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• W3100143645 abstract "Abstract The calculus of variations is employed to find steady divergence-free velocity fields that maximize transport of a tracer between two parallel walls held at fixed concentration for one of two constraints on flow strength: a fixed value of the kinetic energy (mean square velocity) or a fixed value of the enstrophy (mean square vorticity). The optimizing flows consist of an array of (convection) cells of a particular aspect ratio $def xmlpi #1{}def mathsfbi #1{boldsymbol {mathsf {#1}}}let le =leqslant let leq =leqslant let ge =geqslant let geq =geqslant def Pr {mathit {Pr}}def Fr {mathit {Fr}}def Rey {mathit {Re}}varGamma $ . We solve the nonlinear Euler–Lagrange equations analytically for weak flows and numerically – as well as via matched asymptotic analysis in the fixed energy case – for strong flows. We report the results in terms of the Nusselt number ${mathit{Nu}}$ , a dimensionless measure of the tracer transport, as a function of the Péclet number ${mathit{Pe}}$ , a dimensionless measure of the strength of the flow. For both constraints, the maximum transport ${mathit{Nu}}_{mathit{MAX}}({mathit{Pe}})$ is realized in cells of decreasing aspect ratio $varGamma _{mathit{opt}}({mathit{Pe}})$ as ${mathit{Pe}}$ increases. For the fixed energy problem, ${mathit{Nu}}_{mathit{MAX}} sim {mathit{Pe}}$ and $varGamma _{mathit{opt}} sim {mathit{Pe}}^{-1/2}$ , while for the fixed enstrophy scenario, ${mathit{Nu}}_{mathit{MAX}} sim {mathit{Pe}}^{10/17}$ and $varGamma _{mathit{opt}} sim {mathit{Pe}}^{-0.36}$ . We interpret our results in the context of buoyancy-driven Rayleigh–Bénard convection problems that satisfy the flow intensity constraints, enabling us to investigate how the transport scalings compare with upper bounds on ${mathit{Nu}}$ expressed as a function of the Rayleigh number ${mathit{Ra}}$ . For steady convection in porous media, corresponding to the fixed energy problem, we find ${mathit{Nu}}_{mathit{MAX}} sim {mathit{Ra}}$ and $varGamma _{mathit{opt}} sim {mathit{Ra}}^{-1/2}$ , while for steady convection in a pure fluid layer between stress-free isothermal walls, corresponding to fixed enstrophy transport, ${mathit{Nu}}_{mathit{MAX}} sim {mathit{Ra}}^{5/12}$ and $varGamma _{mathit{opt}} sim {mathit{Ra}}^{-1/4}$ ." @default.
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• W3100143645 date "2014-06-24" @default.
• W3100143645 modified "2023-10-16" @default.
• W3100143645 title "Wall to wall optimal transport" @default.
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• W3100143645 doi "https://doi.org/10.1017/jfm.2014.306" @default.
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