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تسجيل مستخدم جديدNew bounds on the vertical heat transport for Benard-Marangoni convection at infinite Prandtl number
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We prove a new rigorous upper bound on the vertical heat transport for Benard-Marangoni convection of a two- or three-dimensional fluid layer with infinite Prandtl number. Precisely, for Marangoni number $Ma gg 1$ the Nusselt number $Nu$ is bounded asymptotically by $Nu lesssim Ma^{2/7}(ln Ma)^{-1/7}$. Key to our proof are a background temperature field with a hyperbolic profile near the fluids surface, and new estimates for the coupling between temperature and vertical velocity.
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The vertical heat transfer in Benard-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 tem
perature field, it has recently been proven by Hagstrom & Doering that $ Nuleq 0.838,Ma^{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 & Doerings formulation at a given $Ma$. Using a piecewise-linear, monotonically decreasing profile we then show that $Nu leq 0.803,Ma^{2/7}$, 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 & Doerings 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 $Nu leq O(Ma^{2/7}(ln Ma)^{-1/2})$ 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.
Using direct numerical simulations, we study the statistical properties of reversals in two-dimensional Rayleigh-Benard convection for infinite Prandtl number. We find that the large-scale circulation reverses irregularly, with the waiting time betwe
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