Physical model of the dependence of the Nusselt number on the Rayleigh number

The dependence of the Nusselt number on the Rayleigh number at the stage preceding the turbulent regime of convection is substantiated and obtained in analytic form. At this stage, the Nusselt number can be described by the power dependence on the Rayleigh number, which contains five constants. One of these constants is the same for all liquids and is the critical Rayleigh number $R_{\mathrm{cr}}$ = 1700 $\pm$ 51. The remaining four constants are strictly individual for each type of liquid. For liquid helium, the critical Rayleigh number $R_{\mathrm{cr}}$ = 35000 $\pm$ 4000 has the threshold value $(R_{1}^{\mathrm{mix}})_{\mathrm{th}}$ = 37 085.75. This value is determined by the small ratio of the diameter of the vessel to its height, as well as the formation of a cylindrical convection cell at the bottom of the vessel with a mode number of 2 and a radial number determined by the fourth root of the firstorder Bessel function of the first kind.