchange of stability holds for (8.7) or for P 1 o1;
a2c4 R21 þ π 2 T 2 r
P 21 ða2c4 þ π 2 Þ3 ; 4ð1 P 1 Þ
ð8:8Þ
or for P 1 ¼ 1;
8 T 2;
8 R21 ;
ð8:9Þ
(iii) the weak principle of exchange of stability holds for P 1 o1;
P 21 ða2c4 þ π 2 Þ3 : a2c4 R21 þ π 2 T 2 4 4ð1 P 1 Þ
Theorem 8.2 (Onset of oscillatory convection). Let P 1 P r ¼ 1. Then oscillatory convection occurs at 2
1
1
ð8:11Þ
c
for P 1 4 1;
a2c4 R21 þ π 2 T 2 4
ðP 1 þ 1Þða2c4 þ π 2 Þ3 : P1 1
8 1 1 X X > > > ∇π ¼ ∇π n ; u ¼ un ; > > > > n¼1 n¼1 > > > 1 1 X X > < θ¼ θn ; Φ ¼ Φn ; n¼1 n¼1 > > > > > ∇ un ¼ 0; > > > > ∂un ∂vn > > ¼ ¼ wn ¼ θn ¼ Φn ¼ 0; : ∂z ∂z
ð9:1Þ
on z ¼ 0; 1
and ð8:10Þ
2 ð1 þ P 1 Þ2 ða c þ π 2 Þ3 þ π 2 T 2 R21 þ 2 ; R2 ¼ R C ¼ 2 P 1 ð1 þ P 1 Þ P ð1 þ P Þa
Let ð∇π ; u; θ; ΦÞ – with w; θ; Φ; A AðΩÞ – be solution of (2.6)– (2.7) with
8 > ð0Þ > > > ð∇π n Þðt ¼ 0Þ ¼ ∇π n ; > > > > > > > > > > ðun Þðt ¼ 0Þ ¼ uð0Þ > n ; > < > ð0Þ > > ðθn Þðt ¼ 0Þ ¼ θn ; > > > > > > > > > ð0Þ > > > > ðΦn Þðt ¼ 0Þ ¼ Φn ; : ð0Þ
ð8:12Þ
Critical values of the Rayleigh number for the onset of convection when P 1 P r ¼ 1 are collected in Tables 4 and 5.
9. Non-linear global stability Our aim is to generalize to rotating layers the linearization principle obtained in [17], via the Auxiliary System Method [13–17].
1 X
ð∇π Þðt ¼ 0Þ ¼ ∇π ð0Þ ¼
∇π nð0Þ ;
n¼1
ðuÞðt ¼ 0Þ ¼ uð0Þ ¼ ðθÞðt ¼ 0Þ ¼ θ
ð0Þ
ðΦÞðt ¼ 0Þ ¼ Φ
¼
ð0Þ
1 X
uð0Þ n ;
n¼1 1 X
θð0Þ n ;
n¼1 1 X
¼
ð9:2Þ
Φnð0Þ ;
n¼1
ð0Þ
∇π ð0Þ ; uð0Þ ; θ ; Φ being assigned initial data such that ∇ uð0Þ ¼ 0. To (2.6)–(2.7), following the guideline of the Auxiliary System Method (AS Method, [17]), we associate 8 n A N the auxiliary system 8 > 1 ∂u n > > > P r ∂t ¼ ∇π n þ Δu n þ T u n k þ Rθ n R1 Φ n k u ∇u n ; > > > > > ∇ u n ¼ 0; > < ∂θ n > ¼ Rw n þ Δθ n P r u ∇θ n ; > > ∂t > > > > > ∂Φ n > > ¼ R1 w n þ ΔΦ n P r P 1 u ∇Φ n ; : P1 ∂t ð9:3Þ
R. De Luca, S. Rionero / International Journal of Non-Linear Mechanics 78 (2016) 121–130
under the initial boundary conditions 8 > ðu n Þðt ¼ 0Þ ¼ uð0Þ ð∇π n Þðt ¼ 0Þ ¼ ∇π ð0Þ n ; n ; > > > < ð0Þ ð0Þ ðθ n Þðt ¼ 0Þ ¼ θn ; ðΦ n Þðt ¼ 0Þ ¼ Φn ; > > > ∂u n ∂v n > ¼ ¼ w n ¼ θ n ¼ Φ n ¼ 0; z ¼ 0; 1: : ∂z ∂z
129
for arbitrary initial data. Then
ð9:4Þ
dEn o 0; 8 t Z0: ð9:14Þ dt Proof. The proof can be obtained by following, step by step, the procedure applied for the proof of Theorem 4.1 in [17].□ 9.2. Global non-linear energy stability
The following theorem holds. Theorem 9.1. Let ð∇π n ; u n ; θ n ; Φ n Þ 8 n A N be solution of (9.3)–(9.4). P1 P1 P1 P1 The series n ¼ 1 ∇π n , n ¼ 1 un, n ¼ 1 θn, n ¼ 1 Φ n are a.e. convergent in Ω and 1 X
1 X
∇π n ¼ ∇π ;
n¼1
u n ¼ u;
By virtue of the linearization principle, it easily follows that conditions guaranteeing linear stability guarantee also the nonlinear global asymptotic stability.
