On thermal convection equations of Oberbeck-Boussinesq type with the dissipation function

Yoshiyuki KAGEI Graduate School of Mathematics Kyushu University Fukuoka 812–8581, JAPAN

In this presentation we consider the stability of the motionless state and the bifurcation problem in the Rayleigh-Bénard convection problem applying a model equation of the Oberbeck-Boussinesq type including the dissipation function.

The Oberbeck-Boussinesq equation is often used as a simplified model equation for the thermomechanical response of linear viscous fluids undergoing isochoric motions in isothermal processes but not necessarily isochoric ones in non-isothermal processes. Its justification from the point of view of continuum mechanics was given by Rajagopal, Růžička and Srinivasa [1] and there is no doubt concerning the usefulness of the Oberbeck-Boussinesq equation. However, there are some phenomena such as the earth’s upper mantle convetcion, convection in fast rotating configurations and etc., in which the Oberbeck-Boussinesq equation seems inappropriate due to the fact that the effect of dissipative heating is not taken into account in the equation. It is thus desirable to derive a model equation including the effect of the dissipative heating.

In [2] we derived a model system of equations including the effect of dissipative heating and investigated the derived system in the context of the Rayleigh-Bénard convection. The system derived in [2] takes the form

div v = 0,
ε^{3}(∂_{t}v + v · ∇v − \frac{1}{\sqrt{Gr}} Δv) + ∇p = (1 − ε^{3}θ)b,
∂_{t}θ + v · ∇θ − ζ(θ + \frac{θ^{b}+θ^{t}}{2(θ^{b}−θ^{t})})v · b − \frac{1}{Pr \sqrt{Gr}} Δθ = \frac{2ζ}{\sqrt{Gr}} D(v) · D(v).

(1)

Here v = (v_{1}, v_{2}, v_{3}) is the velocity field; p is the pressure; θ is the temperature; b = (0, 0, −1); Gr , Pr and ζ are positive physical parameters called the Grashof, Prandtl and dissipation numbers, respectively; and ε > 0 is a small non-dimensional parameter. θ^{t} denotes the temperature at the upper boundary and θ^{b} at the lower boundary. The function 2D(v) · D(v) denotes the dissipation function : 2D(v) · D(v) = \frac{1}{2} ∑_{i,j=1}^{3}(∂_{x_{j}}v_{i} + ∂_{x_{i}}v_{j})^{2}.

As a first step of the mathematical analysis of (1), we consider the stability of the motionless state in the Rayleigh-Bénard convection. After a suitable change of variables we have the following system in the infinite layer {(x_{1}, x_{2}, x_{3}) ∈ R^{3} ; 0 < x_{3} < 1} :

div v = 0,
∂_{t}v − Δv − λθe_{3} + ∇p + v · ∇v = 0,
∂_{t}θ − \frac{1}{Pr} Δθ − \frac{λ}{Pr} v_{3} + \frac{λζ}{Pr} (Θ − x_{3})v_{3} + ζθv_{3} + v · ∇θ = \frac{2ζ}{λ} D(v) · D(v),

(2)

where e_{3} = (0, 0, 1); Θ = \frac{θ^{b}+θ^{t}}{2(θ^{b}−θ^{t})} + \frac{1}{2} ; and λ > 0 is defined by λ^{2} = R = Gr Pr . Here R is the Rayleigh number. The unknown {v, p, θ} now denotes the deviation of the velocity, pressure and temperature from the motionless state \{v̄, p̄, θ̄\} = \{0, −x_{3} + \frac{ε^{3}}{2} x_{3}(1 − x_{3}), \frac{1}{2} − x_{3}\}. The boundary conditions on {x_{3} = 0, 1} are then given by

v = 0 and θ = 0 on {x_{3} = 0, 1}.

We require \{v, p, θ\} to be \frac{2π}{l_{j}}-periodic in x_{j}-direction (j = 1, 2). The initial boundary value problem for (2) is now posed under the initial condition

v|_{t=0} = v_{0}, θ|_{t=0} = θ_{0}.

Notation : We set Ω = T_{l_{1},l_{2}} × (0, 1), T_{l_{1},l_{2}} = R^{2}/(\frac{2π}{l_{1}}Z × \frac{2π}{l_{2}}Z); (·, ·) denotes the scalar product of L^{2}(Ω); H^{m}(Ω) denotes the m-th order L^{2}-Sobolev space on Ω.

In the case of the Oberbeck-Boussinesq equation (ζ = 0) the stability of the motionless state is known to be controlled by the critical Rayleigh number λ_{c}^{2} > 0 which is given by

\frac{1}{λ_{c}} ≡ sup \{\frac{2(v · e_{3}, θ)}{∥∇v∥_{2}^{2} + ∥∇θ∥_{2}^{2}} ; \{v, θ\} ∈ H_{0}^{1}(Ω)^{4} − \{0\}, div v = 0\};

and the motionless state is uncondtionally stable if λ < λ_{c} and unstable if λ > λ_{c}.

In case ζ > 0 we show that the motionless state is (conditionally) asymptotically stable even when λ is slightly beyond λ_{c} for sufficiently small ζ > 0.

Theorem ([2]). (i) For each {v_{0}, θ_{0}} ∈ H_{0}^{1}(Ω)^{3} × L^{2}(Ω) with div v_{0} = 0 there exist T > 0 and a unique solution {v(t), θ(t)} of (2) on [0, T] in the class

v ∈ C([0, T]; (H_{0}^{1})^{3}) ∩ L^{2}(0, T; (H^{2})^{3}), θ ∈ C([0, T]; L^{2}) ∩ L^{2}(0, T; H_{0}^{1}).

(ii) There exist ζ_{0} > 0 and λ_{c}(ζ) such that if 0 ≤ ζ ≤ ζ_{0} and λ < λ_{c}(ζ), then the motionless state is asymptotically stable, namely, there exists δ > 0 such that for each {v_{0}, θ_{0}} ∈ H_{0}^{1}(Ω)^{3} × L^{2}(Ω) with div v_{0} = 0 and ∥v_{0}∥_{H^{1}} + ∥θ_{0}∥_{L^{2}} < δ, the solution {v(t), θ(t)} exists on [0, ∞) and satisfies

∥v(t)∥_{H^{1}} + ∥θ(t)∥_{L^{2}} ≤ Ce^{−αt}(∥v_{0}∥_{H^{1}} + ∥θ_{0}∥_{L^{2}})

for some constants C, α > 0. If λ > λ_{c}(ζ), then the motionless state is unstable.

The number λ_{c}(ζ) satisfies

λ_{c}(0) = λ_{c} and λ_{c}(ζ) > λ_{c} for 0 < ζ ≤ ζ_{0}.

The next question is that what happens after the motioness state becomes unstable when λ is close to the critical number λ_{c}(ζ). It is known that in case ζ = 0 various stationary solutions bifurcate from the motionless state at the critical Rayleigh number λ_{c}. In this case, due to the unconditional stability of the motionless state, bifurcation occurs only supercritically. In contrast to the case ζ = 0, there are some transcritical bifurcation branches when ζ > 0, and, in particular, stationary solutions with hexagonal patterns bifurcate transcritically at λ_{c}(ζ). The details will be given in the talk.

References

[1] Rajagopal, K. R., Růžička, M., Srinivasa, A. R.: On the Oberbeck-Boussinesq Approximation. Math. Models Methods Appl. Sci. 6, 1157– 1167 (1996).

[2] Kagei, Y., Růžička, M., Thäter, G.: Natural convection with dissipative heating, preprint (1999).