The vertical heat transfer in Bénard-Marangoni convection of a fluid layer with infinite Prandtl number is studied by means of upper bounds on the Nusselt number Nu as a function of the Marangoni number Ma. Using the background method for the temperature field, it has recently been proved by Hagstrom & Doering (Phys. Rev. E, vol. 81, 2010, art. 047301) that Nu 0.838Ma 2/7 . In this work we extend previous background method analysis to include balance parameters and derive a variational principle for the bound on Nu, expressed in terms of a scaled background field, that yields a better bound than Hagstrom & Doering's formulation at a given Ma. Using a piecewise-linear, monotonically decreasing profile we then show that Nu 0.803Ma 2/7 , lowering the previous prefactor by 4.2 %. However, we also demonstrate that optimisation of the balance parameters does not affect the asymptotic scaling of the optimal bound achievable with Hagstrom & Doering's original formulation. We subsequently utilise convex optimisation to optimise the bound on Nu over all admissible background fields, as well as over two smaller families of profiles constrained by monotonicity and convexity. The results show that Nu O(Ma 2/7 (ln Ma) −1/2 ) when the background field has a non-monotonic boundary layer near the surface, while a power-law bound with exponent 2/7 is optimal within the class of monotonic background fields. Further analysis of our upper-bounding principle reveals the role of non-monotonicity, and how it may be exploited in a rigorous mathematical argument.

No abstract

In this work, a new software for computer-aided design of composite materials with predefined thermomechanical properties is presented in case of incomplete input data. The mathematical basis of underlying computational method of the properties identification is a modified method of multiscale homogenization named reversible multiscale homogenization method. The system has a modular architecture and includes software implementation of the reversible multiscale homogenization method based on a new technique of construction of software implementations of complex computational methods. The latter was named «Graph-based software engineering» (GBSE) and is based on category and graph theories. The corresponding numerical and experimental results were obtained and compared. The expediency of GBSE approach is discussed for the case of the development of complex computational methods required when solving the applied problems of the design of new heterogeneous materials.

Key words: Authors should not enter keywords on the manuscript, as these must be chosen by the author during the online submission process and will then be added during the typesetting process (see http://journals.cambridge.org/data/relatedlink/jfmkeywords.pdf for the full list) † Email address for correspondence: mmap@leeds.ac.uk arXiv:1806.10097v3 [physics.flu-dyn]

The nonlinear robustness of laminar plane Couette flow is considered under the action of in-phase spanwise wall oscillations by computing properties of the edge of chaos, i.e. the boundary of its basins of attraction. Three measures are used to quantify the chosen control strategy on laminar-to-turbulent transition: the kinetic energy of edge states (local attractors on the edge of chaos), the form of the minimal seed (least energetic perturbation on the edge of chaos), and the laminarization probability (the probability that a random perturbation from the laminar flow of given kinetic energy will laminarize). A novel Bayesian approach is introduced to enable the accurate computation of the laminarization probability at a fraction of the cost of previous methods. While the edge state and the minimal seed provide useful information about the dynamics of transition to turbulence, neither measure is particularly useful to judge the effectiveness of the control strategy since they are not representative of the global geometry of the edge. In contrast, the laminarization probability provides global information about the edge and can be used to evaluate the control effectiveness by computing a laminarization score (the expected laminarization probability) and the associated expected dissipation rate of the controlled flow. These two quantities allow for the determination of optimal control parameter values subject to desired constraints. The results discussed in the paper are expected to be applied to a wide range of transitional flows and control strategies aimed at suppressing or triggering transition to turbulence.