10. Discussion
n¼1
1 X
θ n ¼ θ;
n¼1
1 X
Φ n ¼ Φ:
ð9:5Þ
Double convection in rotating layers, heated and salted from below, is investigated. In particular:
n¼1
Proof. The proof can be obtained by following, step by step, the procedure applied for the proof of Theorem 3.1 in [17].□ 9.1. Linearization principle Let us introduce the energy En of the nth-Fourier component of the perturbation fields h i En ¼ 12 P r 1 J un J 2 þ J θn J 2 þ P 1 J Φn J 2
ð9:6Þ
and the energy E^ n of the solutions of (3.1) h i ^ n J2 E^ n ¼ 12 P r 1 J u^ n J 2 þ J θ^ n J 2 þP 1 J Φ
ð9:7Þ
with J un J 2 ¼ J un J 2 þ J vn J 2 þ J wn J 2 ; ^ n J 2: J u^ n J 2 ¼ J u^ n J 2 þ J v^ n J 2 þ J w
ð9:8Þ
Denoting by Q n ; Q^ n , the quadratic forms Q n ¼ ξn ðu2n þv2n þ w2n þ θn þ Φn Þ þ 2Rθn wn 2
2
(1) the lowest Rayleigh number R C at which stationary convection can occur, in an algebraic closed form is obtained; (2) an algebraic system for finding the lowest Rayleigh number R C at which oscillatory convection can occur, for any values of the fluid and salt Prandtl numbers, is introduced; (3) via the introduction of a new field, conditions guaranteeing the onset of steady or oscillatory convection are given in algebraic closed form in the cases P 1 ¼ 1 or Pr ¼1 or P 1 P r ¼ 1; (4) conditions guaranteeing that R C o R C , i.e. conditions for letting convection occurs via an oscillatory state, are provided; (5) linear stability is recovered as non-linear global asymptotic stability via the AS Method.
Acknowledgments This paper has been performed under the auspices of G.N.F.M. of INdAM. One of the authors (R. De Luca) acknowledges Progetto Giovani GNFM 2015 “Dinamica di sistemi complessi infinito dimensionali con applicazioni in Fluidodinamica, Economia e Biologia”. The accuracy of an anonymous referee is gratefully acknowledged by the authors.
ð9:9Þ References
and 2
2 2 ^ 2 Þ þ 2Rθ^ n w ^ 2n þ θ^ n þ Φ ^n Q^ n ¼ ξn ðu^ n þ v^ n þ w n
ð9:10Þ
it follows that the time derivative of En along the solutions of (9.3)–(9.4) is given by Z dEn ¼ Q n ðun ; vn ; wn ; θn ; Φn Þ dΩ dt Ω
ð9:11Þ
while the time derivative of E^ n along the solutions of (3.1)–(3.2) is given by Z dE^ n ^ n Þ dΩ: ^ n ; θ^ n ; Φ ¼ Q^ n ðu^ n ; v^ n ; w dt Ω
ð9:12Þ
According to [17], the linearization principle is based on the following theorem. Theorem 9.2. Let dE^ n dt
! o 0; ðt ¼ 0Þ
ð9:13Þ
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