Bifurcations and chaos in large-Prandtl number Rayleigh–Bénard convection

Rayleigh–Bénard convection with large-Prandtl number (P) is studied using a low-dimensional model constructed with the energetic modes of pseudospectral direct numerical simulations. A detailed bifurcation analysis of the non-linear response has been carried out for water at room temperature (P=6.8) as the working fluid. This analysis reveals a rich instability and chaos picture: steady rolls, time-periodicity, quasiperiodicity, phase locking, chaos, and crisis. Our low-dimensional model captures the reappearance of ordered states after chaos, as previously observed in experiments and simulations. We also observe multiple coexisting attractors consistent with previous experimental observations for a range of parameter values. The route to chaos in the model occurs through quasiperiodicity and phase locking, and attractor-merging crisis. Flow patterns spatially moving along the periodic direction have also been observed in our model.     

Experiments on the wavy instability in Rayleigh–Bénard–Poiseuille convection: Spatial and temporal features in the linear and nonlinear regime

An experimental study of a Rayleigh–Bénard–Poiseuille air flow in a rectangular channel is presented. The aim of the paper is to characterize a secondary instability, referred to as wavy instability and known to be a convective instability, with the objective to identify the best conditions for an optimal homogenization of heat transfers in the system. A periodic mechanical forcing is introduced at channel inlet and the spatial and temporal evolution of the temperature fluctuations are analyzed, depending on the Rayleigh and Reynolds numbers, the forcing frequency and the forcing amplitude. As the saturation state is a priori the best situation to homogenize the transfers, the objective is to expand the saturation area and to generate a maximum saturation amplitude value by conducting experiments at high Rayleigh numbers. It is shown that the change in the Rayleigh number value influences the saturation length but does not act on the saturation magnitude while the change in the Reynolds number value causes antagonist effects on the saturation parameters. The key parameter acting on the saturation amplitude is the forcing frequency. The most efficient forcing configuration is to introduce the external perturbation into the fully developed region of the longitudinal rolls and to apply a specific low forcing frequency associated with a moderate forcing magnitude.     

On the forward and backward in time problems in the Kelvin–Voigt thermoviscoelastic materials

This paper studies the uniqueness of solutions to the forward and backward in time boundary value problems associated with the Kelvin–Voigt viscoelastic model of the thermoelastic materials. For thermoviscoelastic materials with a center of symmetry, it is shown the uniqueness of solutions to the forward in time boundary value problems without any assumptions upon the thermoviscoelastic constitutive coefficients other than the symmetry properties and those induced by the dissipation inequality. While for the final boundary value problems two uniqueness theorems are presented: the first one is essentially based on the assumption that the specific heat is of negative definite sign, while the second is established in the class of displacement–temperature variation fields whose dissipation energy has a temporal behavior lower than an appropriate growing exponential.     

On an extension of the d’Alembert solution to initial–boundary value problems in multi-layered,multi-material domains

This work introduces a method for the exact solution of initial–boundary value problems for linear, one-dimensional conservation laws in multi-layered, multi-material domains. The method is based on the geometry of the solutions of such conservation laws and represents an extension of the d’Alembert solution to initial–boundary value problems in multi-layered, multi-material domains.     

Average and extremal properties of heat transfer and shear stress on a wall surface in Rayleigh–Bénard convection

In this study surface-averaged and extremal properties of heat transfer and shear stress on the upper wall surface of Rayleigh–Bénard convection are numerically examined. The Prandtl number was raised up to 10³, and the Rayleigh number was changed between 10⁴ and 10⁷. As a result, average Nusselt number Nu and shear rate τ/Pr depends on Pr, Ra, and the entire numerical results are distributed between two correlation equations corresponding to small and large Pr. The small and large Pr equations are closely related to steady and unsteady flow regimes, respectively. Nevertheless, a single relation τ/Pr ~ Nu ^3.0 exists to explain the entire results. Similarly the change of local maximal properties Nu _max and τ _max/Pr depends on Pr, Ra, and these values are also distributed between two correlation equations corresponding to small and large Pr cases. Despite such complicated dependence we can obtain a correlation equation as a form of τ _max/Pr ~ Nu _max^2.6, which has not been obtained theoretically.     

