“…Until now, all the applications of the background method have focused on flows confined between solid boundaries. Examples include bounds on the rate of energy dissipation in surface-velocity-driven flows (Doering & Constantin 1992, 1994; Marchioro 1994; Nicodemus, Grossmann & Holthaus 1997; Wang 1997; Hoffmann & Vitanov 1999; Plasting & Kerswell 2003), pressure-driven flows (Constantin & Doering 1995) and surface-stress-driven flows (Tang, Caulfield & Young 2004; Hagstrom & Doering 2014); bounds on the heat transfer in Rayleigh–Bénard convection in various settings (Doering & Constantin 1996, 2001; Otero et al 2002; Plasting & Ierley 2005; Wittenberg 2010; Whitehead & Doering 2011 b ; Whitehead & Wittenberg 2014; Goluskin 2015; Goluskin & Doering 2016; Fantuzzi 2018) and Bénard–Marangoni convection (Hagstrom & Doering 2010; Fantuzzi, Pershin & Wynn 2018; Fantuzzi, Nobili & Wynn 2020); and bounds on buoyancy flux in stably stratified shear flows (Caulfield & Kerswell 2001; Caulfield 2005).…”