  Ultracold Bose atoms is the physical system, where the quantum and nonlinear
phenomena play crucial role. Ultracold bosons are considered at the small
finite temperatures. Bosons are considered as two different fluids:
Bose-Einstein condensate and normal fluid (the thermal component). An extended
hydrodynamic model is obtained for both fluids, where the pressure evolution
equations and the pressure flux third rank tensor evolution equations are
considered along with the continuity and Euler equations. It is found that the
pressure evolution equation contains zero contribution of the short-range
interaction. The pressure flux evolution equation contains the interaction
which gives the quantum fluctuations in the zero temperature limit. Here, we
obtain its generalization for the finite temperature. The contribution of
interaction in the pressure flux evolution equation which goes to zero in the
zero temperature limit is found. The model is obtained via the straightforward
derivation from the microscopic many-particle Schrodinger equation in the
coordinate representation.
