The paper proves the existence of a unique strong solution of an e-approximation for the equation of heat convection for the Kelvin-Voight fluids in a bounded domain    The convergence of solution  of the e-regularized value problem (6)-(9) to a solution  of the first initial-boundary value problem (1)-(5) as  is proved. The rate of convergence of the strong solution of (6)-(9) to the solution of (1)-(5) is estimated.

an e-approximation, heat convection, Kelvin-Voight fluids, free surface condition.