Fig. 2: Modeling and dimensionless parameter analysis.
From: Thermal Marangoni natural convection enables directional transport across immiscible liquids
a Schematic diagram of the mathematical model for Marangoni natural convection. The axisymmetric system uses cylindrical coordinates with adiabatic upper/lower boundaries. ΔToa and ΔTow are temperature differences between the center and the boundary at the oil-air/oil-water interfaces. ho/hw show oil/water layer thicknesses. H and L denote total height and characteristic length. Navy blue: cooling wall. Arrows: r/Z-axes. b Experimentally measured interfacial temperature differences (ΔTow: circles, ΔToa: hexagons) versus their central temperature Tc during cooling. Color fills represent distinct initial temperatures (T0 = 75, 65, 45, 25 °C). Arrows mark three cooling stages of ΔT evolution. c Evolution of Marangoni numbers (Maow: circles, Maoa: hexagons) with Tc during cooling. Colors correspond to initial temperatures. Ma values are calculated using experimentally measured thermophysical properties. d Evolution of Rayleigh (Ra: hexagons) and Prandtl (Pr: circles) numbers with Tc during cooling. Colors correspond to initial temperatures. Both Pr and Ra values are calculated using experimentally measured thermophysical properties. Numerically simulated spatial-average velocity magnitudes in oil (e) and water (f) flow fields versus dimensionless numbers (Ra: circles, Maow: hexagons, Maoa: squares) under T0 = 75 °C. Colors indicate velocity magnitudes. Color bars: velocity scales. Arrows mark the three ΔT stages from (b). g Schematic of cross-sectional flow field views at selected temperatures during cooling. The gradient color bar denotes velocity magnitude. The initial temperature is 25 °C.
