In atmospheric and oceanographic sciences, fluid phenomena with heat transfer has been extensively studied in a large variety of contexts, see, for instance, [1, 4]. The thermal convection of a fluid powered by the difference of temperature between two horizontal parallel plates, known as Rayleigh-Bénard convection see [2, 4, 5, 6, 7, 8, 9, 10], obeys the rotating Boussinesq system:
where T2>T1, u is the vector velocity field of the fluid, p represents the scalar pressure, Ω is the rotation rate, and e3 is the unit upward vector. As usual, e3:=(0,0,1), ν is the kinematic viscosity, g is the gravity acceleration constant, α is the thermal expansion coefficient, T is the scalar temperature field of the fluid, and κ is the thermal diffusion coefficient. Here we also impose the periodic boundary conditions in the horizontal directions for simplicity.
This system with rotation is a dynamic model having 3D incompressible Navier-Stokes equations via a buoyancy force proportional to temperature coupled with the heat advection-diffusion of the temperature [5,10, 11, 12, 13].
We can use the Boussinesq approximation and non-dimensionalization to obtain the simplification of Boussinesq system, namely,
with the boundary and initial conditions:
where is the torus in is the Prandtl number, is the Ekman number and is the Rayleigh number.
This system is different from the nondimensional form in [12,13]. Encouraged by the results on the global existence and the regularities of the suitable weak solution in [12,13], and the related models, see [2,5,7,10,14, 15, 16, 17, 18], this system has also suitable weak solution by adopting Galerkin approximation method.
By using the asymptotic expansion methods of the singular perturbation theory and the Stokes operator [9,19,20,21,22], we construct an exact approximating solution and study the infinite Prandtl number limit Pr→∞ (i.e., ε→0), of Rayleigh-Bénard convection (1.1)-(1.6).
The main purpose of this paper is to show that the solutions of Boussinesq system for Rayleigh-Bénard convection converge to those of the infinite Prandtl number limit Pr→∞ (i.e., ε→0) model. It is a singular perturbation problem.
The rest of this paper is outlined as follows. The derivation of initial layer is stated in Section 2. The main convergence results are stated in section 3. The approximating solution is constructed and the properties of approximating solution are showed in section 4. The proofs of main convergence results are showed in Section 5. The conclusion is stated in Section 6.
2 The derivation of initial layer
for (x,y,z,t)∈Q×(0,S), S>0.
Then we impose the initial condition of TO,0 as follws:
where is the limit of as ε → 0
We now turn to derive the boundary conditions of TO,0.
Restricting (2.3) to z=0,1, one gets
The equation (2.10) is a stationary Stokes equation with rotation, we solve (2.10), (2.11) and (2.12) and know that the value uO,0(t=0) is determined by the initial data TO,0(t=0) of the temperature. But can be given arbitrarily and independently of TO,0(t=0).
Thus, an initial layer occurs. We observe that the infinite Prandtl number limit of the Boussinesq system only has an initial layer, which is a singular perturbation problem.
3 Main convergence results
Assume that the initial data have an expansion up to the 1st order as follows
where are all C∞(Q) functions, and satisfy
for some positive constant C independent of ε.
Assume that (3.1) holds. Also, assume that , satisfy the suitable compatibility conditions like etc. Then, as ε→0, for any 0<S<∞, we have the following convergence:
where H1(Q)=W1,2(Q), for some positive constant C independent of ε.
The functions are given in Section 3.
Due to the assumption (3.2), we can get the optimal convergence rate by adding assumption
Then we have the following theorem.
for some positive constant C independent of ε, where , is the solution of the following linear problem
The functions uO,0, TI,1, uI,0, TO,1, TO,0, uI,1 are given in Section 4.
4 Approximating solutions and the properties
In this section, we carry out the method of matched asymptotic expansions [25,26] and the two-time-scale approach [2,26]. We construct the approximating solution including the outer one away from t=0 and the initial layer expansion near t=0. We also derive the corresponding properties of this approximating solution.
where ε is the length of the initial layers, is the fast time variable; (uO,i,pO,i,TO,i)(x,y,z,t) are the outer functions for the velocity field, pressure and temperature field, respectively, which are independent of ε; (uI,i,pI,i,TI,i)(x,y,z,τ) are the initial layer functions for the velocity field, pressure and temperature field, respectively. The initial layer functions satisfy: uI,i,pI,i,TI,i decay to zero exponentially, as τ→ ∞.
We discuss in detail the construction of the outer and initial layer functions here as
4.1 Outer functions
with (uO,i,pO,i,TO,i)(x,y,z,t) to be determined later.
where the remainders and satisfy the estimates
for any fixed S>0 and any s≥1.
Now, we first consider the coefficient of leading order O(ε0) in the outer equations. We set the coefficient of O(ε0) in the system (4.4)-(4.6) as zero and use the boundary conditions (4.7)-(4.8) and the initial data (2.5).
