Notes on symmetry in convective flows
Introduction
Normal modes, decoupling of unknowns and symmetries are main tools to achieve solutions for systems of differential equations arising in mathematical physics. Linear systems, even coupled ones, can often be decoupled by simply projecting the data into suitable subspaces. In case of nonlinear PDE’s this procedure is, in general, not working, and it is not possible to split the solution into two components satisfying a decoupled system, even though there are cases where this might indeed occur. A most notable example is given by the Navier–Stokes equations in a sufficiently smooth domain (): where and are unknown pressure and velocity fields and is the constant kinematic viscosity. Then, the well-known Helmholtz–Weyl decomposition theorem allows one to solve (1.1) first for and then recovering as the solution to a suitable Neumann problem for the Laplace operator [1], [2], [3].
It must be added that the tools mentioned above, typically, also lead to find some physically significant exact solutions that may constitute the basic state for stability analysis. Well-known examples are Couette and Poiseuille flow, Asymptotic Suction Profile etc.,1 and, in a non-isothermal situations, Bénard purely conducting steady-state. Concerning the latter, we recall that the Bénard problem in the Oberbeck–Boussinesq (O–B) approximation regards the study of the following set of equations (in dimensionless form) where is temperature field, is a unit upright vector parallel to the gravity, and and are Prandtl and Rayleigh number, respectively.
In the first part of this note (Section 2) we show by a general symmetry argument (Proposition 2.1) the existence of a particular subclass of solutions to (1.1) and its non-isothermal counterpart in the Boussinesq approximation. Of course, for this argument to work we need that the flow domain as well as the driving mechanism satisfy suitable symmetry conditions. We prove that these solutions belong to a subspace, (say), of the underlying Banach space that remains invariant under the relevant dynamics and is stable to every perturbation with data in . As a consequence the generic solution can be split into two components: one, , in , and the other, , in its complement. Thus, it is the latter that produces possible instability and symmetry breaking of the bifurcating (stable) solution, due to the contribution of the convective (nonlinear) term given by .
The second part (Section 3) concerns the instability of a thermal flow between two horizontal coaxial cylinders kept at different temperatures, with (say) being their difference. As shown in [6], for arbitrary values of physical and geometric parameters, there exists a steady-state flow, say s , which, unlike the classical Bénard problem between horizontal planes, has a nontrivial velocity field. Moreover, is symmetric around the vertical diameter of the common cross-section of the cylinders. As expected on mathematical ground and suggested by numerical and experimental tests, this solution becomes non-unique and unstable if is sufficiently large, all other parameters being fixed. However, the interesting question not entirely yet clarified is whether bifurcation occurs in a steady-state or time-periodic fashion and whether symmetry is preserved in the transition. We then show (Proposition 3.1) that symmetry breaking can occur only in a range of physical parameters which makes the smallest eigenvalue of a suitable eigenvalue problem strictly greater than 1.
Section snippets
Continuous symmetries
We give sufficient conditions for the occurrence of decoupling. These conditions are based on symmetries considerations. More precisely, we are interested in how symmetry properties of the domain (and of the driving mechanism), may naturally lead to find exact solutions, or to prove their existence in subspaces possessing that particular symmetry. In order to reach this goal, we briefly recall, for completeness, some basic concepts of differential geometry.
Given a -differentiable scalar
A discrete symmetry problem
We are interested in the 2D-convective motion of a Navier–Stokes liquid between two horizontal coaxial cylinders with radii and , when temperature distributions , on the inner jacket, and , on the outer jacket, are prescribed, with see [6], [15], [16], [17], [18], [19]. In such a case, one observes steady-state flow, no matter how small . The precise mathematical formulation of the problem goes as follows. Denote by a set of polar coordinates in the
Acknowledgments
We would like to thank the reviewers for their very useful comments and suggestions that led to a rather improved version of the original manuscript.
References (22)
- et al.
Stability in the rotating Bénard problem with Newton–Robin and fixed heat flux boundary conditions
Mech. Res. Commun.
(2010) - et al.
The Bénard problem for quasi-thermal-incompressible materials: a linear analysis
Int. J. Non-Linear Mech.
(2014) - et al.
Non-linear approximations for natural convection in a horizontal annulus
Int. J. Non-Linear Mech.
(2007) Prandtl number effected on bifurcation and dual solutions in natural convection in a horizontal annulus
Int. J. Heat Mass Transfer
(1999)The Navier–Stokes Equations. An Elementary Functional Analytic Approach
(2001)An Introduction to the Mathematical Theory of the Navier–Stokes Equation. I and II
(1998)A note on the Prodi-Serrin conditions for the regularity of a weak solution to the Navier–Stokes equations
J. Math. Fluid Mech.
(2018)Intégration des Équations du mouvement d’un guide visqueux incompressible
Exact solutions to the steady-state Navier–Stokes equations
Annu. Rev. Fluid Mech.
(1991)- et al.
Theoretical results on steady convective flows between horizontal coaxial cylinders
SIAM J. Appl. Math.
(2011)