Influence of thermal radiation and Joule heating in the Eyring–Powell fluid flow with the Soret and Dufour effects

A two-dimensional magnetohydrodynamic boundary layer flow of the Eyring–Powell fluid on a stretching surface in the presence of thermal radiation and Joule heating is analyzed. The Soret and Dufour effects are taken into account. Partial differential equations are reduced to a system of ordinary differential equations, and series solutions of the resulting system are derived. Velocity, temperature, and concentration profiles are obtained. The skin friction coefficient and the local Nusselt and Sherwood numbers are computed and analyzed.     

On a Darcy–Stefan Problem arising in freezing and thawing of saturated porous media

A model with phase change for material convection in a saturated porous medium with a frozen region is formulated as a Darcy-Stefan problem. We propose a new generalized formulation for this Stefan-type problem with convection governed by Darcy's law. This approach, which is valid for irregular geometries with irregular subregions, has the advantage of not requiring the smoothness of the temperature, that restricted previous mathematical works to two-dimensional particular cases. We show existence of generalized solutions, passing to the limit in suitable approximated problems, which in principle can be solved numerically by the finite element method. Received October 20, 1998     

On the First Critical Field in Ginzburg–Landau Theory for Thin Shells and Manifolds

In this article, we investigate the response of a thin superconducting shell to an arbitrary external magnetic field. We identify the intensity of the applied field that forces the emergence of vortices in minimizers, the so-called first critical field H _c1 in Ginzburg–Landau theory, for closed simply connected manifolds and arbitrary fields. In the case of a simply connected surface of revolution and vertical and constant field, we further determine the exact number of vortices in the sample as the intensity of the applied field is raised just above H _c1. Finally, we derive via Γ-convergence similar statements for three-dimensional domains of small thickness, where in this setting point vortices are replaced by vortex lines.     

Patterns and heat transfer in Rayleigh-Bénard thermal gravitational convection in helium (<Superscript>3</Superscript>He) near the critical point

Thermal gravitational convection in a bottom-heated layer of near-critical ³He is considered. The range of criteria determining the convection parameters beyond the stability threshold is discussed. The specific features of 2D and 3D supercritical structures, the adiabatic compression effect, and heat transfer are considered.     

Flow structures of turbulent Rayleigh–Bénard convection in annular cells with aspect ratio one and larger

Acta Mechanica Sinica - We present an experimental study of flow structures in turbulent Rayleigh–Bénard convection in annular cells of aspect ratios $$\varGamma =1$$ , 2 and 4, and...     

On a Degenerate Free Boundary Problem and Continuous Subsonic–Sonic Flows in a Convergent Nozzle

This paper concerns the well-posedness of a boundary value problem for a quasilinear second order elliptic equation which is degenerate on a free boundary. Such problems arise when studying continuous subsonic–sonic flows in a convergent nozzle with straight solid walls. It is shown that for a given inlet being a perturbation of an arc centered at the vertex of the nozzle and a given incoming mass flux belonging to an open interval depending only on the adiabatic exponent and the length of the arc, there is a unique continuous subsonic–sonic flow from the given inlet with the angle of the velocity orthogonal to the inlet and the given incoming mass flux. Furthermore, the sonic curve of this continuous subsonic–sonic flow is a free boundary, where the flow is singular in the sense that while the speed is C ^1/2 Hölder continuous at the sonic state, the acceleration blows up at the sonic state.     

On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases

This paper is motivated by the study of a version of the so-called Schrödinger–Poisson–Slater problem: $- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,$ where ${u \in H^{1}(\mathbb {R}^3)}This paper is motivated by the study of a version of the so-called Schr?dinger–Poisson–Slater problem:

- Du + wu + l( u2 *\frac1|x| ) u=|u|p-2u,- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,      18. Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic–plastic bar      《International Journal of Solids and Structures》2003,40(6):1369-1391 We propose a fundamentally new concept to the treatment of material instabilities and localization phenomena based on energy minimization principles in a strain-softening elastic–plastic bar. The basis is a recently developed incremental variational formulation of the local constitutive response for generalized standard media. It provides a quasi-hyperelastic stress potential that is obtained from a local minimization of the incremental energy density with respect to the internal variables. The existence of this variational formulation induces the definition of the material stability of inelastic solids based on convexity properties in analogy to treatments in elasticity. Furthermore, localization phenomena are understood as micro-structure development associated with a non-convex incremental stress potential in analogy to phase decomposition problems in elasticity. For the one-dimensional bar considered the two-phase micro-structure can analytically be resolved by the construction of a sequentially weakly lower semicontinuous energy functional that envelops the not well-posed original problem. This relaxation procedure requires the solution of a local energy minimization problem with two variables which define the one-dimensional micro-structure developing: the volume fraction and the intensity of the micro-bifurcation. The relaxation analysis yields a well-posed boundary-value problem for an objective post-critical localization analysis. The performance of the proposed method is demonstrated for different discretizations of the elastic–plastic bar which document on the mesh-independence of the results.      19. On the relative position of twist and shear centres in the orthotropic and fiberwise homogeneous Saint–Venant beam theory      Raffaele Barretta 《International Journal of Solids and Structures》2012,49(21):3038-3046       20. On the General Ericksen–Leslie System: Parodi’s Relation, Well-Posedness and Stability      Hao Wu  Xiang Xu  Chun Liu 《Archive for Rational Mechanics and Analysis》2013,208(1):59-107 In this paper we investigate the role of Parodi’s relation in the well-posedness and stability of the general Ericksen–Leslie system modeling nematic liquid crystal flows. First, we give a formal physical derivation of the Ericksen–Leslie system through an appropriate energy variational approach under Parodi’s relation, in which we can distinguish the conservative/dissipative parts of the induced elastic stress. Next, we prove global well-posedness and long-time behavior of the Ericksen–Leslie system under the assumption that the viscosity μ 4 is sufficiently large. Finally, under Parodi’s relation, we show the global well-posedness and Lyapunov stability for the Ericksen–Leslie system near local energy minimizers. The connection between Parodi’s relation and linear stability of the Ericksen–Leslie system is also discussed.

Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic–plastic bar

We propose a fundamentally new concept to the treatment of material instabilities and localization phenomena based on energy minimization principles in a strain-softening elastic–plastic bar. The basis is a recently developed incremental variational formulation of the local constitutive response for generalized standard media. It provides a quasi-hyperelastic stress potential that is obtained from a local minimization of the incremental energy density with respect to the internal variables. The existence of this variational formulation induces the definition of the material stability of inelastic solids based on convexity properties in analogy to treatments in elasticity. Furthermore, localization phenomena are understood as micro-structure development associated with a non-convex incremental stress potential in analogy to phase decomposition problems in elasticity. For the one-dimensional bar considered the two-phase micro-structure can analytically be resolved by the construction of a sequentially weakly lower semicontinuous energy functional that envelops the not well-posed original problem. This relaxation procedure requires the solution of a local energy minimization problem with two variables which define the one-dimensional micro-structure developing: the volume fraction and the intensity of the micro-bifurcation. The relaxation analysis yields a well-posed boundary-value problem for an objective post-critical localization analysis. The performance of the proposed method is demonstrated for different discretizations of the elastic–plastic bar which document on the mesh-independence of the results.     

On the General Ericksen–Leslie System: Parodi’s Relation, Well-Posedness and Stability

In this paper we investigate the role of Parodi’s relation in the well-posedness and stability of the general Ericksen–Leslie system modeling nematic liquid crystal flows. First, we give a formal physical derivation of the Ericksen–Leslie system through an appropriate energy variational approach under Parodi’s relation, in which we can distinguish the conservative/dissipative parts of the induced elastic stress. Next, we prove global well-posedness and long-time behavior of the Ericksen–Leslie system under the assumption that the viscosity μ ₄ is sufficiently large. Finally, under Parodi’s relation, we show the global well-posedness and Lyapunov stability for the Ericksen–Leslie system near local energy minimizers. The connection between Parodi’s relation and linear stability of the Ericksen–Leslie system is also discussed.