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In this paper, linear and nonlinear properties of thermohaline convection at the onset with the stress-free boundary conditions are investigated using perturbation analysis relevant to oceanic water and ground water. The nonlinear governing equations are expanded using linear stability solutions, related to the temperature, motion, and concentration in the series of nonhomogeneous linear equations. The proposed method yields finite amplitude steady solutions by means of successive approximations. The rate of heat transfer is studied up to sixth-order using an expansion of thermal Rayleigh number (Ra1) as proposed by Kuo (1961, "Solution of the Non-Linear Equations of the Cellular Convection and Heat Transport," J. Fluid Mech., 10(04), pp. 611-630). The heat transfer rate of the system depends on Lewis number, Prandtl number, and thermal (Ra1) as well as solutal (Ra2) Rayleigh number. The result for the heat transport (N) in the system fits a power law of exponent of 0.288, i.e., N≃(Ra1/Ra1o)0.288 for Ra2=0 ⁠, Pr = 10 and Ra1>4Ra1o ⁠. This power law of exponent value of 0.288 is close to the results (0.25) obtained by Kuo (1961, "Solution of the Non-Linear Equations of the Cellular Convection and Heat Transport," J. Fluid Mech., 10(04), pp. 611-630), and also to the experimental results for laminar convection by Jakob (1949, Heat Transfer, Vol. 1, Wiley, New York.). The characteristics of heat transfer and flow field results are depicted by means of isotherms and streamlines, respectively. The path of convective heat transport and the comprehensive analysis of energy distribution by means of heatlines are explained using the concept of heat function. Laminar natural convection using entropy generation analysis due to fluid friction and heat transfer is also being studied. The numerical simulation for total entropy generation has been carried out for Prandtl numbers pertaining to ground water for different Ra1 and Ra2. [ABSTRACT FROM AUTHOR]