At first order, (uO,1,pO,1,TO,1) satisfy the following system:
The infinite Prandtl number rotating system (2.1)-(2.5) has stationary Stokes equations via a buoyancy force proportional to temperature coupled with heat advection of the temperature. The linearized infinite Prandtl number type rotating system (4.10)-(4.14) has Stokes equations via a buoyancy force proportional to temperature coupled with linearized heat advection of the temperature. Therefore, the existence of the smooth solutions is the same as the incompressible Stokes equations. We find that:
The proof of Proposition 4.1 is elementary and we omit it.
Now we turn to the construction of the initial layer functions.
4.2 Initial layer functions
Near t=0, we will approximate the solution uniformly up to t=0 by the two-scale expansions (4.2)
We use the Taylor series expansion
Now we compare the coefficients of O(εi), i≥0 in the resulting system and derive the systems satisfying the initial layer functions.
this show that the temperature has no zero order initial layer.
Then, setting the coefficients of O(ε0) in (4.16)-(4.18) as zero, using (4.24) and requiring that the approximating solution satisfies the boundary and initial conditions (4.19) and (4.22), the initial layer functions (uI,0,pI,0,TI,0) satisfy the system as
Now, we turn to derive the initial and boundary conditions of TI,1.
In fact, we restrict (4.30) to τ=0, and replace the right term of result by , that is,
Now we turn to state the exponentially decay properties of the initial layer functions.
Let the assumptions of Theorem 3.1 hold. Then there exist a unique and smooth solution (uI,0,pI,0) to the system (4.24)-(4.26), (4.28) and (4.29) and a unique and smooth solution (uI,1,pI,1,TI,1) to the system (4.27) and (4.34)-(4.37) satisfying the exponential decay to zero as τ→∞ , namely,
for some positive constants C,β and any s≥1.
We summarize the approximating solution in the next subsection.
4.3 Approximating solution
With outer functions and initial layer functions defined in section 4.1 and 4.2, one gets
where the remainders and , caused by the initial layer, are given exactly by
Hence, the previous computations show that solves the following initial-boundary problem:
We now turn to the proofs of convergence results.
5 The proofs of main convergence results
Without loss of generality, we denote C by a positive generic constant independent of ε. Noting that C may depend upon S for any fixed S>0. Let t∈[0,S]. We use the standard L2-energy method to prove Theorems 3.1 and 3.5.
5.1 The proof of Theorem 3.1
In this subsection we assume that (3.1) holds and define error functions
Step 2. Taking the L2-inner product of temperature error equation (5.3) with and integrating over Q with respect to (x,y,z) yield
We first estimate I1 by Green’s first formula and the boundary condition (5.5). We have that
where Γ is the boundary surface.
Here η1 is a small constant, C(η1)>0 is a constant, independent of ε. We have used the estimate
Similarly, we estimate the last integral term I4 by using same method in estimating I3. We obtain that
With the help of Poincaré inequality and taking η1 to be sufficiently small but independent of ε, one gets
Step 3. Similarly, testing the velocity equation (5.1) by and integrating over Q with respect to (x,y,z).
Next, we deal with the right-hand side terms of (5.13) as follows:
where we use and the boundary condition (5.4).
Then, putting the above derivation equations into (5.13), we obtain that
With the help of the Poincaré inequality, restricting ε to be sufficiently small such that and taking η3, η4 to be sufficiently small but independent of ε, one gets
Integrating (5.15) with respect to t over [0,t] for any t∈[0,S] and any fixed S>0, one gets
where , C2=2C(η1)C+2C(η4)C.
Using Gronwall’s lemma and the assumption (3.2) yield
where we have used the estimate
We deduce from (5.18) that
The proof of Theorem 3.1 is complete.
Obviously, the convergence rate is not optimal one, so we derive the optimal convergence rate in the next subsection.
5.2 The proof of Theorem 3.5
Step 1. We cancel the order O(ε) term of (4.42) to get the optimal convergence rate, and regard new result as in the remainder. Moreover, by virtue of another initial layer function TI,2, we define it as .
We define TI,2 to be the solution of the system (3.8)-(3.9), which can be solved by
By the exponential decay of the initial layer functions (uI,0,uI,1,TI,1), it follows that
for some positive constants C,Γ and any s≥1.
Step 2. Now set .
Then satisfies the following error equations
The proof of Theorem 3.5 is complete.
In this paper, we have used matched asymptotic expansion analysis to study the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit, which involves initial layers. It is a singular perturbation problem. We have derived the convergence of the solution of the Boussinesq system for Rayleigh-Bénard convection to that of the infinite Prandtl number limit system by adopting the effective approximating expansion.
The boundary value of the limit limε→0(uε,Tε) are not equal to (uO,0,TO,0), due to the initial and boundary conditions effect, the boundary layer occurs. This need an extra correction term of boundary layer with two fast variables. We will discuss it in the future.
The initial data satisfies a higher order correction in powers of ε, then the similar higher-order correction result in powers of ε can be obtained in the same way. We leave it for further investigation.
This work is supported by National Natural Science Foundation of China (No. 11771031, 11531010) and by Natural Science Fund of Qinghai Province (2017-ZJ-908).
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About the article
Published Online: 2018-10-29
Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1145–1160, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0094.
© 2018 Fan et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0