The approximate analytical solution of a problem about nonstationary free convection in the conductive and laminar mode of the Newtonian liquid in square area at the instantaneous change of temperature of a sidewall and lack of heat fluxes is submitted on top and bottom the bases. The equations of free convection in an approximation of Oberbeka-Bussinesk are linearized due to neglect by convective items. For reduction of number of hydrothermal parameters the system is given to the dimensionless look by introduction of scales for effect and explanatory variables. Transition from classical variables to the variables "whirlwind-a flow function" allowed to reduce system to a nonstationary heat conduction equation and a nonstationary nonuniform biharmonic equation, and the first is not dependent on the second. The decision in the form of a flow function is received by application integral a sine - Fourier transforms with terminating limits to a biharmonic equation at first on a variable x, and then on a variable y. The flow function has an appearance of a double series of Fourier on sine with coefficients in an integral form. Coefficients of a row represent integrals from unknown functions. On the basis of a hypothesis of an express type of integrals coefficients are calculated from the linear equation system received from boundary conditions on partial derivatives of function. Dependence of structure of a current on Prandtl's number is investigated. The cards of streamlines and isolines of components of speed describing development of a current from the moment of emergence before transition to a stationary state are received. The schedules of a field of vectors of speeds in various time illustrating dynamics of a current are provided. Reliability of a hypothesis of an express type of integral coefficients is confirmed by adequacy to physical sense and coherence of the received results with the numerical solution of a problem.

Oberbeck-Boussinesq equation, biharmonic equation, conductive and laminar free convection, sine integral Fourier transform with finite limits

1. 1 Terehov V.I., Ekaid A.L. Laminar free convection between vertical parallel plates with various temperatures. Teplofizika I aeromehanika [Thermal physics and air mechanics] 2012, vol. 19, no. 4, pp. 415–429.(in Russian).

2. 2 Sheremet M.A. The laminar and turbulent modes of the conjugate free convection in square area.Vyshislitelnaya mehanika sploshnyh sred [Computing mechanics of continua] 2012, vol. 5, no. 3, pp. 327–338. (in Russian).

3. 3 Terehov V.I., Ekaid A.L. Three-dimensional laminar convection in a parallelepiped with heating of sidewalls. Teplofizika vysokih temperature [Thermal physics of high temperatures] 2011, vol. 49, no. 6, pp. 905–911.(in Russian).

4. 4 Jani S., Mahmoodi M., Amini M., Jam J. Numerical investigation of natural convection heattransfer in a symmetrically cooled square cavity witha thin fin on its bottom wall.Thermal science, 2014, vol. 18, no. 4, pp. 1119–1132.

5. 5 Gros T., Revnic C., Pop I., Ingham D.B. Free convection heat transfer in a square cavity filled with a porousmedium saturated by ananofluid. International Journal of Heat and Mass Transfer, 2015, vol. 87, pp. 36–41.

6. 6 Ryzhskih V.I., Popov M.I. About a numerical integration of a nonstationary nonhomogeneous biharmonic equation in problems of conductive free convection. Vestnik Voronezhsko gogosudarstvennogo tehnichesko gosuniversiteta [Bulletin of the Voronezh state technical university] 2014, vol. 10, no. 1, pp. 56–62.(in Russian).

7. 7 Ustinov A.S.,Savin I.K. Convective Heat Transfer in a joint dei tion forced and free convection and measurable sculpt in time the boundary conditions on the wall. Vestnik Mezhdunarodnoj akademii holoda. [Journal of the International Academy of Refrigeration]. 2009, no. 3, pp. 8–10.(in Russian)