The approximate analytical solution of the internal problem of conductive and laminar free convection

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.