On the mechanism of fluctuation formation in hydrodynamic models
Peoples' Friendship University of Russia Russia, 117198, Moscow, Miklouho-Maclay St., 6.
The possible mechanism of the formation of fluctuations in hydrodynamic models is discussed. To this end, we consider solutions of the Burgers equation
w_ (t) + ww_x = w_xx
with the complex-valued function w (x, t) = u (x, t) + iv (x, t) and the complex viscosity = + i in the autowave approximation, i.e., in the form of a solution of a plane traveling wave w (x, t) = w ̃ (), where = x - c0t, c0 is the propagation velocity of disturbances in the medium. In this approximation, the solutions of the corresponding ordinary differential equation are investigated
w ̃ "() = (w ̃ () -c_0) w '(). (1)
It is established that equation (1) for a fixed value of the parameter has two special solutions in the phase space of the variables ((u) ̃, (v) ̃): a stable fixed point w ̃_01 = (u ̃_01, v ̃_01) and an unstable limit cycle w ̃ ^ * = (u ̃ ^ *, v ̃ ^ *). The point w ̃_01 corresponds to the fulfillment of the conditions u u01, v v01 for , where (u01, v01) is a uniform stationary state of rest, corresponding to the motion of a continuous medium (liquid or gas) at a constant speed. The solution w ̃ ^ * corresponds to an unstable traveling wave in the variables (t, x).
It is shown that at a flow rate less than a certain value corresponding to the appearance of an unstable limit cycle, the solution tends to another stable fixed point w ̃_02 = (u ̃_02, v ̃_02), in the vicinity of which an unstable limit cycle is generated. At values of the flow velocity exceeding the indicated value, the oscillation amplitude of the unstable limit cycle increases to a certain value, and then the solution passes to another stationary state corresponding to the fixed point w ̃_01 = (u ̃_01, v ̃_01). It is shown that the maximum value of the perturbation caused by the unstable limit cycle substantially and nonlinearly depends on the difference between the flow velocity and the propagation velocity of the perturbation.
In our opinion, the results obtained can explain the appearance of fluctuations of various amplitudes, including large velocity fluctuations in hydrodynamic models.