The formation of the melt meniscus and convection heat transfer in the melt is a major issue in the float zone technique for growing single crystals. The present study investigates the issues of existence, multiplicity and physical realizability of the solutions, based on the concept of free energy minimization, to the Young-Laplace equation, representing the liquid menisci formed under equilibrium conditions. The physical realizability of the meniscus profile is sharply affected by the material properties and geometry, through the dimensionless parameters including Bond number and aspect ratio. A typical meniscus shape obtained at Bond numbers commonly encountered in practical configurations is selected for buoyancy driven convection and thermocapillary convection heal transfer studies. For the materials and geometries of interest, thermocapillary effects dominate over buoyancy effects. The impact of the convection pattern on the meniscus shape is reflected by the magnitude of the capillary number, which is found to be small in this case. Calculations were conducted over a range of Grashof numbers, Marangoni numbers and Prandtl numbers to gage the sensitivity of the solutions to the physical properties of the molten material. The relative strength of buoyancy-driven convection and thermocapillary convection has the greatest impact on the heat transfer results. The results reported here can help assess the suitability of a given float zone design.